Free Electron Theory of Metals & Fermi Surfaces

1. Postulates of the Drude Model (1900)

Paul Drude treated valence electrons in a metal as an ideal classical gas of non-interacting point particles of mass $m$ and charge $-e$, moving through a static background of positive ion cores:

1. Independent & Free Electron Approximation: Electron-electron and electron-ion interactions are neglected between collisions.
2. Relaxation-Time Approximation: Collisions are instantaneous, random events occurring with an average probability per unit time $1/\tau$, where $\tau$ is the relaxation time (mean free time between collisions).
3. Thermal Equilibrium: Electrons emerge from each collision in thermal equilibrium with the local lattice temperature, with zero average drift velocity.

2. Derivation of Ohm's Law and DC Electrical Conductivity

Under an applied macroscopic electric field $\vec{E}$, the equation of motion for the average drift velocity $\vec{v}_d$ of an electron is:

In steady state ($d\vec{v}_d/dt = 0$):

The macroscopic electric current density $\vec{J}$ carried by electron density $n$ is:

This derives microscopic Ohm's Law, with DC electrical conductivity $\sigma_0$ given by the Drude formula:

where $\mu = e\tau/m$ is the electron drift mobility.

3. The Classical Wiedemann-Franz Law & Failure of Drude Model

Treating electrons as a classical Maxwell-Boltzmann gas with thermal conductivity $\kappa = \frac{1}{3} C_v v_{\text{th}} \ell$ and heat capacity $C_v = \frac{3}{2} n k_B$:

While this qualitatively explained the empirical Wiedemann-Franz law, Drude's classical model suffered from a catastrophic failure: it predicted that the electronic heat capacity should contribute $\frac{3}{2} R$ per mole, which was completely absent in room-temperature measurements.

1. Schrödinger Equation in a 3D Potential Well

Arnold Sommerfeld (1928) resolved the Drude paradox by applying quantum mechanics and Fermi-Dirac statistics to the valence electrons. Consider $N$ free electrons confined within a box of volume $V = L_x L_y L_z$ with periodic Born-von Kármán boundary conditions:

The energy eigenvalues are continuous parabolic free-particle dispersions:

where allowed wavevectors form a uniform grid in reciprocal $\vec{k}$-space: $k_i = \frac{2\pi n_i}{L_i}$ ($n_i \in \mathbb{Z}$). Each allowed state occupies a volume in $k$-space of $\Delta k^3 = \frac{(2\pi)^3}{V}$.

2. Derivation of the Density of States $g(E)$ in 3D

Taking into account electron spin degeneracy ($g_s = 2$):

Substituting $k = \left(\frac{2mE}{\hbar^2}\right)^{1/2}$:

The Density of States (DOS) $g(E) = \frac{dN}{dE}$ per unit volume in three dimensions is:

3. Dimensionality Comparison of Density of States

• 1D (Quantum Wire): $g_{\text{1D}}(E) = \frac{\sqrt{2m}}{\pi \hbar} E^{-1/2} \propto E^{-1/2}$ (Van Hove singularity).
• 2D (Quantum Well): $g_{\text{2D}}(E) = \frac{m}{\pi \hbar^2} = \text{constant}$ (energy-independent step functions).
• 3D (Bulk Metal): $g_{\text{3D}}(E) = \frac{m}{\pi^2 \hbar^3} \sqrt{2mE} \propto E^{1/2}$.

1. The Fermi-Dirac Distribution Function

Because electrons are indistinguishable fermions obeying the Pauli exclusion principle, the probability of an electronic state of energy $E$ being occupied at absolute temperature $T$ is given by the Fermi-Dirac distribution:

where $\mu(T)$ is the chemical potential. At $T = 0\text{ K}$, the chemical potential is defined as the Fermi Energy $E_F \equiv \mu(0)$. The distribution becomes an exact step function:

2. Calculation of the Fermi Energy $E_F$ and Fermi Surface

At $T = 0\text{ K}$, $N$ electrons fill all available states in $k$-space up to a maximum radius, the Fermi wavevector $k_F$, enclosing a sphere known as the Fermi Sphere:

The fundamental ground-state parameters of the free Fermi gas are:

• Fermi Energy:
  $$E_F = \frac{\hbar^2 k_F^2}{2m} = \frac{\hbar^2}{2m} (3\pi^2 n)^{2/3}$$
• Fermi Velocity:
  $$v_F = \frac{\hbar k_F}{m} = \frac{\hbar}{m} (3\pi^2 n)^{1/3}$$
• Fermi Temperature:
  $$T_F = \frac{E_F}{k_B} \sim 10^4 - 10^5\text{ K}$$

Because room temperature ($T \approx 300\text{ K}$) satisfies $T \ll T_F$, valence electrons in typical metals form a highly degenerate Fermi gas.

1. Quantum Suppression of Electronic Heat Capacity

When a metal is heated from $T = 0\text{ K}$ to temperature $T$, the Pauli exclusion principle dictates that electrons with energies deep below the Fermi surface ($E < E_F - k_BT$) cannot absorb thermal energy, because all states within $\sim k_BT$ above them are already occupied. Only electrons within a narrow thermal layer of width $\sim k_BT$ around $E_F$ can undergo transitions.

The fraction of thermally active electrons is approximately $\frac{k_B T}{E_F} = \frac{T}{T_F} \sim 10^{-2}$. Each of these excited electrons acquires thermal energy $\sim k_B T$, yielding an excess internal energy:

Differentiating with respect to $T$ yields a heat capacity linear in temperature:

2. Rigorous Sommerfeld Expansion for $C_{el}$

Performing a rigorous Sommerfeld expansion of the internal energy integral $U(T) = \int_0^\infty E g(E) f(E) dE$ leads to the exact theoretical formula for the electronic heat capacity:

where $\gamma = \frac{\pi^2}{3} g(E_F) k_B^2$ is the Sommerfeld coefficient.

At liquid helium temperatures ($T < 10\text{ K}$), the total heat capacity of a normal non-magnetic metal is the sum of electronic and phononic (Debye) contributions:

Plotting $C_V/T$ against $T^2$ yields a straight line whose $y$-intercept is $\gamma$ and whose slope determines the Debye temperature $\Theta_D$.

3. Pauli Paramagnetism of Conduction Electrons

In an external magnetic field $B$, electron spin magnetic moments $\mu_B$ align parallel or antiparallel to the field, shifting the up- and down-spin sub-bands by $\mp \mu_B B$. At $T = 0\text{ K}$, electrons near $E_F$ flip their spins into the lower energy sub-band until the Fermi levels equalize:

The resulting net magnetic magnetization is $M = \Delta N \mu_B = g(E_F) \mu_B^2 B$. The Pauli paramagnetic susceptibility is:

Unlike classical Curie paramagnetism ($\chi \propto 1/T$), Pauli paramagnetism is completely temperature-independent, in perfect agreement with experimental data for alkali and noble metals.

1. The Hall Effect & Hall Coefficient $R_H$

When an electric current density $J_x$ flows along the $x$-direction of a conductor immersed in a transverse magnetic field $B_z$ in the $z$-direction, the magnetic Lorentz force $\vec{F}_B = q (\vec{v}_d \times \vec{B})$ deflects charge carriers along the $y$-direction:

Charge accumulates on the lateral boundaries, generating a transverse electrostatic Hall electric field $E_y$. In steady state, the transverse electrostatic force exactly balances the magnetic Lorentz force ($F_y = 0$):

Since $J_x = n q v_{d,x} \implies v_{d,x} = \frac{J_x}{n q}$:

where $R_H$ is the Hall Coefficient:

The Hall effect provides an unambiguous experimental measurement of both the sign of the charge carriers (negative for electrons, positive for holes) and the carrier concentration $n$.

2. Quantum Wiedemann-Franz Law & The Lorenz Number

Applying Sommerfeld's degenerate Fermi gas theory to electronic thermal conduction ($\kappa = \frac{1}{3} C_{el} v_F^2 \tau$ with $C_{el} = \frac{\pi^2}{3} g(E_F) k_B^2 T$) and electrical conductivity ($\sigma = \frac{n e^2 \tau}{m}$) yields the quantum Wiedemann-Franz law:

where $L = \frac{\pi^2}{3}(k_B/e)^2$ is the universal Lorenz number, completely independent of carrier density, mass, and material parameters.

3. Matthiessen's Rule for Electrical Resistivity

Electrons in a real metal are scattered by both static structural crystal defects/impurities and dynamic thermal lattice vibrations (phonons). By Matthiessen's Rule, independent scattering rates add linearly:

Consequently, the total electrical resistivity $\rho = m / (ne^2\tau)$ separates into temperature-independent and temperature-dependent terms:

where $\rho_{\text{residual}}$ is determined by impurity concentration, and $\rho_{\text{ideal}}(T) \propto T^5$ at low temperatures (Bloch-Grüneisen law) and $\rho_{\text{ideal}}(T) \propto T$ at high temperatures ($T > \Theta_D$).