5.1 Free electron model

The free electron model simplifies how electrons behave in metals by treating them as a gas of non-interacting particles moving through a uniform positive background. This approach explains a surprising number of metallic properties, including electrical conductivity, heat capacity, and thermal transport. It also serves as the foundation for the more complete band theory you'll encounter next.

That said, the model has clear limitations. It fails for transition metals, semiconductors, and insulators, which is exactly why band theory exists.

Free electron gas model

The free electron gas model starts from a bold simplification: ignore the details of the ionic potential and electron-electron repulsion, and just treat conduction electrons as free particles in a box. Despite how crude this sounds, it captures the essential physics of simple metals remarkably well.

Electrons in periodic potential

In a real metal, conduction electrons move through a periodic potential created by the regularly spaced positive ions in the crystal lattice. This periodicity has deep consequences for the allowed energy states.

Bloch's theorem tells you that electron wavefunctions in a periodic potential take the form of a plane wave modulated by a periodic function with the same periodicity as the lattice:

$\psi_k(\vec{r}) = e^{i\vec{k}\cdot\vec{r}} u_k(\vec{r})$

where $u_k(\vec{r})$ has the periodicity of the lattice. The free electron model is the limiting case where $u_k$ is constant, meaning the periodic potential is effectively zero.

Assumptions of the model

• Electrons are independent particles: electron-electron interactions are neglected.
• The positive ions form a uniform background potential (the "jellium" model), so there's no lattice structure.
• Electron-ion interactions are either ignored or treated as a weak periodic perturbation.
• The energy-momentum relation is parabolic (like a free particle):

$E = \frac{\hbar^2 k^2}{2m}$

where $\hbar$ is the reduced Planck's constant, $k$ is the electron wavevector, and $m$ is the free electron mass. This parabolic dispersion is the signature of the model and the first thing that breaks down when you move to real band structures.

Electron density

The electron density $n$ is the number of conduction electrons per unit volume. For a free electron gas at absolute zero, all states up to the Fermi wavevector $k_F$ are filled, and the relationship is:

$n = \frac{k_F^3}{3\pi^2}$

This expression comes from counting the number of allowed $k$-states inside a Fermi sphere of radius $k_F$ in reciprocal space (including a factor of 2 for spin). The electron density sets the scale for nearly every other quantity in the model: Fermi energy, density of states, and conductivity all trace back to $n$.

Fermi energy

The Fermi energy $E_F$ is the energy of the highest occupied state at absolute zero. Substituting $k_F$ into the free electron dispersion gives:

$E_F = \frac{\hbar^2 k_F^2}{2m}$

Typical Fermi energies in metals are a few eV (for example, ~7.0 eV for copper, ~11.7 eV for aluminum). Higher valence electron counts push $n$ up, which raises $k_F$ and therefore $E_F$. The Fermi energy is the single most important energy scale in the model: it determines which electrons participate in conduction, heat capacity, and thermal transport.

Density of states

The density of states $g(E)$ counts how many electron states exist per unit energy per unit volume. For a 3D free electron gas:

$g(E) = \frac{1}{2\pi^2}\left(\frac{2m}{\hbar^2}\right)^{3/2}\sqrt{E}$

The $\sqrt{E}$ dependence means there are more available states at higher energies. This function is critical for computing thermodynamic quantities because integrals over energy always weight by $g(E)$. At the Fermi level, $g(E_F)$ directly determines the electronic heat capacity coefficient.

Electron dynamics

This section covers how free electrons respond to applied fields. The results here connect directly to measurable transport properties like electrical conductivity and resistivity.

Equation of motion

Under an applied electric field $\vec{E}$, a free electron obeys:

$m\frac{d\vec{v}}{dt} = -e\vec{E}$

The negative sign reflects that electrons (negative charge) accelerate opposite to the field direction. In steady state, scattering events balance the acceleration, producing a constant drift velocity $\vec{v}_d$ superimposed on the random thermal motion.

Electron effective mass

In the free electron model, the effective mass $m^*$ equals the bare electron mass $m$. In real crystals, the periodic potential modifies the dispersion relation, and the effective mass is defined by the band curvature:

$\frac{1}{m^*} = \frac{1}{\hbar^2}\frac{d^2E}{dk^2}$

A strongly curved band gives a small $m^*$ (light, mobile electrons), while a flat band gives a large $m^*$ (heavy, sluggish electrons). This concept becomes essential once you move beyond the free electron picture.

Electron mobility

Electron mobility $\mu$ quantifies how easily electrons drift through the metal:

$\mu = \frac{v_d}{E} = \frac{e\tau}{m}$

where $\tau$ is the relaxation time (average time between scattering events). Higher mobility means electrons respond more readily to an applied field. Mobility is reduced by scattering from lattice vibrations (phonons) and from impurities.

Conductivity of metals

The electrical conductivity $\sigma$ in the Drude/free-electron picture is:

$\sigma = \frac{ne^2\tau}{m}$

This ties together three factors: how many carriers there are ($n$), how strongly they couple to the field ($e$), and how long they travel before scattering ($\tau$). You can also write $\sigma = ne\mu$, making the connection to mobility explicit. Metals like copper have both high $n$ and relatively long $\tau$, giving them excellent conductivity.

Matthiessen's rule

Different scattering mechanisms contribute independently to the total resistivity:

$\rho_{total} = \rho_{phonon} + \rho_{impurity}$

• $\rho_{phonon}$ increases with temperature (more lattice vibrations means more scattering).
• $\rho_{impurity}$ is roughly temperature-independent (impurities don't go away when you cool the sample).

At high temperatures, phonon scattering dominates. At very low temperatures, impurity scattering sets a residual resistivity floor. The ratio of room-temperature to residual resistivity (the residual resistance ratio) is a common measure of sample purity.

Heat capacity

The heat capacity of a metal has two contributions: one from the conduction electrons and one from lattice vibrations (phonons). The free electron model's treatment of the electronic part was one of its early triumphs.

Classical vs quantum behavior

Classical statistical mechanics (the equipartition theorem) predicts each electron contributes $\frac{3}{2}k_B$ to the heat capacity, independent of temperature. This is the Dulong-Petit prediction for the electronic part, and it dramatically overestimates the measured values.

The resolution is quantum mechanical. The Fermi-Dirac distribution dictates that at temperature $T$, only electrons within ~$k_BT$ of the Fermi energy can be thermally excited. Since $k_BT \ll E_F$ at room temperature (by a factor of ~100), the vast majority of electrons are "frozen out" and don't contribute to the heat capacity at all.

Electronic heat capacity

The electronic heat capacity for a free electron gas is:

$C_e = \gamma T$

where the Sommerfeld coefficient $\gamma$ is:

$\gamma = \frac{\pi^2 k_B^2}{3} g(E_F)$

The linear-in-$T$ dependence is the key result. It reflects the fact that the number of thermally active electrons grows linearly with temperature (the "thermal window" around $E_F$ widens as $k_BT$).

Low temperature approximation

At low temperatures (well below the Debye temperature $\Theta_D$), the total heat capacity of a metal is commonly written as:

$C = \gamma T + AT^3$

The first term is the electronic contribution; the second is the phonon (Debye) contribution. Because the phonon term drops as $T^3$, the electronic term dominates at sufficiently low $T$. Plotting $C/T$ vs. $T^2$ gives a straight line whose intercept is $\gamma$ and whose slope is $A$. This is the standard experimental method for extracting the Sommerfeld coefficient.

Linear temperature dependence

The linear $C_e = \gamma T$ behavior has been confirmed experimentally across many metals. Deviations from the predicted $\gamma$ value reveal physics beyond the free electron model:

• In heavy-fermion compounds, $\gamma$ can be orders of magnitude larger than expected, indicating a very large effective mass.
• In transition metals, d-band electrons produce a higher density of states at $E_F$, also enhancing $\gamma$.

These deviations don't invalidate the functional form (still linear in $T$), but they show that the free electron value of $g(E_F)$ needs correction.

Thermal conductivity

Metals are good thermal conductors primarily because conduction electrons carry heat efficiently. The free electron model connects thermal and electrical transport in a clean, testable way.

Wiedemann-Franz law

The Wiedemann-Franz law states that the ratio of thermal to electrical conductivity is proportional to temperature:

$\frac{\kappa}{\sigma} = LT$

where $L$ is the Lorenz number. This relationship holds because the same electrons carry both charge and heat. If scattering reduces electrical conductivity, it reduces thermal conductivity by the same factor, and the ratio depends only on fundamental constants and temperature.

Lorenz number

The free electron model predicts:

$L = \frac{\pi^2}{3}\left(\frac{k_B}{e}\right)^2 = 2.44 \times 10^{-8} \text{ W}\Omega\text{K}^{-2}$

Experimental values for most metals at room temperature cluster near this prediction, which is strong evidence for the free electron picture. Deviations occur at intermediate temperatures where different scattering mechanisms affect thermal and electrical transport unequally (inelastic vs. elastic scattering).

Electron mean free path

The mean free path $l$ is the average distance between scattering events:

$l = v_F \tau$

where $v_F$ is the Fermi velocity (since electrons near $E_F$ dominate transport). Typical values in clean metals at room temperature are on the order of tens of nanometers. At low temperatures in very pure samples, $l$ can reach millimeters or more, limited only by impurity scattering and sample boundaries.

Phonon contribution

Phonons contribute to thermal conductivity through lattice vibrations, independent of the electronic channel:

$\kappa_{total} = \kappa_{electronic} + \kappa_{phonon}$

In metals, the electronic term usually dominates. The phonon contribution becomes more significant at higher temperatures (above the Debye temperature) and in poor metals or alloys where electronic transport is suppressed. Phonons also scatter electrons, reducing $\tau$ and therefore reducing $\kappa_{electronic}$ as temperature rises.

Failures of the free electron model

The free electron model works surprisingly well for simple metals, but it breaks down in predictable ways. Understanding where it fails is just as important as understanding where it succeeds, because those failures motivate band theory.

Alkali metals

Alkali metals (Li, Na, K, etc.) with one valence electron per atom are the closest real systems to a free electron gas. Even here, the model doesn't get everything right. The measured electronic heat capacity coefficients deviate from free electron predictions by 10-30%, indicating that the true band structure is not perfectly parabolic. Electron-electron interactions (captured in Fermi liquid theory) also play a role.

Transition metals

Transition metals (Fe, Cu, Ni, etc.) have partially filled d-orbitals that are much more localized than s- or p-electrons. The free electron model cannot account for:

• The high density of states from narrow d-bands
• Magnetic ordering (ferromagnetism in Fe, Ni, Co)
• The complex, multi-sheeted Fermi surfaces observed experimentally

These properties require a full band structure calculation that includes the d-electron character.

Semiconductors and insulators

The free electron model assumes a partially filled band, so it has no mechanism to produce an energy gap. Semiconductors (Si, Ge) and insulators (diamond, NaCl) have a filled valence band separated from an empty conduction band by a band gap. The model cannot explain:

• Why some materials are insulating despite having electrons
• The exponential temperature dependence of conductivity in semiconductors
• Doping and the creation of electron/hole carriers

Band theory of solids

Band theory resolves these failures by incorporating the periodic lattice potential. Solving the Schrödinger equation with a periodic potential (via Bloch's theorem) produces energy bands separated by forbidden gaps. This framework:

• Explains metals (partially filled bands), semiconductors (small gap), and insulators (large gap) within a single theory
• Predicts effective masses that vary with $k$-direction
• Accounts for complex Fermi surface geometries

The free electron model is the $V(\vec{r}) \to 0$ limit of band theory. It remains the starting point for understanding electronic structure, but the periodic potential is what gives solids their rich variety of electronic behavior.