5.3 Nearly free electron model

Origin of the Nearly Free Electron Model

The free electron model treats conduction electrons as if they move through a perfectly uniform potential, ignoring the ions entirely. That works surprisingly well for some properties of simple metals, but it can't explain why solids have energy gaps or why some materials are insulators. The nearly free electron model fixes this by adding a weak periodic potential that represents the crystal lattice.

The key idea: the lattice potential is treated as a small perturbation on top of the free electron picture. Because the perturbation is weak, you can use perturbation theory to calculate how the electron energies and wave functions change. Those small corrections turn out to produce energy bands and band gaps, which are central to understanding electronic behavior in solids.

Assumptions of the Model

Weak Periodic Potential

The model assumes the periodic potential $V(\mathbf{r})$ from the ion cores is small compared to the kinetic energy of the electrons. This is a reasonable approximation for simple metals like sodium or aluminum, where the conduction electrons are loosely bound and the ion cores are well-screened.

Because the potential is weak, you can apply degenerate perturbation theory at points in $k$-space where free electron states are nearly degenerate. Away from those special points, the free electron dispersion $E = \hbar^2 k^2 / 2m$ barely changes.

Bloch's Theorem and Electron Wave Functions

Since the potential has the periodicity of the lattice, Bloch's theorem applies. The electron wave functions take the form of Bloch functions:

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

where $u_k(\mathbf{r})$ is a function with the same periodicity as the lattice. In the nearly free electron picture, these look almost like plane waves, with $u_k(\mathbf{r})$ deviating only slightly from a constant. The deviations grow larger near Brillouin zone boundaries, where the perturbation has its strongest effect.

Consequences of the Weak Periodic Potential

Bragg Reflection and Bragg Planes

When an electron's wave vector $\mathbf{k}$ satisfies the Bragg condition:

$2\mathbf{k} \cdot \mathbf{G} = G^2$

(where $\mathbf{G}$ is a reciprocal lattice vector), the electron wave undergoes Bragg reflection off the lattice planes. At these wave vectors, the forward-traveling and backward-traveling waves have the same energy in the free electron picture, so they mix strongly under even a weak perturbation.

This mixing produces two standing wave solutions rather than traveling waves. One standing wave concentrates electron density on the ions (lower energy), and the other concentrates it between the ions (higher energy). The energy difference between these two solutions is the band gap.

Energy Gaps

The band gaps open at the Brillouin zone boundaries, exactly where the Bragg condition is satisfied. For a 1D lattice with lattice constant $a$, the first gaps appear at $k = \pm \pi/a$.

The size of the gap at a given zone boundary is determined by the relevant Fourier component of the periodic potential. Specifically, for a reciprocal lattice vector $\mathbf{G}$, the gap is:

$E_g = 2|V_\mathbf{G}|$

where $V_\mathbf{G}$ is the Fourier coefficient of the lattice potential corresponding to $\mathbf{G}$. Stronger potentials produce larger gaps; weaker potentials produce narrower ones.

Brillouin Zones

The Bragg planes in reciprocal space carve out regions called Brillouin zones:

• The first Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice. It contains all wave vectors that can be reached from the origin without crossing a Bragg plane.
• Higher-order Brillouin zones are the successive shells beyond the first zone, each bounded by additional Bragg planes.

Every distinct electronic state can be labeled by a wave vector within the first Brillouin zone plus a band index. This is the basis of the reduced zone scheme.

The E-k Diagram

Reduced and Extended Zone Schemes

The relationship between electron energy $E$ and wave vector $k$ can be displayed in two standard ways:

• Extended zone scheme: Plot $E(k)$ over all of $k$-space. The free electron parabola gets broken into segments by gaps at each zone boundary. This view makes it easy to see how the nearly free electron bands deviate from the free electron parabola.
• Reduced zone scheme: Fold all bands back into the first Brillouin zone using reciprocal lattice translations. Each band gets its own index (1st band, 2nd band, etc.). This is the most common representation because it exploits the periodicity of the lattice.

There's also a periodic zone scheme, where the reduced zone picture is repeated across all of reciprocal space. This can be useful for visualizing Fermi surfaces.

Energy Bands and Gaps

In the E-k diagram:

• Energy bands are the continuous ranges of allowed electron energies. Within a band, $E(k)$ varies smoothly.
• Energy gaps are the forbidden ranges where no electron states exist. They appear at zone boundaries.
• Near the zone center (away from boundaries), the bands closely follow the free electron parabola. Near the boundaries, the bands flatten and split apart by $2|V_\mathbf{G}|$.

Effective Mass

Definition

The effective mass captures how an electron in a periodic potential accelerates in response to an external force. It's defined through the band curvature:

$m^* = \hbar^2 \left(\frac{\partial^2 E}{\partial k^2}\right)^{-1}$

This replaces the bare electron mass $m$ in equations of motion. The effective mass accounts for the internal forces from the lattice, so you can treat the electron as if it were a free particle with mass $m^*$.

• Positive $m^*$ (upward curvature): the electron accelerates in the direction of the applied force, behaving like a normal particle.
• Negative $m^*$ (downward curvature): the electron accelerates opposite to the applied force. This is the origin of hole behavior near the top of a valence band.

Calculating from the E-k Diagram

1. Identify the band extremum you care about (a band minimum for electrons, a band maximum for holes).
2. Fit a parabola to $E(k)$ near that extremum.
3. Extract $m^*$ from the curvature of the fit using the formula above.

For parabolic bands, $m^*$ is constant. For non-parabolic bands (common further from the extrema), $m^*$ becomes $k$-dependent and you need to evaluate the second derivative numerically at each point.

Density of States

Effect of Energy Bands on the DOS

The density of states $g(E)$ counts the number of available electronic states per unit energy. In the free electron model, $g(E) \propto \sqrt{E}$ in 3D. The periodic potential reshapes this.

• Near band edges, the bands flatten (small $\partial E / \partial k$), which piles up many states in a narrow energy range. The DOS is high here.
• In the middle of bands, the dispersion is steep and the DOS is lower.
• Inside band gaps, the DOS drops to zero since no states exist.

Van Hove Singularities

At certain critical points in the Brillouin zone, the group velocity $\nabla_k E$ vanishes. These produce sharp features in the DOS called Van Hove singularities. The type of singularity depends on the local band topology:

• Band minima or maxima: produce a step-like onset or cutoff in the DOS (in 3D).
• Saddle points: produce logarithmic divergences in 2D, or kinks in 3D.

Van Hove singularities have measurable consequences. They show up in optical absorption spectra, tunneling measurements, and can enhance electron-phonon coupling at specific energies.

Fermi Surfaces

Construction

The Fermi surface is the constant-energy surface in $k$-space at the Fermi energy $E_F$. At zero temperature, all states inside the Fermi surface are occupied and all states outside are empty.

To construct it in the nearly free electron model:

1. Start with the free electron Fermi sphere (a sphere of radius $k_F$ in reciprocal space).
2. Identify where this sphere intersects or approaches Brillouin zone boundaries.
3. Near those boundaries, the bands distort: the Fermi surface bulges toward the zone boundary (attracted by the lower band) or pulls away from it, depending on the band filling.
4. The result is a Fermi surface that deviates from a perfect sphere, with distortions concentrated near zone boundaries.

The shape of the Fermi surface controls many physical properties, including electrical conductivity, magnetoresistance, and de Haas-van Alphen oscillations.

Electron and Hole Pockets

When the Fermi surface crosses into higher Brillouin zones, it can break into disconnected pieces:

• Electron pockets: small closed surfaces around band minima where the curvature is positive. Carriers here behave as electrons with positive effective mass.
• Hole pockets: small closed surfaces around band maxima where the curvature is negative. Carriers here behave as holes.

The coexistence of electron and hole pockets means the material has multiple types of charge carriers, which affects Hall effect measurements and magnetotransport. The topology of these pockets can be mapped experimentally through de Haas-van Alphen or Shubnikov-de Haas oscillations.

Limitations of the Model

Strongly Correlated Systems

The nearly free electron model assumes electrons interact weakly with each other and with the lattice potential. This breaks down in strongly correlated systems like transition metal oxides (e.g., $\text{NiO}$) and heavy fermion compounds, where electron-electron interactions dominate. Phenomena like Mott metal-insulator transitions and unconventional superconductivity are completely outside the scope of this model.

Band Gap Predictions

The model systematically underestimates band gaps in semiconductors and insulators. The actual ionic potentials in materials like silicon or diamond are not weak, so treating them as small perturbations on free electrons is a poor approximation. For quantitative band gap predictions, you need methods like the tight-binding model (which starts from atomic orbitals rather than free electrons) or density functional theory (DFT), though even DFT has its own well-known band gap problem.

Applications

Semiconductors and Insulators

The nearly free electron model gives a qualitative picture of how bands and gaps form in semiconductors and insulators. It explains why these materials have forbidden energy ranges and how the gap size relates to the lattice potential. You can also use it to estimate effective masses near band edges. For quantitative work on real semiconductors, though, tight-binding or ab initio methods are standard.

Simple Metals and Alloys

This is where the model shines. For alkali metals (Li, Na, K) and other simple metals with weak pseudopotentials, the nearly free electron picture gives accurate predictions of:

• Band structure and Fermi surface shape
• Density of states near the Fermi level
• The origin of electron and hole pockets

The model also serves as the conceptual foundation for pseudopotential theory, which extends these ideas to more complex metals and alloys by replacing the true ionic potential with a smoother effective potential.