2.2 Nearly free electron model

Nearly free electron approximation

The nearly free electron model starts from the free electron picture and adds a weak periodic potential from the ion cores. This small addition turns out to have dramatic consequences: it explains how energy gaps form and why some materials are metals while others are semiconductors. The model uses perturbation theory to get there, making it one of the most tractable analytic approaches to real band structures.

Assumptions and limitations

The central assumption is that electrons in a solid experience a periodic potential from the ion cores, but that this potential is weak compared to the electron's kinetic energy. You treat it as a small perturbation to the free electron Hamiltonian.

This works well for:

• Simple metals (Na, K, Al) where valence electrons are highly delocalized
• Some semiconductors where the pseudopotential is weak

It breaks down for:

• Materials with strongly localized electrons (transition metal oxides, rare earths)
• Systems where electron-electron correlations are strong
• Deep core electrons, which feel a much stronger ionic potential

Perturbation theory approach

Because the periodic potential is weak, you can apply perturbation theory to the free electron Schrödinger equation. The periodic potential is expanded as a Fourier series over reciprocal lattice vectors $\mathbf{G}$:

$V(\mathbf{r}) = \sum_{\mathbf{G}} V_{\mathbf{G}} \, e^{i\mathbf{G} \cdot \mathbf{r}}$

Each Fourier coefficient $V_{\mathbf{G}}$ quantifies how strongly the potential couples free electron states whose wavevectors differ by $\mathbf{G}$. Away from zone boundaries, first-order perturbation theory gives small corrections to the free electron energies. The real action happens when two free electron states are nearly degenerate, which occurs exactly at Brillouin zone boundaries. There, you must use degenerate perturbation theory, and the result is the opening of energy gaps of magnitude $2|V_{\mathbf{G}}|$.

Bloch's theorem

Bloch's theorem is the foundational result for electrons in any periodic potential, not just weak ones. It constrains the form of every electronic wavefunction in a crystal and makes band theory possible.

Periodic potential effects

A periodic potential imposes its symmetry on the electronic states. Instead of the simple parabolic dispersion $E = \hbar^2 k^2 / 2m$ of free electrons, the dispersion relation develops:

• Energy bands: continuous ranges of allowed energies
• Band gaps: forbidden energy regions where no electronic states exist
• Modified scattering and transport behavior, since electrons in Bloch states propagate without scattering off the perfect periodic lattice itself

Wavefunction in a periodic lattice

Bloch's theorem states that the wavefunction for an electron in a periodic potential takes the form:

$\psi_{\mathbf{k}}(\mathbf{r}) = e^{i\mathbf{k} \cdot \mathbf{r}} \, u_{\mathbf{k}}(\mathbf{r})$

• $e^{i\mathbf{k} \cdot \mathbf{r}}$ is a plane wave envelope
• $u_{\mathbf{k}}(\mathbf{r})$ is a function with the full periodicity of the lattice: $u_{\mathbf{k}}(\mathbf{r} + \mathbf{R}) = u_{\mathbf{k}}(\mathbf{r})$ for any lattice vector $\mathbf{R}$

The quantity $\hbar\mathbf{k}$ is the crystal momentum. It is not the electron's true momentum (the electron also scatters off the lattice), but it plays an analogous role in conservation laws. Electronic states are labeled by the wavevector $\mathbf{k}$ and a band index $n$.

Energy band structure

The energy band structure $E_n(\mathbf{k})$ maps out how electron energy depends on crystal momentum for each band. It governs nearly every electronic, optical, and thermal property of a crystalline solid.

Formation of energy gaps

Energy gaps open at Brillouin zone boundaries through a specific mechanism:

1. A free electron state with wavevector $\mathbf{k}$ near a zone boundary is nearly degenerate with another state at $\mathbf{k} - \mathbf{G}$.

2. The periodic potential couples these two states through the matrix element $V_{\mathbf{G}}$.

3. Degenerate perturbation theory gives two new energy levels split apart by $2|V_{\mathbf{G}}|$.

4. The lower state has its wavefunction probability concentrated on the ion cores (lower energy), while the upper state has probability concentrated between the ions (higher energy).

Physically, this is Bragg reflection: electron waves satisfying the Bragg condition form standing waves rather than traveling waves, and the two possible standing waves have different energies because they sample the potential differently. The size of the gap depends directly on the Fourier component $V_{\mathbf{G}}$ of the potential. Whether a material is a metal, semiconductor, or insulator depends on where the Fermi level falls relative to these gaps.

Brillouin zones

Brillouin zones are constructed in reciprocal space ($\mathbf{k}$-space) as Wigner-Seitz cells of the reciprocal lattice.

• The first Brillouin zone contains all unique $\mathbf{k}$ values needed to describe every electronic state (thanks to the periodicity of $E_n(\mathbf{k})$ in reciprocal space).
• Higher-order zones correspond to regions reached by crossing one or more zone boundaries. In the extended zone scheme, each band occupies a different zone. In the reduced zone scheme, all bands are folded back into the first zone.
• High-symmetry points ($\Gamma$, $X$, $L$, etc.) and lines connecting them are where band structures are conventionally plotted.

Effective mass

When an external force acts on an electron in a crystal, the electron doesn't accelerate like a free particle with mass $m_e$. Instead, the periodic potential modifies its response, and you capture this through the effective mass.

Concept and significance

The effective mass is defined from the curvature of the energy band:

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

This tells you how "heavy" or "light" an electron feels in a given band. Near the bottom of a band, $d^2E/dk^2 > 0$, so $m^* > 0$: the electron accelerates in the expected direction. Near the top of a band, the curvature is negative, giving $m^* < 0$. This is where the hole picture becomes useful.

Effective mass directly controls:

• Carrier mobility: lighter effective mass means higher mobility
• Density of states: heavier effective mass means more states per unit energy
• Optical transition strengths and exciton binding energies

Calculation methods

• From band structure: compute $d^2E/dk^2$ numerically or analytically at the band extremum
• Experimentally: cyclotron resonance measures $m^*$ directly, since the cyclotron frequency $\omega_c = eB/m^*$
• Tensor form: in anisotropic crystals, effective mass is a tensor $m^*_{ij} = \hbar^2 \left(\frac{\partial^2 E}{\partial k_i \partial k_j}\right)^{-1}$, meaning the electron responds differently to forces in different directions (e.g., Si has longitudinal and transverse effective masses)
• Computational: density functional theory and related methods provide $E(\mathbf{k})$ from which $m^*$ is extracted

Density of states

The density of states $g(E)$ counts the number of available electronic states per unit energy per unit volume. It determines how many electrons can occupy a given energy range and directly enters calculations of heat capacity, electrical conductivity, and optical absorption.

Free electron vs. nearly free electron

For free electrons in 3D, the density of states follows:

$g(E) \propto \sqrt{E}$

The nearly free electron model modifies this in two important ways:

• Band gaps create regions where $g(E) = 0$. The DOS drops to zero inside a gap and reappears when the next band starts.
• Van Hove singularities appear at critical points in the Brillouin zone where $\nabla_{\mathbf{k}} E = 0$ (band edges, saddle points). These produce kinks, jumps, or divergences in $g(E)$ depending on the dimensionality and the nature of the critical point.

These modifications have measurable consequences for the temperature dependence of electronic heat capacity and for optical absorption spectra.

Energy dependence by dimensionality

The functional form of $g(E)$ depends strongly on the dimensionality of the system:

• 3D (bulk): $g(E) \propto \sqrt{E}$ for free electrons, modified by band structure effects
• 2D (quantum wells, graphene): constant $g(E)$ within each subband, with step-like jumps at subband edges
• 1D (quantum wires, nanotubes): $g(E) \propto 1/\sqrt{E}$, diverging at each subband edge
• 0D (quantum dots): discrete delta-function peaks, like an atom

Fermi surface

The Fermi surface is the constant-energy surface in $\mathbf{k}$-space at the Fermi energy $E_F$. At absolute zero, all states inside this surface are occupied and all states outside are empty. The geometry of the Fermi surface controls most transport and thermodynamic properties of metals.

Shape and properties

• For a free electron gas, the Fermi surface is a perfect sphere.
• A weak periodic potential distorts this sphere, especially near Brillouin zone boundaries where energy gaps open. The Fermi surface bends to meet zone boundaries at right angles (a consequence of the vanishing group velocity at the boundary).
• In real metals like Cu or Al, the Fermi surface can develop "necks" or "bellies" that reflect the crystal symmetry.
• Nesting: when large parallel sections of the Fermi surface can be connected by a single wavevector $\mathbf{q}$, instabilities like charge density waves or spin density waves can occur.

Experimental observations

• de Haas-van Alphen effect: oscillations in magnetization as a function of $1/B$ reveal extremal cross-sectional areas of the Fermi surface
• ARPES (angle-resolved photoemission spectroscopy): directly maps $E(\mathbf{k})$ and thus the Fermi surface
• Compton scattering and positron annihilation: probe the electron momentum distribution, giving complementary Fermi surface information
• Comparing experimental Fermi surfaces with calculated ones is one of the best tests of band structure theory

Electron dynamics

Understanding how electrons move through a crystal under applied fields connects band theory to measurable transport properties like conductivity and the Hall effect.

Group velocity vs. phase velocity

The physically meaningful velocity of an electron wave packet is the group velocity:

$\mathbf{v}_g = \frac{1}{\hbar} \nabla_{\mathbf{k}} E(\mathbf{k})$

This is the velocity at which the electron actually moves through the crystal. It depends on the slope of the band, not the absolute energy. At a band edge where the band is flat, $\mathbf{v}_g = 0$.

The phase velocity $\mathbf{v}_p = E(\mathbf{k}) / \hbar k$ describes the propagation of individual plane wave components but doesn't correspond to the motion of a localized electron.

Acceleration in an electric field

Under an applied electric field $\mathbf{E}$, the semiclassical equation of motion gives:

$\hbar \frac{d\mathbf{k}}{dt} = -e\mathbf{E}$

The electron's wavevector changes at a constant rate, but its real-space acceleration depends on the effective mass:

$\mathbf{a} = -\frac{e\mathbf{E}}{m^*}$

Several consequences follow:

• Near the top of a band where $m^* < 0$, the electron accelerates opposite to what you'd expect. This is the physical basis for treating holes as positive charge carriers.
• Bloch oscillations: in a perfect crystal with no scattering, an electron in a uniform electric field would oscillate rather than accelerate indefinitely, because $\mathbf{k}$ sweeps across the entire Brillouin zone periodically. In practice, scattering prevents this in bulk materials, but Bloch oscillations have been observed in semiconductor superlattices.
• Scattering (from phonons, impurities, defects) limits the acceleration and establishes a steady-state drift velocity.

Optical properties

The band structure determines how a material interacts with light. Photon absorption and emission involve transitions between electronic states, and the selection rules and available states come directly from the bands.

Interband vs. intraband transitions

• Interband transitions: an electron jumps from one band to another (typically valence to conduction). These require the photon energy to exceed the band gap and are responsible for the absorption edge in semiconductors.
• Intraband transitions (free carrier absorption): an electron is excited to a higher-energy state within the same band. These dominate in metals and heavily doped semiconductors, and they require a momentum source (phonon or impurity) since photons carry negligible momentum.

Absorption and reflection

The absorption coefficient depends on the joint density of states (the number of valence-conduction state pairs separated by the photon energy) and the transition matrix elements. In a direct-gap semiconductor, absorption rises sharply at the gap energy. In an indirect-gap material, absorption near the gap is weaker because a phonon must assist the transition.

Reflection is governed by the complex dielectric function $\epsilon(\omega)$. In metals, the plasma frequency $\omega_p$ sets a dividing line: below $\omega_p$, the metal reflects strongly; above it, the metal becomes transparent. For simple metals, $\omega_p$ falls in the ultraviolet, which is why metals are shiny in visible light.

Applications in solids

The nearly free electron model provides a clear framework for understanding why different materials have such different electronic properties.

Metals vs. semiconductors

• Metals have partially filled bands, with the Fermi level cutting through one or more bands. There are always empty states available just above $E_F$, so electrons can easily respond to applied fields. This gives high electrical and thermal conductivity.
• Semiconductors have a fully occupied valence band separated from an empty conduction band by a gap (typically 0.1 to 3 eV). At zero temperature they're insulators; at finite temperature, thermal excitation across the gap creates a small number of carriers. Doping with impurities allows precise control of carrier concentration and type.

The distinction comes down to where $E_F$ sits relative to the band gaps predicted by the model.

Alloys and impurities

• Alloying (e.g., mixing GaAs with AlAs to form AlGaAs) modifies the periodic potential and therefore the band structure. This is used to engineer specific band gaps for optoelectronic applications.
• Impurities in semiconductors introduce localized energy levels within the band gap. Donor impurities (like P in Si) add states just below the conduction band; acceptor impurities (like B in Si) add states just above the valence band. This is the basis of all semiconductor device technology.

Limitations and extensions

The nearly free electron model is analytically powerful but limited in scope. Knowing where it fails points you toward more sophisticated methods.

Beyond the nearly free electron model

• Tight-binding model: starts from atomic orbitals rather than plane waves. Better suited for materials with more localized electrons (d-band metals, covalent semiconductors).
• Pseudopotential methods: replace the strong true ionic potential with a weaker effective potential that reproduces the correct valence electron behavior. This is why the nearly free electron model works better than you might expect for many materials.
• Density functional theory (DFT): a first-principles computational method that handles electron-electron interactions approximately. The workhorse of modern electronic structure calculations.
• $\mathbf{k} \cdot \mathbf{p}$ theory: expands the band structure around a specific $\mathbf{k}$-point (usually $\Gamma$) using perturbation theory. Useful for calculating effective masses and band dispersions near band extrema.

Strongly correlated systems

In some materials, electron-electron interactions are so strong that single-particle band theory fails qualitatively. Transition metal oxides, heavy fermion compounds, and high-$T_c$ cuprate superconductors fall into this category.

• Mott insulators have half-filled bands that band theory predicts should be metallic, but strong on-site Coulomb repulsion (the Hubbard $U$) prevents double occupancy and opens a gap.
• Theoretical approaches include the Hubbard model and dynamical mean-field theory (DMFT), which incorporate local correlations beyond what DFT captures.
• This remains one of the most active areas of condensed matter research.