2.4 Energy bands

Fundamentals of energy bands

Energy bands describe the ranges of energy that electrons can (or cannot) occupy inside a crystalline solid. They arise because atoms in a crystal are packed so closely together that their electron wavefunctions overlap and interact, splitting discrete atomic energy levels into broad, quasi-continuous bands. This framework is what lets us explain why copper conducts electricity, why diamond doesn't, and why silicon sits in between.

Crystal structure and periodicity

Atoms in a crystalline solid arrange themselves in a periodic lattice, which means the same structural motif repeats over and over in three dimensions. This periodicity creates a repeating potential energy landscape that electrons must navigate.

• There are 14 Bravais lattices that describe every possible three-dimensional crystal symmetry.
• Lattice constants set the spacing between repeating units. For example, silicon has a lattice constant of about 5.43 Å.
• Symmetry operations (translation, rotation, reflection) leave the crystal structure unchanged. These symmetries simplify the math enormously because you only need to solve the problem in one repeating unit.

Bloch's theorem

Bloch's theorem is the key mathematical result that makes band theory work. It states that an electron moving through a perfectly periodic potential has a wavefunction of the form:

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

• $e^{i\mathbf{k} \cdot \mathbf{r}}$ is a plane wave that carries the crystal momentum $\mathbf{k}$.
• $u_k(\mathbf{r})$ is a function with the same periodicity as the lattice, so $u_k(\mathbf{r} + \mathbf{R}) = u_k(\mathbf{r})$ for any lattice vector $\mathbf{R}$.

The physical consequence: electron wavefunctions in a crystal aren't localized on single atoms. They spread through the entire lattice as modulated plane waves, and their allowed energies depend on $\mathbf{k}$.

Brillouin zones

Brillouin zones live in reciprocal space (k-space), which is the Fourier transform of the real-space lattice.

• The first Brillouin zone is the Wigner-Seitz cell of the reciprocal lattice. It contains all the unique $\mathbf{k}$-points you need to fully describe the electronic structure.
• Higher-order Brillouin zones are obtained by translating the first zone by reciprocal lattice vectors. Because of periodicity, they don't contain new physics.
• Zone boundaries correspond to the Bragg diffraction condition, which is exactly where energy gaps open up.
• High-symmetry points like $\Gamma$ (zone center), $X$, and $L$ label special locations in k-space. Band structure diagrams plot energy along paths connecting these points.

Band formation mechanisms

Different models explain how continuous energy bands emerge from discrete atomic levels. Each model starts from a different physical picture and works best for a different class of materials.

Nearly free electron model

This model treats electrons as nearly free particles experiencing a weak periodic perturbation from the ion cores. It works well for simple metals like sodium or aluminum, where valence electrons are loosely bound.

• Far from Brillouin zone boundaries, the dispersion is approximately parabolic: $E \approx \frac{\hbar^2 k^2}{2m}$.
• At zone boundaries, electron waves satisfy the Bragg condition and undergo strong reflection. The resulting standing waves have different energies depending on whether the electron density piles up on the ions or between them, creating an energy gap.
• The size of the gap is proportional to the Fourier component of the periodic potential at that reciprocal lattice vector.

Tight-binding approximation

This model starts from the opposite limit: electrons are tightly bound to individual atoms, and you build bands by letting neighboring atomic orbitals overlap slightly.

• You begin with isolated atomic orbitals and then "turn on" the coupling between neighbors.
• The resulting bandwidth depends on the overlap integral between adjacent atoms. Stronger overlap produces wider bands; weaker overlap gives narrow, flat bands.
• This approach is natural for describing insulators, transition metal d-bands, and covalent semiconductors where electrons retain significant atomic character.

Kronig-Penney model

This is a simplified 1D model that captures the essential physics of band formation using a periodic array of rectangular potential barriers.

1. Set up a repeating sequence of square-well potentials with barrier height $V_0$, well width $a$, and barrier width $b$.
2. Solve the Schrödinger equation in each region (well and barrier) and match boundary conditions.
3. Apply Bloch's theorem to enforce periodicity, which yields a transcendental equation relating energy to $k$.
4. Solutions exist only for certain energy ranges (allowed bands), separated by gaps where no solutions exist (forbidden bands).

The model shows clearly that stronger potentials and wider barriers produce larger band gaps. Though it's one-dimensional, the qualitative insights transfer directly to real 3D crystals.

Band structure characteristics

A band structure diagram plots electron energy $E$ versus crystal momentum $\mathbf{k}$. It's essentially the "roadmap" of what energies and momenta are available to electrons in a given material.

Allowed vs. forbidden energy states

• Allowed bands are continuous ranges of energy where electron states exist. They result from constructive interference of electron waves in the periodic lattice.
• Band gaps (forbidden regions) are energy ranges with zero available states. They arise from destructive interference at specific energies, particularly near Brillouin zone boundaries.
• The pattern of allowed and forbidden states is what ultimately determines whether a material is a metal, semiconductor, or insulator.

Conduction and valence bands

• The valence band is the highest-energy band that is fully (or nearly fully) occupied at zero temperature. Electrons here are involved in bonding.
• The conduction band is the next band up, which is empty or partially filled. Electrons in this band are free to carry current.
• The valence band maximum (VBM) and conduction band minimum (CBM) are the critical energy points that define the band gap: $E_g = E_{\text{CBM}} - E_{\text{VBM}}$.
• In metals, these bands overlap or the highest band is only partially filled, so electrons can move freely without needing extra energy.

Band gaps and types

The band gap $E_g$ is the minimum energy an electron needs to jump from the valence band to the conduction band.

• Direct band gap: The CBM and VBM occur at the same $\mathbf{k}$-point. An electron can transition between bands by absorbing a photon alone. Example: GaAs ($E_g \approx 1.42$ eV).
• Indirect band gap: The CBM and VBM sit at different $\mathbf{k}$-values. Transitions require both a photon and a phonon (lattice vibration) to conserve momentum. Example: silicon ($E_g \approx 1.12$ eV).
• Zero band gap: Materials like graphene have the valence and conduction bands touching at specific points (Dirac points), giving rise to massless-fermion-like behavior.

The distinction between direct and indirect gaps has huge practical consequences for optoelectronics, which is covered further below.

Electronic properties of materials

Metals vs. insulators vs. semiconductors

Band theory gives a clean classification of solids based on band filling and gap size:

Doping introduces impurity atoms into a semiconductor to add extra electrons (n-type) or create holes (p-type), shifting the balance of charge carriers without changing the band gap itself.

Fermi level and Fermi surface

The Fermi level ($E_F$) is the chemical potential of electrons at absolute zero. It marks the boundary between occupied and unoccupied states.

• In a metal, $E_F$ lies inside a band, so there are always states available for conduction.
• In a semiconductor, $E_F$ sits inside the band gap (roughly mid-gap for intrinsic material).
• In an insulator, $E_F$ also lies in the gap, but the gap is so large that thermal excitation across it is negligible.

The Fermi surface is the constant-energy surface in k-space at $E = E_F$. Its shape governs many metallic properties: electrical conductivity, thermal conductivity, and the de Haas–van Alphen oscillations used to map it experimentally. Temperature and doping shift $E_F$ in semiconductors, which is how you control device behavior.

Density of states

The density of states $g(E)$ counts how many electron states are available per unit energy per unit volume. It connects the band structure to measurable quantities like heat capacity and optical absorption.

• In 3D, the free-electron density of states goes as $g(E) \propto \sqrt{E}$.
• In 2D (quantum wells), $g(E)$ is a step function: each subband contributes a constant value.
• In 1D (quantum wires), $g(E) \propto 1/\sqrt{E}$, diverging at each subband edge.
• Van Hove singularities are peaks or kinks in $g(E)$ that occur at critical points in the band structure where $\nabla_k E = 0$. They show up as sharp features in optical spectra.

Band structure calculation methods

Calculating band structures from first principles is computationally demanding. Different methods trade off between accuracy, computational cost, and the types of materials they handle well.

k·p method

A perturbation theory approach that expands the Hamiltonian around a known high-symmetry point (usually $\Gamma$) in powers of $\mathbf{k}$.

• Most useful near band edges, where you care about effective masses and optical transition strengths.
• Produces analytical expressions, which makes it fast and physically transparent.
• Works best for direct-gap semiconductors like GaAs and InP.
• Less reliable far from the expansion point, where higher-order terms become important.

Pseudopotential method

The core electrons of heavy atoms create a deep, rapidly oscillating potential that's expensive to represent numerically. The pseudopotential method replaces this with a smoother, weaker pseudopotential that reproduces the same scattering properties for valence electrons.

• Dramatically reduces the computational cost compared to all-electron calculations.
• Produces accurate band structures for sp-bonded semiconductors and metals.
• Empirical pseudopotentials are fitted to experiment; ab initio pseudopotentials are derived from atomic calculations with no fitting parameters.

Density functional theory

DFT reformulates the many-electron problem in terms of the electron density $n(\mathbf{r})$ rather than the many-body wavefunction. You solve self-consistent Kohn-Sham equations that look like single-particle Schrödinger equations with an effective potential.

• It's the workhorse of modern computational materials science, applicable to metals, semiconductors, insulators, and complex heterostructures.
• Standard DFT (using LDA or GGA functionals) systematically underestimates band gaps, often by 30–50%. This is a well-known limitation.
• Corrections like hybrid functionals (HSE06) or the GW approximation improve gap predictions at higher computational cost.
• DFT struggles with strongly correlated systems (e.g., Mott insulators, heavy-fermion materials) where electron-electron interactions dominate.

Experimental techniques

Theory and computation need experimental validation. Several spectroscopic techniques directly probe the electronic structure of solids.

Photoemission spectroscopy

A photon with known energy $h\nu$ hits the sample surface and ejects an electron. By measuring the kinetic energy of the emitted electron, you can determine its original binding energy:

$E_{\text{binding}} = h\nu - E_{\text{kinetic}} - \phi$

where $\phi$ is the work function of the material.

• UPS (ultraviolet photoemission) uses UV light and probes valence band states.
• XPS (X-ray photoemission) uses X-rays and probes deeper core-level states, providing chemical composition information.
• Both techniques are surface-sensitive (probing depth of ~1–10 nm).

X-ray absorption spectroscopy

XAS measures how strongly a material absorbs X-rays as a function of photon energy, probing unoccupied states above the Fermi level.

• XANES (near-edge structure) reveals the local electronic environment and oxidation state.
• EXAFS (extended fine structure) provides information about bond lengths and coordination numbers around a specific element.
• Because you tune the X-ray energy to a specific absorption edge, XAS is element-specific, which is powerful for studying multi-component materials.

Angle-resolved photoemission spectroscopy

ARPES is the most direct way to measure band structure experimentally. It measures both the energy and the emission angle of photoelectrons, and from the angle you extract the in-plane crystal momentum $\mathbf{k}_{\parallel}$.

• The result is a direct map of $E$ vs. $\mathbf{k}$, which you can compare side-by-side with calculated band structures.
• ARPES can image Fermi surfaces and detect many-body effects like electron-phonon coupling (seen as "kinks" in the dispersion).
• Spin-resolved ARPES adds spin detection, which is essential for studying topological insulators and magnetic materials.

Band structure effects

Effective mass

Inside a crystal, electrons don't respond to external forces the way free electrons do. The effective mass $m^*$ captures how the band curvature modifies the electron's inertia:

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

• A strongly curved band (large $d^2E/dk^2$) gives a small effective mass and high carrier mobility. GaAs electrons have $m^* \approx 0.067 \, m_e$, which is why GaAs devices are fast.
• A flat band gives a large effective mass and sluggish transport.
• Effective mass can be anisotropic: different along different crystal directions. Silicon has longitudinal and transverse electron masses that differ by a factor of ~5.

Hole concept

When an electron is excited out of the valence band, it leaves behind an empty state. Rather than tracking all the remaining electrons, it's far simpler to treat this vacancy as a hole: a quasiparticle with positive charge $+e$ and its own effective mass.

• Holes move in the opposite direction to the missing electron, so they behave like positive charge carriers.
• Hole effective masses are typically larger than electron effective masses in most semiconductors (because valence bands tend to be flatter than conduction bands).
• p-type doping (e.g., boron in silicon) introduces acceptor levels near the valence band edge, creating extra holes.

Band bending at interfaces

When two materials with different Fermi levels are brought into contact, charge flows across the interface until the Fermi levels align. This charge redistribution creates a space charge region and bends the bands near the junction.

• In a p-n junction, band bending creates a built-in electric field that separates electrons and holes, which is the basis of diode behavior.
• In a metal-semiconductor contact, band bending determines whether you get an ohmic contact or a Schottky barrier.
• Applying an external voltage modifies the band bending, which is how field-effect transistors (MOSFETs) control current.

Advanced band concepts

Indirect vs. direct band gaps

This distinction, introduced earlier, has major consequences for device design:

• Direct gap materials (GaAs, InP, GaN) can emit and absorb light efficiently because transitions only need a photon. This makes them ideal for LEDs and laser diodes.
• Indirect gap materials (Si, Ge) require phonon assistance for optical transitions, making them poor light emitters but giving them longer carrier lifetimes, which benefits solar cells and transistors.
• Band structure engineering can tune the gap type. For example, in $\text{GaAs}_{1-x}\text{P}_x$ alloys, the gap transitions from direct to indirect as the phosphorus fraction $x$ increases past ~0.45.

Band structure engineering

You can deliberately modify a material's band structure through several techniques:

• Alloying: Mixing semiconductors (e.g., $\text{Al}_x\text{Ga}_{1-x}\text{As}$) to tune the band gap continuously between the endpoint values.
• Strain engineering: Applying mechanical strain shifts band edges and can split degenerate bands. Strained silicon in modern transistors has ~30% higher electron mobility than unstrained silicon.
• Quantum confinement: Reducing dimensions to nanoscale sizes modifies the effective band structure (see below).

Quantum confinement effects

When a material dimension shrinks to the scale of the electron's de Broglie wavelength (typically a few nanometers), the continuous band structure breaks into discrete sub-levels.

• Quantum wells (confinement in 1 dimension): Energy levels become quantized along the confined direction while remaining band-like in the other two. Used in laser diodes and high-electron-mobility transistors (HEMTs).
• Quantum wires (confinement in 2 dimensions): Further quantization narrows the density of states.
• Quantum dots (confinement in 3 dimensions): Fully discrete, atom-like energy levels. The emission wavelength depends on dot size, which is why quantum dots are used for tunable-color displays and biological imaging.

Applications of band theory

Semiconductor devices

• Transistors (MOSFETs, HBTs): Gate voltage modulates band bending to switch current on and off. Band engineering optimizes channel mobility and threshold voltage.
• Diodes and LEDs: The p-n junction's band alignment controls rectification. In LEDs, electrons and holes recombine across a direct band gap, emitting photons with energy $\approx E_g$.
• Solar cells: Photons with energy $\geq E_g$ excite electrons across the gap. The Shockley-Queisser limit (~33% efficiency for a single junction at $E_g \approx 1.34$ eV) comes directly from band theory considerations.

Optoelectronic materials

• Laser diodes require population inversion between conduction and valence band states, achieved by strong carrier injection into a direct-gap active region.
• Photodetectors absorb photons via band-to-band transitions; the cutoff wavelength is set by $\lambda_{\text{max}} = hc / E_g$.
• Quantum well and quantum dot structures provide sharp density-of-states features that improve device efficiency and enable wavelength tunability.

Thermoelectric materials

Thermoelectric devices convert temperature differences into voltage (Seebeck effect) or vice versa (Peltier effect). Band structure plays a central role:

• Good thermoelectrics need high electrical conductivity but low thermal conductivity. Narrow-gap semiconductors ($E_g \sim 0.1$–$0.3$ eV) like $\text{Bi}_2\text{Te}_3$ often hit this sweet spot.
• The Seebeck coefficient depends on how the density of states and carrier mobility vary with energy near $E_F$, both of which are band structure properties.
• Band engineering strategies (e.g., band convergence, resonant doping) aim to maximize the thermoelectric figure of merit $ZT$.