Electronic Band Structure

Electronic band structure is the set of allowed electron energies in a crystalline solid, expressed as functions $E_n(\mathbf{k})$ of wave vector $\mathbf{k}$ and band index $n$ . It describes both the dispersion of electronic states and the energy intervals in which no states occur. A central concept in condensed-matter physics, it connects microscopic quantum mechanics with the distinction between metals and insulators and with the motion of charge carriers. (davidtong.org)

Periodicity and Bloch states

In an ideal crystal, an effective single-electron Hamiltonian is invariant under translations by lattice vectors $\mathbf{R}$ . For a local periodic potential, this means

V(\mathbf{r}+\mathbf{R})=V(\mathbf{r}).

Bloch’s theorem states that its energy eigenstates can be chosen with wave functions of the form

\psi_{n\mathbf{k}}(\mathbf{r}) =e^{i\mathbf{k}\cdot\mathbf{r}}u_{n\mathbf{k}}(\mathbf{r}), \qquad u_{n\mathbf{k}}(\mathbf{r}+\mathbf{R}) =u_{n\mathbf{k}}(\mathbf{r}).

The plane-wave factor specifies the response to translation; the periodic factor contains the structure within a unit cell. Different solutions at the same $\mathbf{k}$ produce different bands. (ocw.mit.edu)

The quantity $\hbar\mathbf{k}$ is called crystal momentum. Unlike ordinary mechanical momentum, it is defined modulo $\hbar\mathbf{G}$ , where $\mathbf{G}$ belongs to the reciprocal lattice. Consequently, distinct wave vectors can be restricted to the first Brillouin zone. The corresponding band energies repeat in reciprocal space. (qs3.mit.edu)

Formation of bands

Two complementary models explain how electronic bands arise.

The nearly-free-electron model begins with the parabolic dispersion of free electrons and introduces a weak periodic potential. This potential couples states whose wave vectors differ by a reciprocal-lattice vector. Near degeneracies, particularly at Brillouin-zone boundaries, coupling can split their energies and open gaps. A gap along one direction does not necessarily imply an energy interval devoid of states throughout the entire zone. (davidtong.org)

The tight-binding model begins with localized atomic orbitals. Hopping between sites turns their initially discrete energies into dispersive bands. For an ideal one-dimensional chain with one orbital per site and nearest-neighbour hopping,

E(k)=\varepsilon_0-2t\cos(ka),

where $\varepsilon_0$ is the on-site energy, $a$ the lattice spacing, and $t$ the hopping parameter in this sign convention. The bandwidth is $4|t|$ . Additional orbitals and hopping processes produce more complex dispersions. (www-thphys.physics.ox.ac.uk)

These models describe different starting limits rather than mutually exclusive explanations: localized orbital combinations and extended Bloch states can represent the same periodic electronic system. (davidtong.org)

Occupation and material classification

A band structure specifies available states, not their occupation. Electrons fill states subject to the Pauli exclusion principle. For a spin-degenerate band in a crystal containing $N$ primitive cells, there are $N$ allowed wave vectors and room for $2N$ electrons. A completely filled band cannot produce ordinary longitudinal conduction simply by redistributing its occupations under a weak field; a partially filled band can. (www-thphys.physics.ox.ac.uk)

Within the independent-electron description:

Metals have partially occupied bands, with states available arbitrarily close to the zero-temperature Fermi energy.
Band insulators have completely occupied lower bands separated from empty upper bands by an energy gap.
Semimetals can have slightly overlapping valence and conduction bands, producing electron and hole pockets. (davidtong.org)

In a gapped material, the occupied bands nearest the gap are called valence bands, and the empty bands above it conduction bands. Their edge separation is the band gap:

E_g=\min_{\mathbf{k}}E_c(\mathbf{k}) -\max_{\mathbf{k}}E_v(\mathbf{k}).

A semiconductor has the same basic band-filling pattern as a band insulator; the terminology reflects the accessibility of charge carriers under relevant conditions rather than a universal numerical boundary between two fundamentally different band structures. (vasp.at)

Reading a band-structure plot

A conventional plot displays energy vertically and a sequence of wave vectors horizontally. The path usually connects high-symmetry points in the Brillouin zone. $\Gamma$ denotes $\mathbf{k}=0$ ; other labels depend on the crystal symmetry and coordinate convention. The horizontal axis is a path through reciprocal space, not a distance through the material. (vasp.at)

Each curve traces a band along that path. A horizontal reference often marks the Fermi energy or a valence-band edge. Because the plot samples only selected lines, it may miss extrema or crossings elsewhere in the zone. Establishing the global gap therefore requires adequate reciprocal-space sampling, not merely inspection of a standard high-symmetry path. (vasp.at)

The density of states gives the number of states per energy interval after summing over wave vectors. It complements a band plot but discards momentum-resolved information. Orbital-projected plots instead indicate how strongly selected atomic orbitals contribute to each state. (quantum-espresso.org)

Carrier dynamics

Band slopes determine the velocity of an electron wave packet:

\mathbf{v}_n(\mathbf{k}) =\frac{1}{\hbar}\nabla_{\mathbf{k}}E_n(\mathbf{k}).

Near a nondegenerate, approximately parabolic extremum, the curvature defines an effective-mass tensor:

(m^{*-1})_{ij} =\frac{1}{\hbar^2} \frac{\partial^2E_n} {\partial k_i\,\partial k_j}.

This mass describes acceleration within a band, not a change in the electron’s intrinsic mass. Near a valence-band maximum, missing electrons can be treated as positively charged holes. In metals, the Fermi surface identifies the wave vectors whose energies equal the Fermi energy and organizes the low-energy carrier dynamics. (davidtong.org)

Direct and indirect gaps

A direct gap has its valence-band maximum and conduction-band minimum at the same wave vector. For an indirect gap, these extrema occur at different wave vectors. The distinction matters because optical transitions must satisfy both energy and momentum constraints; transitions across an indirect gap ordinarily require an additional momentum-transfer process involving a phonon. (vasp.at)

Band-edge separation alone does not determine optical intensity. Transition matrix elements and symmetry also matter, so identifying a direct gap is not equivalent to establishing that its lowest-energy transition is strongly allowed. Computational methods can evaluate momentum matrix elements alongside band energies. (quantum-espresso.org)

Calculation and experimental measurement

Realistic band structures are commonly calculated using density-functional theory (DFT). A typical workflow first determines a self-consistent electron density on a reciprocal-space mesh and then evaluates the resulting Kohn–Sham Hamiltonian along a plotting path. The plotted curves are its eigenvalues. (vasp.at)

Kohn–Sham eigenvalues are auxiliary quantities and should not automatically be identified with measured electron-addition or electron-removal energies. Even the exact local Kohn–Sham potential need not yield the fundamental gap directly from its eigenvalue separation. Generalized Kohn–Sham approaches, including hybrid functionals, can give more realistic gaps, but their results remain dependent on the approximation. (arxiv.org)

Angle-resolved photoemission spectroscopy (ARPES) measures the energies and emission directions of electrons released by light, allowing occupied-state dispersions to be reconstructed. More precisely, it probes the electronic spectral function, with experimental weighting factors. Interactions can shift and broaden its peaks; well-defined quasiparticles appear as relatively sharp dispersive features. (arpes.stanford.edu)

Conventional ARPES primarily accesses occupied states. Time-resolved and two-photon photoemission methods can populate and investigate otherwise unoccupied states, extending the experimentally accessible band structure. (arpes.stanford.edu)

Topology and limitations

Band energies do not exhaust the information contained in Bloch states. The way their wave functions vary across reciprocal space can support topological invariants. Topological insulators illustrate this distinction: a gapped bulk band structure can coexist with boundary states associated with the topology and symmetries of the occupied states. (arxiv.org)

The simplest band picture assumes periodicity and independent electrons in an effective potential. In strongly interacting systems, spectral weight can move into incoherent features, and sharp quasiparticle bands may cease to provide an adequate description. A measured spectral function then contains information that cannot be represented faithfully by a collection of infinitely sharp curves. (arpes.stanford.edu)

Historical development

Felix Bloch developed the quantum description of electrons in periodic crystals in 1928. Alan Wilson subsequently used band occupation and energy gaps to explain the distinction between metals and insulators in 1931. These developments established the conceptual framework connecting crystal periodicity with electronic conduction. (davidtong.org)

References

Solid State Physics — David Tongdavidtong.org
Band Structure — David Tongdavidtong.org
Electron Dynamics in Solids — David Tongdavidtong.org
Lecture Notes for Solid State Physics — Steve Simonwww-thphys.physics.ox.ac.uk
lecture_17b.pdf — Physics for Solid-State Applicationsocw.mit.edu
Introduction to Topological Bands — Jennifer Canoqs3.mit.edu
Band-structure calculation using density-functional theory — VASP Wikivasp.at
BANDGAP — VASP Wikivasp.at
Quantum ESPRESSO Post-processing User’s Guidequantum-espresso.org
bands.x: input descriptionquantum-espresso.org
Understanding Band Gaps of Solids in Generalized Kohn-Sham Theoryarxiv.org
Angle-resolved Photoemission Spectroscopy — Shen Laboratoryarpes.stanford.edu