| band structure | |
|---|---|
| Name | Band structure |
| Caption | Schematic of electronic bands and a band gap in a crystalline solid |
| Field | Solid-state physics |
| Related | Bloch's theorem, Band gap, Density functional theory |
band structure
Band structure describes the allowed energy levels of electrons in a periodic solid as a function of crystal momentum; it determines whether a material is a conductor, semiconductor, or insulator. In quantum mechanics and solid-state physics the distribution and dispersion of electronic bands control electrical, optical, and thermal properties, and underpin technologies such as semiconductor devices and photovoltaic cells.
Band structure is the mapping E(k) of electronic energy eigenvalues versus crystal wavevector k in a periodic lattice such as a crystal. The arrangement of occupied and unoccupied bands, including the band gap, sets carrier concentrations, effective masses, and transport coefficients. Practical outcomes include the operation of p–n junctions, metallicity, and the optical absorption spectra exploited in light-emitting diodes and solar cells. Band structure links microscopic quantum solutions to macroscopic observables measured in laboratories like Bell Labs and research centers such as CERN for materials studies.
The origin of band structure lies in solving the single-particle Schrödinger equation with a periodic potential given by the ionic lattice. Bloch's theorem (formulated by Felix Bloch) states that eigenfunctions take the form of Bloch functions ψ_{n,k}(r)=e^{ik·r}u_{n,k}(r), reducing the problem to a single Bravais cell and producing a discrete set of bands indexed by band number n and crystal momentum k in the Brillouin zone. The periodicity follows from the lattice vectors of a Bravais lattice, while symmetry considerations use group theory and the space group of the crystal. Concepts from the Born–von Karman boundary conditions and Bloch wave propagation are essential for understanding band dispersion and degeneracies such as those enforced by time-reversal symmetry and inversion symmetry.
Bands arise from the splitting and hybridization of atomic orbitals when atoms form a solid; prototypical examples include s- and p-derived bands in silicon and d-bands in transition metals like iron and nickel. The Fermi level separates occupied from unoccupied states at zero temperature. Direct and indirect band gaps (e.g., germanium vs. gallium arsenide) determine optical transition probabilities. Defects, doping by elements such as phosphorus or boron, and disorder produce localized states within gaps described by models like Anderson localization. Quasiparticle effects modify single-particle bands; these are captured by GW approximation and many-body perturbation theory.
A hierarchy of theoretical methods is used to compute band structures. The nearly free electron model treats electrons as weakly perturbed plane waves and explains free-electron-like metals and Bragg reflection at zone boundaries. The tight-binding model (linear combination of atomic orbitals) captures strong localization and hopping in covalent solids; it underlies models for graphene and the Hubbard model. The predominant ab initio approach is density functional theory (DFT) developed by Walter Kohn and Pierre Hohenberg, implemented in codes like VASP, Quantum ESPRESSO, and WIEN2k. DFT often requires corrections from hybrid functionals (e.g., HSE06) or many-body methods such as the GW approximation and Bethe–Salpeter equation for excitons. Low-energy effective descriptions use the k·p perturbation method and models such as the Dirac equation in graphene or the Bernevig–Hughes–Zhang model for topological insulators.
Experimental access to band structure employs techniques that resolve energy and momentum. Angle-resolved photoemission spectroscopy (ARPES) measures occupied band dispersions and has been pivotal at facilities like Stanford University and Lawrence Berkeley National Laboratory. Scanning tunneling microscopy and scanning tunneling spectroscopy (STM/STS) probe local density of states and surface bands. Optical spectroscopy, including photoluminescence and absorption spectroscopy, reveals band gaps and excitons, while cyclotron resonance and quantum oscillation experiments (Shubnikov–de Haas, de Haas–van Alphen) determine effective masses and Fermi surfaces. Electrical transport and Hall measurements provide complementary information about carrier density and mobility in materials developed by companies such as Intel and tested in device laboratories.
Band structure dictates electronic conductivity through available states at the Fermi level and scattering processes described by Boltzmann transport equation treatments. Optical responses depend on interband transitions and selection rules derived from symmetry; semiconductors with suitable gaps are used in lasers and LEDs. Thermal properties involve electronic contributions to heat capacity and thermal conductivity, as in metals where electrons dominate, and in thermoelectrics where band engineering (e.g., in Bi2Te3) improves the ZT. Engineering band extrema, effective masses, and valley degeneracy underpins modern applications in spintronics and valleytronics.
Advances in topology have reinterpreted band structure through invariants such as the Chern number and Z2 topological invariant, predicting phases like quantum Hall effect, topological insulators, and Weyl semimetals. Key theoretical and experimental progress has involved figures and institutions like Charles Kane, Shoucheng Zhang, and groups at MIT and Princeton University. The interplay of spin–orbit coupling, symmetry breaking, and band inversions leads to protected surface or edge states with potential for low-dissipation electronics and quantum computation. Contemporary research integrates machine learning for materials discovery, high-throughput DFT databases such as the Materials Project and AFLOW to screen band structures, and experimental platforms including two-dimensional materials (e.g., graphene, transition metal dichalcogenides) and engineered photonic crystal analogues that emulate electronic band phenomena.
Category:Solid-state physics Category:Condensed matter physics