# Electronic band structure The **electronic band structure** of a solid is the set of energies an [[Electron|electron]] is allowed to have inside it, drawn as a function of the electron's wavevector. In a free electron gas every energy above zero is available; in a [[Crystal_structure|crystal]] the periodic potential of the ions leaves continuous *bands* of allowed energies separated by *gaps* in which no travelling state exists.[^likharev-comb][^openstax-v3-ch9] Almost every electrical, optical and thermal property of a solid follows from the shape of those bands and from how far up them the available electrons reach. In the microsim below the reader raises the lattice strength μ of a one-dimensional chain of delta barriers — the Kronig–Penney or Dirac comb model — and watches a single free-particle parabola break into bands.[^kronig-penney] The equation that answers is the master dispersion relation `cos(q·a) = cos(k·a) + (mu/(k·a))·sin(k·a)`, with μ = m·a·W/ħ², a the lattice period, W the barrier weight, k the electron's own wavenumber and q the quasimomentum.[^likharev-comb] The sim samples in k rather than solving for it: wherever the right-hand side falls within ±1 the energy (ħk)²/2m is allowed, and wherever it does not, a gap is shaded.[^spec-m37] A second control sets the number of electrons per unit cell, which decides whether the topmost occupied state lands in the middle of a band — a [[Metallic_bonding|metal]] — or exactly at a band top, leaving a gap above it: an [[Insulator_(electricity)|insulator]] or a [[Semiconductor|semiconductor]]. The readouts are the band widths, the gaps, and the effective mass at the bottom of band 1. On the [[Materials_science]] flagship this article is the root of the *Electronic, optical, and magnetic* section of Part VII — Research, and the parent of the shared C31 sim set: [[Doping_(semiconductor)|doping]] fills the gap this page opens, and [[Band_gap|band gap]] turns its width into a colour. ## Why bands and band gaps occur Two arguments arrive at the same picture from opposite ends. Start from isolated atoms and bring them together: each atomic [[Energy_level|energy level]] splits into as many closely spaced levels as there are atoms, and those levels form a band whose width measures how strongly neighbouring atoms overlap.[^openstax-v3-ch9] Start instead from a [[Free_electron_model|free electron]] and switch on a weak periodic potential: the electron is Bragg-reflected wherever its wavelength fits the lattice, standing waves form, and the two standing waves — one piling charge on the ions, one between them — have different energies. A gap of width 2|U_n| opens at each zone boundary, U_n being the n-th Fourier component of the potential.[^likharev-nfe] The Dirac comb of the sim shows both limits on one dial. With μ = 0 the dispersion relation reduces to cos(qa) = cos(ka), the free parabola, and there is no gap. Turning μ up opens gaps and narrows the bands, and the band tops stay pinned at ka = nπ for every μ, because sin(ka) vanishes there and the right-hand side equals cos(nπ) = ±1 whatever the lattice strength.[^derived-ebs] At μ = 5 band 1 runs from ka = 2.2845 to ka = π, band 2 begins at ka = 4.7613, and in units of ħ²/(2ma²) the first band is 4.65 wide while the first gap is 12.80.[^derived-ebs] With a lattice period of 0.5 nm that energy unit is 0.152 [[Electronvolt|eV]], so μ = 5 describes a wide-gap material: a first band 0.71 eV wide beneath a gap of 1.95 eV.[^derived-ebs] Raising μ to 30 narrows band 1 to 1.19 units, 0.18 eV, and widens the gap to 3.79 eV — the isolated-atom limit, where the band is collapsing back toward a sharp atomic level.[^derived-ebs] One sign convention must be watched. The extraction of the Portal Book prints the relation with a minus sign, `cos(qa) = cos(ka) − β·sin(ka)`; a direct derivation for a repulsive barrier gives the plus sign used here, and the sub-manual records the discrepancy.[^manual04-sign] The test is at ka → 0, where the right-hand side tends to 1 + μ: with a repulsive comb there must be a gap at the bottom of the spectrum, so an allowed band that reaches k = 0 means the attractive comb has been coded by mistake.[^manual04-sign] ## Basic concepts A band structure is a statement about a single electron moving in a fixed, perfectly periodic potential, and every word of that description is an approximation to be checked. Four ideas do most of the work: what the model assumes, why periodicity turns the wavevector into a good label, how states are counted at each energy, and how many electrons there are to place. The first three fix the drawing; the fourth decides what kind of material it describes. ### Assumptions and limits of band structure theory Three assumptions are built in. The lattice is perfect and infinite, so that a wavevector is a good [[Quantum_number|quantum number]]; the ions are fixed, so that the potential does not move; and the electrons are independent, each moving in an average potential rather than seeing one another individually. None is exactly true of any real specimen, and each fails in a characteristic way. [[Crystallographic_defect|Defects]], surfaces and [[Dislocation|dislocations]] destroy strict periodicity and smear the sharp band edges. Lattice vibrations — [[Phonon|phonons]] — scatter electrons between states and shift the gaps with [[Temperature|temperature]]. Strong correlation defeats the independent-electron picture outright: several transition-metal oxides have partly filled bands, which should make them metals, and are insulators because the cost of putting two electrons on one site exceeds the energy gained by hopping. Band theory is nevertheless the working language of [[Solid-state_physics|solid-state physics]], because for the ordinary metals, [[Semiconductor|semiconductors]] and closed-shell insulators it is close to exact.[^openstax-v3-ch9] ### Crystalline symmetry and wavevectors Periodicity is what makes the problem solvable. [[Bloch's_theorem|Bloch's theorem]] states that a solution in a periodic potential can be written as ψ(x + a) = ψ(x)·exp(i·q·a): the wavefunction repeats from cell to cell up to a phase, and q, the quasimomentum, labels the state.[^likharev-bloch][^bloch1929] Because exp(i·q·a) is unchanged when q shifts by 2π/a, every observable is periodic in q with that period, and all distinct states can be drawn within one interval — the first Brillouin zone, from −π/a to π/a in one dimension. This is why band diagrams are always drawn in a reduced zone: the sim folds its k-sampling back into that interval, and the free parabola appears as a set of folded arcs rather than as one curve.[^likharev-bloch][^spec-m37] In three dimensions the same argument runs over the [[Reciprocal_lattice|reciprocal lattice]], and the zone becomes a polyhedron whose symmetry points carry conventional names. ### Density of states The density of states g(E) counts how many states lie in each interval of energy, and it is the quantity that most physical properties actually integrate over. In one dimension the states are evenly spaced in q, dN = (L/2π)·dq, so g(E) = (L/2π)·(dq/dE): wherever the band flattens and dE/dq goes to zero, the density of states diverges.[^likharev-dos] That happens at every band edge, which is why band edges dominate optical absorption and carrier statistics even though they occupy a vanishing range of q. In three dimensions the divergences are softened into kinks, but the principle survives — a flat band means many states at one energy, and a steep band means few. ### Filling of bands Bands decide what is possible; filling decides what happens. Each band holds two electrons per unit cell, one for each [[Spin_(physics)|spin]], so a crystal with an even number of electrons per cell can fill an integer number of bands exactly and a crystal with an odd number cannot. A completely filled band carries no current: for every state moving right there is an occupied state moving left, and an applied field cannot change the balance because there is nowhere to move an electron to. A partly filled band conducts, because states just above the highest occupied one are empty and an infinitesimal field can shift the distribution into them. This is the second control of the sim: with one electron per cell the [[Fermi_level|Fermi level]] sits in the middle of band 1 and the readout says metal; with two it sits at the top of band 1 with the gap above, and the same lattice is an insulator or, if the gap is small enough that [[Temperature|thermal]] excitation across it matters, a semiconductor.[^spec-m37] The divide between the last two is quantitative, not structural: both have a filled valence band and an empty conduction band, and only the size of the gap differs.[^openstax-v3-ch9] ## Theory in crystals Calculating a band structure means solving a one-electron [[Schrödinger_equation|Schrödinger equation]] in a periodic potential, and the methods below differ in where they start and in what they are willing to approximate. Two are analytic limits, useful for understanding and for one-dimensional models like the sim's; the rest are numerical schemes built to be run on real crystals, and they are ordered here roughly by how much electron–electron interaction they keep. ### Nearly free electron approximation If the potential is weak, the free-electron states are almost right, and degenerate perturbation theory fixes them where they fail. Two plane waves differing by a reciprocal-lattice vector are degenerate at the zone boundary; mixing them gives `E = E_ave ± sqrt(((E_l − E_l')/2)^2 + |U_n|^2)`, so the degeneracy is lifted and a gap of 2|U_n| opens at q = πm/a.[^likharev-nfe] The approximation explains why simple metals have nearly spherical Fermi surfaces and why their gaps are small, and it is the μ → 0 end of the sim's dial, where the bands are wide, the gaps are narrow, and the dispersion is barely bent away from the [[Free_electron_model|free-electron]] parabola. ### Tight binding model At the opposite end, the electron is nearly bound to one atom and only occasionally hops to a neighbour. Keeping nearest-neighbour hopping only gives a band of pure cosine shape, `E = E_n + 2·hbar·eta_n·cos(q·a)`, centred on the atomic level E_n and of total width 4ħ|η_n|, valid while ħ|η_n| is much smaller than E_n.[^likharev-tb] Two things follow at once. The band width is set by the hopping rate and therefore falls off exponentially with atomic spacing, which is why core levels stay sharp while valence levels broaden into wide bands. And the effective mass at the band bottom is inversely proportional to the band width, so narrow bands carry heavy carriers. The sim's large-μ end is this limit: at μ = 30 the first band has narrowed to about an eighth of its μ = 1 width, and the cosine shape the model predicts is visible in the curve the sim draws.[^derived-ebs] ### KKR model The Korringa–Kohn–Rostoker method treats the crystal as a lattice of scatterers rather than as a potential to be diagonalized. Each atom is given a spherical muffin-tin potential, its scattering properties are computed once, and the band condition becomes the requirement that the waves scattered by all the sites interfere constructively. Because the scattering and the geometry separate, the same atomic input can be reused for different lattices, and the Green's-function formulation extends naturally to disordered [[Alloy|alloys]], where no single periodic cell exists. ### Density-functional theory [[Density_functional_theory|Density-functional theory]] is the workhorse of modern band-structure calculation. It replaces the many-electron problem by a fictitious system of non-interacting electrons moving in an effective potential chosen so that the ground-state density is reproduced exactly, and the resulting single-particle eigenvalues are plotted as bands. The theory is exact in principle for the ground-state density and energy, but the eigenvalues are not guaranteed to be the energies of adding or removing an electron, and with the usual local and semi-local approximations to the exchange–correlation energy the calculated gaps come out systematically too small. That the method still predicts band shapes, effective masses and Fermi surfaces well is the reason it dominates practice; the gap error is why it is usually corrected before an optical property is quoted. ### Green's function methods and the <i>ab initio</i> GW approximation The GW approximation attacks precisely the quantity density-functional theory gets wrong. Instead of an effective potential it computes the self-energy of an electron as the product of the one-particle Green's function G and the screened Coulomb interaction W, giving the energy of a real quasiparticle — an electron dressed by the response of all the others. Gaps calculated this way are much closer to those measured by photoemission, at a cost in computer time an order of magnitude or more above a density-functional calculation. The same machinery, extended to electron–hole pairs, is what allows absorption spectra to be predicted rather than fitted. ### Dynamical mean-field theory Where correlation is strong, neither of the above works. Dynamical mean-field theory maps the lattice onto a single site exchanged with a self-consistent bath, keeping the local repulsion between two electrons on the same atom exactly while treating the rest of the lattice as an average. It reproduces the correlation-driven metal-to-insulator transition that band theory misses and the narrow, heavy bands seen in transition-metal and rare-earth compounds. The price is that a band is no longer a sharp line: the spectral function has finite width because a quasiparticle in a correlated metal has a finite lifetime. ### Others Several other approaches are in routine use. The k·p method expands the bands around a single point of the zone and, with a few measured parameters, gives accurate effective masses for semiconductor device work. Pseudopotential methods remove the tightly bound core electrons and the rapid oscillations of the valence wavefunctions near the nucleus, leaving a smooth problem that plane waves can solve efficiently. Empirical tight-binding and empirical pseudopotential schemes fit their parameters to measured gaps rather than deriving them, which is what the sim's single μ does in its own one-dimensional way.[^spec-m37] ## Band diagrams A band diagram plots energy against wavevector along chosen lines through the Brillouin zone, and reading one is a skill with a few fixed rules. The vertical axis is energy, usually with the [[Fermi_level|Fermi level]] or the valence-band maximum set to zero. A curve's curvature gives the effective mass through `1/m_ef = (1/hbar^2)·d2E/dq2`, so a sharply curved band is light and a flat one heavy, and the curvature is negative at a band top, which is the formal meaning of a positive-mass hole.[^likharev-meff] The Portal Book's reference values are silicon at 0.26 electron masses in the conduction band and 0.39 in the valence band, with indium antimonide as low as 0.0145.[^likharev-meff] The sim reports the same quantity at the bottom of its band 1, and the comparison is instructive rather than flattering: the one-dimensional repulsive comb gives 1.019 electron masses at μ = 1 and 1.275 at μ = 5, rising with lattice strength and never falling below one.[^derived-ebs] The toy model produces bands and gaps faithfully but not light carriers, and the 0.26 figure is on the readout as a scale check, not as a target the model can hit. In a real diagram the further things to read are where the valence maximum and conduction minimum sit relative to one another — the [[Direct_and_indirect_band_gaps|direct or indirect]] distinction — and how many bands are degenerate at a symmetry point, which symmetry alone fixes. ## See also - [[Bloch's_theorem]] — the periodicity theorem the whole subject rests on - [[Particle_in_a_one-dimensional_lattice]] — the sim's own model, treated as its own pair - [[Free_electron_model]] — the μ → 0 limit of the dial - [[Fermi_level]] — where the filling stops - [[Semiconductor]] - [[Band_gap]] — the width of the forbidden range - [[Doping_(semiconductor)]] — filling the gap with carriers - [[Density_functional_theory]] ## References [^likharev-comb]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part QM: Quantum Mechanics*. Chapter 2 (pp. 31–106), the Dirac comb: the master dispersion relation `cos(q*a) = cos(k*a) + (mu/(k*a))*sin(k*a)` with `mu = m*a*W/hbar^2` (pp. 72–73, 83–84). https://open.umn.edu/opentextbooks/textbooks/part-qm-quantum-mechanics [^likharev-bloch]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part QM: Quantum Mechanics*. Chapter 2, Bloch's theorem `psi(x + a) = psi(x)*exp(i*q*a)` and the 2π/a periodicity of all observables in the quasimomentum (pp. 74–75). https://open.umn.edu/opentextbooks/textbooks/part-qm-quantum-mechanics [^likharev-tb]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part QM: Quantum Mechanics*. Chapter 2, tight binding: `E = E_n + 2*hbar*eta_n*cos(q*a)`, band width 4ħ|η_n|, valid for ħ|η_n| ≪ E_n (pp. 78–79). https://open.umn.edu/opentextbooks/textbooks/part-qm-quantum-mechanics [^likharev-nfe]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part QM: Quantum Mechanics*. Chapter 2, the weak-potential limit: gaps of width Δ_n = 2|U_n| at q = πm/a, from `E = E_ave ± sqrt(((E_l − E_l')/2)^2 + |U_n|^2)` (pp. 81–82). https://open.umn.edu/opentextbooks/textbooks/part-qm-quantum-mechanics [^likharev-dos]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part QM: Quantum Mechanics*. Chapter 2, the one-dimensional density of states dN = (L/2π)dq and its divergence at band edges (p. 86). https://open.umn.edu/opentextbooks/textbooks/part-qm-quantum-mechanics [^likharev-meff]: Likharev, Konstantin (2013). *Essential Graduate Physics, Part QM: Quantum Mechanics*. Chapter 2, effective mass `1/m_ef = (1/hbar^2)*d2E/dq2`, negative at a band top; silicon 0.26 mₑ (conduction) and 0.39 mₑ (valence), InSb down to 0.0145 mₑ (pp. 87–89). https://open.umn.edu/opentextbooks/textbooks/part-qm-quantum-mechanics [^openstax-v3-ch9]: Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 3* (OpenStax). Chapter 9, "Condensed Matter Physics" (pp. 393–440), §9.5 Band Theory of Solids and §9.6 Semiconductors and Doping: the broadening of atomic levels into bands, and the filled-band/empty-band classification of metals, semiconductors and insulators (page to pin). https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-3 [^manual04-sign]: Wikitube MICROSIM_GUIDE sub-manual 04, *Atomic, Quantum, Statistical and Electromagnetic Physics*, §3.5 "Energy bands of a Dirac comb" and Appendix A: the text extraction of [^likharev-comb] prints the dispersion relation with a minus sign, while a direct derivation for a repulsive barrier gives the plus sign; the stated test is that the right-hand side tends to 1 + μ as ka → 0, so a repulsive comb must have a gap at the bottom of the spectrum. [^bloch1929]: Bloch, Felix (1929). "Über die Quantenmechanik der Elektronen in Kristallgittern." *Zeitschrift für Physik* 52. Pages and DOI to pin. [^kronig-penney]: Kronig, Ralph de Laer; Penney, William G. (1931). "Quantum Mechanics of Electrons in Crystal Lattices." *Proceedings of the Royal Society of London A* 130. Pages and DOI to pin. The model is also called the Dirac comb when the barriers are taken as delta functions, as in [^likharev-comb] and in this page's sim. [^spec-m37]: Matter & Energy Cluster contract, `_registry/plans/MATERIALS_SCIENCE_SECTIONS.md` row M37: new root sim (`wt-solid.transport`, sub-manual 04 §3.5 on the framework), the Dirac-comb dispersion sampled in k at 4,096 points per change and never inverted, bands 1–4 drawn in the reduced zone with the gaps shaded, the lattice strength μ (0–30) as first control and electrons per cell as the second, and readouts of band widths, gaps and m_eff at the band-1 bottom with silicon's 0.26 mₑ as the sanity value. [^derived-ebs]: Computed for this article by sampling the dispersion relation of [^likharev-comb] on 4×10⁶ points over ka ∈ (0, 4π] and taking the allowed set |RHS| ≤ 1, in units of ħ²/(2ma²). Band-1 bottoms: ka = 1.3065 (μ = 1), 2.2845 (μ = 5), 2.6277 (μ = 10), 2.9458 (μ = 30); every band top at ka = nπ. At μ = 5, band 1 is 4.651 wide and the first gap 12.800; at μ = 30, band 1 is 1.192 and the gap 24.883; band 2 begins at ka = 4.7613 for μ = 5. These reproduce the derived values recorded in sub-manual 04 §3.5. With a = 0.5 nm, ħ²/(2ma²) = 0.1524 eV, giving 0.709 eV and 1.951 eV at μ = 5 and 0.182 eV and 3.792 eV at μ = 30. Effective mass at the band-1 bottom, from m_ef/m = −R′(ka₀)/ka₀ with R the right-hand side: 1.0186 (μ = 1), 1.2751 (μ = 5), 3.6853 (μ = 30), approaching 1 as μ → 0 and never falling below it. ## Further reading - Likharev, Konstantin (2013). *Essential Graduate Physics, Part QM: Quantum Mechanics*, Chapter 2 — the Dirac comb, Bloch's theorem, tight binding and effective mass in closed form. https://open.umn.edu/opentextbooks/textbooks/part-qm-quantum-mechanics - Sanny, Jeff; Ling, Samuel (2016). *University Physics Volume 3*, Chapter 9 — band theory at first-course level, with the metal/semiconductor/insulator classification. https://open.umn.edu/opentextbooks/textbooks/university-physics-volume-3 <!-- MATTERSIM:BEGIN g33 — Matter & Energy Cluster microsim (framework build, specs/sims/Electronic_band_structure.json); do not hand-edit inside --> **Microsim — three.js (Wikitube framework):** *Electronic band structure* <div class="wt-sim" data-src="https://wikitube-3d-microsims.netlify.app/matter/Electronic_band_structure.html" data-title="Electronic band structure"></div> *Built from `MICROSIM_GUIDE/specs/sims/Electronic_band_structure.json`; part of the [[PORTAL_Matter|Matter portal]] spine (section sims and See-also variants).* <!-- MATTERSIM:END --> ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Electronic_band_structure) : [Wikitube](https://en.wikitube.io/wiki/Electronic_band_structure) · pinned revision [1316527141](https://en.wikipedia.org/w/index.php?oldid=1316527141) · 2026-09-11 ## Previous hub tags Hubs: `Life_Physics`. Portals: [[PORTAL_Materials_science]], [[PORTAL_Physics]]. --- *Matter & Energy Cluster child articles, wave 1 · 2026-09-11 · drafted · Materials_science row M37 · sim pending (matter/Electronic_band_structure).*