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The wavelength- or frequency-dependent absorption of a specified specimen. Composition, optical path length, concentration, linewidths and illumination determine its transmitted colour; a single band gap does not specify that colour.

An allowed energy band is an interval of energies supporting extended Bloch states. Adjacent bands may be separated by a band gap.

A semiconductor solid solution whose composition changes its band gap.

For $V(x)=\alpha {\sum }_j\delta (x-ja)$ with $\alpha >0$, put $P=m\alpha a/{\hslash }^2$ and $x=ka$. The Floquet discriminant of the delta-comb Kronig-Penney model is $f(x)=\cos x+P\sin x/x$. Extended Bloch states require $|f(x)|\le 1$. Since $f(0)=1+P>1$, there is a positive lower band edge; a repulsive potential does not support negative-energy states in this energy convention.

The $j$th positive-energy band ends at $x=j\pi$. Its lower edge lies in $((j-1)\pi ,j\pi )$ and solves $f(x)=(-1{)}^{j-1}$. A finite periodic ring selects only the discrete allowed values of crystal momentum within these bands. Nonzero band gaps do not by themselves imply an insulator: the band filling must also place the Fermi energy in a gap.

For fixed band effective masses and band gap $E_g$, the intrinsic carrier concentration obeys $n_i=C{T}^{3/2}{e}^{-E_g/(2k_{B}T)}$. ThusA base-ten plot against $x=1000\mathrm{K}/T$ instead has slope $-E_g/(2000k_B\mathrm{K}\ln 10)$ after the prefactor correction. The intrinsic branch must be used, rather than the extrinsic plateau or dopant freeze-out branch. A changing gap or effective mass makes this an activation-energy estimate, and a small gap uncertainty can produce a much larger multiplicative uncertainty in a room-temperature minority density.

Band hybridization mixes two orbital-derived bands through an off-diagonal Hamiltonian matrix element. For a real coupling $\alpha$,An allowed bare crossing becomes an avoided crossing with direct separation $2|\alpha |$. Orbital character transfers continuously between branches through the crossing. A direct separation alone does not guarantee a global band gap; the extrema at different wavevectors must also be compared.

A band insulator has an integer number of completely filled energy bands, with the Fermi energy inside a positive band gap. At zero temperature, no arbitrarily low-energy charged excitation is then available.

A band offset is an interface discontinuity in a conduction band or valence band edge.^[ref]

For a chosen two-material example, let the central material A have $E_{c,A}=1$ and $E_{v,A}=0$, and the surrounding material B have $E_{c,B}=1.6$ and $E_{v,B}=-0.4$, in stipulated energy units. Define positive confinement barriers by $\Delta E_c=E_{c,B}-E_{c,A}$ and $\Delta E_v=E_{v,A}-E_{v,B}$. Then $\Delta E_c=0.6$, $\Delta E_v=0.4$, and the band gaps obeyThis identity determines the sum of the band offsets; a band gap difference alone cannot determine their partition. The chosen numbers illustrate the definition and do not specify a measured material pair.

Blue transitions motivate wide-bandgap semiconductors, but a large band gap alone cannot guarantee efficient emission. Gallium nitride has a direct band gap; indium gallium nitride composition tunes the transition energy.^[ref]

For an emitted photon, with frequency $\nu$ and vacuum wavelength $\lambda$,where $h$ is the Planck constant and $c$ the vacuum speed of light. Multiplying energy by wavelength gives $hc=1239.841984\mathrm{eV}\mathrm{nm}$. Thus chosen wavelengths $450\mathrm{nm}$ and $630\mathrm{nm}$ correspond to $2.75520\mathrm{eV}$ and $1.96800\mathrm{eV}$: blue requires more energy per photon than red. A stipulated transition energy $3.40\mathrm{eV}$ instead gives $364.659\mathrm{nm}$, in the ultraviolet. These are conversions, not measured emission lines.

For near-edge emission, $E_{\gamma }\approx E_g$ estimates the photon energy from the emitting material's band gap. Quantum well confinement or impurity transitions can change it. The 1994 blue light-emitting diode used zinc-doped indium gallium nitride, emitting below its near-edge transition.^[ref]

The measured forward terminal electric potential difference $V_f$ is also not fixed by $E_g/q$, where $q$ is the elementary charge. It depends on electric current, temperature, contacts and series electrical resistance. Even the simple Shockley diode equation with series electrical resistance has $V_f=V_j+I{R}_s$, whereas an optical transition does not acquire an extra energy $qI{R}_s$ merely because the contact drop increases. Consequently, extracting a band gap from one terminal-voltage reading is not justified.

A Bragg point is a crystal momentum at which two plane waves differing by a reciprocal lattice vector are degenerate. A periodic potential mixes the waves there, producing an avoided crossing and opening a band gap.

A Chern insulator is a two-dimensional band insulator whose occupied energy bands have a nonzero total Chern number. It has a bulk band gap, yet an interface with a topologically trivial band insulator carries a net excess of quantum Hall edge states propagating in one direction. Time-reversal symmetry must be broken. The Haldane model shows that a nonzero average magnetic flux is unnecessary.

Why distinguish two solids that both have a band gap? The energy eigenvalues tell us whether an excitation costs energy; the energy eigenstates also describe how the state changes around the Brillouin zone. Their Berry curvature can integrate to a nonzero integer. A small deformation changes the Berry curvature continuously, but cannot continuously change an integer. A Chern insulator and a trivial band insulator therefore cannot be connected through a path of isolated occupied energy bands with a common bulk band gap. The distinction survives sufficiently weak local perturbations. It does not make every microscopic quantity or every boundary state immune to every disturbance.

For a single filled energy band, set the Berry connection to ${\mathcal{A}}_j=i⟨u_{-}|{\partial }_{k_j}{u}_{-}⟩$, and orient the momentum plane by $(k_x,k_y)$. ThenThis is the First Chern class evaluated on the Brillouin zone, a torus. Smooth eigenvectors may require multiple patches; their transition phases have integer winding numbers. For charge $-e$ and conventional electric current $j_i={\sigma }_{ij}{E}_j$, these conventions give ${\sigma }_{yx}=C{e}^2/h$ and ${\sigma }_{xy}=-C{e}^2/h$. The measurable magnitude agrees with Chern number and integer Hall conductivity. Reversing the momentum orientation or the sign of the Berry connection reverses the displayed $C$.

The initially empty energy band above the band gap in the simple semiconductor description. Excited electrons in it can carry electric current; they can also bind to valence band electron holes as excitons.

A nanocrystal with one semiconducting composition surrounded by another. band offsets, interface quality and shell thickness govern carrier confinement and coupling to external traps. type-I band alignment favors electron and hole localization together; type-II band alignment can separate them. A larger shell band gap does not by itself determine those offsets.

Thermal generation of electron-hole pairs within a reverse-biased depletion region can supply a leakage electric current in addition to neutral-region diffusion. For a simple midgap trap with equal electron and hole lifetime parameters, its scale is $e\mathcal{A}W{n}_i/(2{\tau }_g)$, where $W$ is depletion width and ${\tau }_g$ the lifetime parameter. The electric field separates generated carriers. Unlike ideal diode minority-carrier diffusion current, this contribution scales approximately with $n_i$ rather than $n_i^2$ and can vary with reverse voltage through $W$. Consequently a measured activation energy need not equal the full band gap.

For spin-degenerate electrons at atomic spacing $a$, alternate the hopping amplitudes $-(t+\delta )$ and $-(t-\delta )$, with $t>0$ and $|\delta |<t$. The doubled primitive unit cell gives an off-diagonal Bloch Hamiltonian entry $-(t+\delta )-(t-\delta )e^{-2ika}$. Its two eigenvalues are the displayed energy bands, for $|k|\le \pi /(2a)$. The band gap at the edge of the Brillouin zone is $4|\delta |$. At half filling there are two electrons per doubled primitive unit cell, so the lower band is full and the upper empty. This explicit model realizes the period-doubling form of a Peierls distortion.

A band gap whose conduction band minimum and valence band maximum occur at the same crystal momentum, allowing an optical transition without a momentum-balancing phonon.

For a vertical optical transition between parabolic energy bands with common carrier temperature, net stimulated emission has occupation factor $f_c(E_c)-f_v(E_v)$. The monotonic Fermi-Dirac distribution makes this positive exactly when the quasi-Fermi-level splitting exceeds the transition photon energy. An allowed transition must also lie above the band gap. Broadening, band degeneracy and a zero-temperature occupation window require their own qualifications; this condition alone does not establish laser threshold.

At sufficiently low temperature, donor or acceptor ionization becomes energetically unfavourable. Carriers remain bound to dopants rather than contributing to conduction, reducing the mobile carrier density. This dopant freeze-out is distinct from the high-temperature intrinsic regime, where thermal excitation across the band gap overwhelms the net dopant contribution.

An electron-hole pair consists of an excited electron and the unoccupied valence state described as an electron hole. Promoting a valence electron across a semiconductor band gap creates such a pair without changing the total electric charge. The carriers can separate under an electric field, or undergo electron-hole recombination. A bound electron-hole pair is an exciton; an arbitrary unbound pair need not be an exciton.

The range of energy within one electronic energy band, obtained by varying its wavevector. This differs from an interband band gap.