A band gap is the range of energies in a solid where no electron states exist: the gap between the top of the filled valence band and the bottom of the empty conduction band (Wikipedia: Band gap). Its size, measured in electronvolts (eV), largely decides whether a material behaves as a metal, a semiconductor or an insulator, which colors of light it absorbs or emits, and how it can be used in electronics. Measured gaps come from optical or photoemission experiments; computed gaps in large databases usually come from density functional theory (DFT), which systematically underestimates them.

• Metals have no gap; semiconductors have small to moderate gaps (silicon: 1.12 eV); insulators typically have gaps above about 4 eV (Ioffe Institute; Wikipedia).
• In a direct gap material such as GaAs, electrons can emit light efficiently; in an indirect gap material such as silicon, a lattice vibration (phonon) must take part (Wikipedia: Direct and indirect band gaps).
• Gaps are measured mainly by optical absorption, often analyzed with a Tauc plot, and by photoemission spectroscopy.
• DFT with the common PBE functional underestimates gaps; the Materials Project reports an average factor of about 1.6 (Materials Project docs).
• Strongly correlated oxides such as CuO can even come out metallic in standard DFT, although the real material is a semiconductor (Ekuma et al. 2014).

What is a band gap, in plain terms?

Electrons in an isolated atom occupy discrete energy levels. When many atoms bond into a crystal, those levels broaden into continuous bands. In many solids, the band formed from bonding states (the valence band) is completely full, and the next band up (the conduction band) is empty, with an energy range between them where no electron states exist. That range is the band gap.

A full band cannot carry net current, because every state is occupied and electrons have nowhere to move. To conduct, an electron must be promoted across the gap into the conduction band, leaving behind a vacancy (a "hole") in the valence band that also carries current. The energy for that promotion can come from heat, light or an applied field. The bigger the gap, the harder this is.

The gap is not fixed. It generally shrinks as temperature rises, a trend often described by Varshni's empirical relation (Wikipedia). For silicon, the Ioffe Institute's semiconductor database gives 1.12 eV at 300 K and a temperature formula that extrapolates to about 1.17 eV at absolute zero (Ioffe Institute). Reported values should therefore always state their temperature.

How does the band gap separate metals, semiconductors and insulators?

In metals, the bands overlap or a band is only partly filled, so there is no gap and electrons move freely. Semiconductors have gaps small enough that some electrons are thermally excited at room temperature and that doping can control conductivity precisely. Insulators have large gaps, typically above about 4 eV (Wikipedia), so almost no carriers are excited. The boundaries are conventions rather than physical laws: wide bandgap semiconductors such as GaN and SiC, usually defined as having gaps above about 2 eV (Wikipedia), have very few thermally excited carriers when pure yet become useful semiconductors when doped (see GaN vs SiC).

Sources: Ge and diamond, Wikipedia: Band gap; Si, Ioffe; GaAs, Ioffe; ZnO, Wikipedia: Zinc oxide; GaN, Ioffe; gap types, Wikipedia and the same pages.

The gap also sets optical behavior. A photon can only be absorbed across the gap if its energy exceeds the gap. Silicon's 1.12 eV gap corresponds to near-infrared light, so silicon absorbs visible light and appears opaque, while ZnO's gap of about 3.3 eV lies in the ultraviolet, so ZnO is transparent to visible light. That combination of transparency and conductivity is the basis of transparent conducting oxides.

What is the difference between a direct and an indirect band gap?

Electron states in a crystal are labeled by energy and by crystal momentum. If the lowest point of the conduction band sits at the same crystal momentum as the highest point of the valence band, the gap is direct. If they sit at different momenta, the gap is indirect (Wikipedia).

The difference matters because a photon carries almost no momentum. In a direct gap material, an electron can drop from the conduction band to the valence band and emit a photon in one step. In an indirect gap material, the transition also needs a phonon to supply or absorb the momentum mismatch, which makes both light emission and absorption near the gap much weaker (Wikipedia).

The practical consequences are large:

• Light emitters. LEDs and laser diodes are made from direct gap materials such as GaAs and GaN, not silicon (Wikipedia).
• Solar cells. Crystalline silicon absorbs weakly near its gap, so wafer cells must be hundreds of micrometers thick, whereas thin-film cells made from direct gap absorbers such as CdTe can work at thicknesses below one micrometer (Wikipedia).

How is a band gap measured?

There is no single "band gap measurement"; different experiments probe slightly different quantities.

• Optical absorption. Light of increasing photon energy is shone through a sample, and the absorption coefficient rises sharply near the gap. A Tauc plot, introduced by Jan Tauc in 1966 for amorphous germanium, plots a power of the absorption coefficient against photon energy and extrapolates the straight part to zero (Wikipedia: Tauc plot). For crystalline semiconductors the exponent depends on the transition: absorption squared is plotted for direct allowed transitions and its square root for indirect ones (Wikipedia; Wikipedia). Choosing the wrong exponent is a common source of inconsistent literature values.
• Photoluminescence. Light emitted when excited carriers recombine gives the gap directly in good direct gap emitters, though defect and exciton states can shift the peak.
• Photoemission and inverse photoemission. Photoemission removes electrons to map the occupied states, and inverse photoemission adds electrons to map the empty states. Combined, they give the fundamental (electronic) gap.

The optical gap and the fundamental gap are not always the same. Light can create a bound electron-hole pair, an exciton, at an energy slightly below the fundamental gap. The difference is negligible in most inorganic semiconductors but substantial in organic semiconductors and carbon nanotubes (Wikipedia). ZnO is an intermediate case, with an exciton binding energy of about 60 meV (Wikipedia: Zinc oxide).

Why does DFT underestimate band gaps?

Most computed gaps in public materials databases come from Kohn-Sham DFT using a generalized gradient approximation (GGA), most often the Perdew-Burke-Ernzerhof (PBE) functional. These calculations are good at total energies and structures, but the gap between Kohn-Sham eigenvalues is not the true fundamental gap. A key missing piece is the "derivative discontinuity" of the exchange-correlation energy, which is exactly zero in local and semilocal functionals for solids; this is often given as the reason for their systematic underestimation of gaps (Borlido et al., npj Comput. Mater. 2020).

The size of the error is well documented:

• The Materials Project, comparing 237 compounds with experiment, found computed gaps underestimated by an average factor of 1.6, with a residual mean absolute error of 0.6 eV even after correcting for that shift (Materials Project documentation).
• A benchmark of 473 materials found a mean absolute percentage error of 46% for PBE, with a mean percentage error of -41%, meaning gaps are almost always too small (Borlido et al. 2020).
• The same group's benchmarks found that LDA, PBE and PBEsol each predict roughly 30 to 35 "false metals," materials with a real gap that come out gapless (Borlido et al. 2020).

The ordering of gaps across materials is usually more reliable than their absolute values, which is why PBE gaps remain useful for screening.

How big is the error for real materials?

Comparing measured gaps with the representative computed (PBE) gaps shown on LatticeGraph compound pages at the time of writing makes the pattern concrete. The computed values originate from Materials Project calculations.

The errors are not a constant offset. Silicon and GaN are underestimated by roughly half, while GaAs and ZnO are off by a much larger fraction. A simple scissors shift or a single multiplicative factor cannot fix every compound.

CuO is the instructive failure. Experimentally it is a p-type semiconductor with a reported gap of 1.0 to 1.9 eV, depending on the measurement and sample, yet standard DFT with local exchange-correlation functionals generally predicts a nonmagnetic metal (Ekuma et al., arXiv:1305.6283). The problem is not just the missing derivative discontinuity but the poor treatment of strongly interacting, localized copper 3d electrons. Adding an on-site Coulomb correction (DFT+U) opens a gap and recovers an antiferromagnetic insulator in reasonable agreement with experiment (Ekuma et al.). A computed gap of zero for a transition-metal oxide should therefore be treated as a warning sign, not a finding.

How do HSE and GW fix the band gap problem?

Several methods improve on PBE, at increasing cost:

• DFT+U adds a Hubbard-like penalty for partially filled localized d or f shells. It is cheap and helps for correlated oxides such as CuO, but the U value must be chosen.
• Meta-GGA potentials such as the modified Becke-Johnson (mBJ) potential cost more than PBE but far less than hybrids; in one large benchmark, mBJ runs were 10 to 50 times slower than LDA/GGA runs (Borlido et al. 2020).
• Hybrid functionals such as HSE06 mix in a fraction of exact exchange. The 2020 benchmark confirmed mBJ, the HLE16 GGA and HSE06 as the most accurate functionals for band gaps, but hybrids were about two orders of magnitude slower than LDA or GGA calculations (Borlido et al. 2020).
• GW many-body perturbation theory, introduced by Hedin in 1965, computes quasiparticle energies directly and is the usual reference for accurate gaps, at a much higher cost still.

Because hybrid and GW calculations are expensive, high-throughput databases mostly report PBE gaps, sometimes alongside a smaller set of higher-level values.

How should you read computed band gaps in a materials database?

• Check the method. A gap labeled PBE or GGA is a lower-bound estimate for most semiconductors and insulators, not a measurement.
• Use it for ranking and screening. Comparing materials computed with the same functional is more reliable than comparing a computed value with an experimental one.
• Be wary of zero gaps in d- and f-electron compounds. A metallic PBE result for a transition-metal oxide may be a false metal, as with CuO.
• Look across polymorphs and databases. Different structures of the same formula can have very different gaps, and databases differ in settings and corrections (see why DFT databases disagree and how the major databases compare).
• Find an experimental value before making design decisions, and note its temperature and measurement method.

Frequently asked questions

What is a band gap in simple terms?

It is the energy an electron needs to jump from the filled valence band into the empty conduction band, where it can carry current. Metals have no gap, semiconductors have a small one, and insulators have a large one.

Is silicon a direct or indirect band gap semiconductor?

Silicon has an indirect gap of about 1.12 eV at room temperature (Ioffe). That is why it absorbs light weakly near the gap and is a poor light emitter.

Why are DFT band gaps too small?

Semilocal functionals such as PBE lack the derivative discontinuity of the exact functional and suffer from self-interaction error, so Kohn-Sham gaps fall short of true fundamental gaps. Benchmarks show PBE underestimates gaps by roughly 40% on average (Borlido et al. 2020; Materials Project).

What is the most accurate way to calculate a band gap?

GW calculations are the usual high-accuracy reference. Among DFT approaches, the mBJ potential, the HLE16 GGA and the HSE06 hybrid performed best in a large benchmark (Borlido et al. 2020).

What does a Tauc plot measure?

It estimates the optical band gap by plotting a power of the absorption coefficient against photon energy and extrapolating the linear region to zero absorption. The exponent depends on whether the transition is direct or indirect (Wikipedia).

Explore computed band gaps on LatticeGraph

Compound pages such as Si, GaAs, ZnO and CuO show computed (mostly PBE) gaps across polymorphs, alongside structures and stability. The wide bandgap oxides class page helps with screening; check experimental values before relying on absolute numbers.