The emission wavelength of an LED or laser, and the longest wavelength a photodetector can sense, are roughly set by the band gap of the material. Enter a band gap to get the wavelength, or a wavelength to get the band gap you need.
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Wavelength λ
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Photon energy (= Eg)
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Wavelength (µm)
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Frequency
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Wavenumber
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Spectral region
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The scale is in nm and drawn logarithmically. Visible light spans roughly 380–750 nm, and optical communication commonly uses wavelengths near 1310 nm and 1550 nm. The colored strip above shows the approximate color for visible wavelengths.
Calculated as λ [nm] = hc/(q·Eg[eV]) ≈ 1239.84/Eg. Preset band gaps are representative material values at 300 K. Actual emission wavelengths shift by a few nm to a few tens of nm because of spectral width, temperature, alloy composition, quantum-well confinement and so on. Indirect-gap materials emit light weakly and are not normally used for LEDs.
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The conversion λ ≈ 1240/Eg
The energy of a single photon is E = hν = hc/λ. Inserting Planck's constant h = 6.626×10⁻³⁴ J·s, the speed of light c = 2.998×10⁸ m/s and the elementary charge q = 1.602×10⁻¹⁹ C, and working in eV and nm, gives E[eV] × λ[nm] ≈ 1239.84, which is easy to remember as λ ≈ 1240/Eg. When an electron at the bottom of the conduction band recombines with a hole at the top of the valence band and emits light, the photon energy is almost equal to the band gap, so this formula gives an estimate of the emission wavelength. Conversely, a photodetector does not absorb light with wavelengths longer than 1240/Eg, which sets its sensitivity limit (the cutoff wavelength).
Assumptions
The photon energy is taken to be equal to the band gap, so only transitions between the bottom of the conduction band and the top of the valence band are considered. The emission peak of a real LED appears at a slightly higher energy, by about the thermal energy of the carriers (of order kT), and has a finite width. As temperature rises, the band gap shrinks and the wavelength shifts longer. In alloys such as InGaN and AlGaAs the band gap depends on composition, and quantum wells add confinement energy, so the emission wavelength differs from product to product even for the same material name.
Worked example
With the default GaAs (Eg = 1.42 eV), λ ≈ 1239.84/1.42 ≈ 873 nm, a near-infrared wavelength used in LEDs and similar devices. Si (1.12 eV) gives about 1107 nm, which explains why silicon photodetectors are almost insensitive beyond about 1.1 µm. GaN (3.4 eV) gives about 365 nm in the ultraviolet; blue LEDs add In to form InGaN, lowering the band gap to about 2.7 eV to emit blue light near 460 nm. Conversely, the band gap corresponding to the 1550 nm optical-communication wavelength is about 0.80 eV, which is provided by alloys such as InGaAsP grown on InP.
Values used and their sources
The preset band gaps are representative values at 300 K found in semiconductor textbooks and materials handbooks (Si 1.12, Ge 0.66, GaAs 1.42, InP 1.34, In₀.₅₃Ga₀.₄₇As 0.75, GaP 2.26, 4H-SiC 3.26, GaN 3.4, ZnO 3.37, diamond 5.47, AlN 6.2 eV). Values vary between references by about the second decimal place. The direct/indirect classification is also the standard one. The conversion constant 1239.84 eV·nm is derived from the physical constants h, c and q.
Photon energy is E = hc/λ, or E ≈ 1240/λ with E in eV and λ in nm. For λ = 620 nm (red), E ≈ 2.0 eV. The same conversion quickly shows that silicon's absorption edge (1.12 eV) is about 1.1 µm and GaN's (3.4 eV) about 365 nm in the ultraviolet. Since an LED's color is set mainly by its band gap, red LEDs use materials such as AlGaInP with about 1.9–2.0 eV.
Wavelength λ(nm) ≈ 1240/Eg(eV) gives 1240/1.42 ≈ 873 nm—invisible near-infrared light. GaAs LEDs are used in TV remote controls and photoelectric sensors, and GaAs/AlGaAs VCSELs near 850 nm serve smartphone face authentication and short-reach optical links. Emitting visible light (about 400–700 nm) requires wider-gap materials such as GaAsP or AlGaInP.
CThe band gap of the semiconductor used in the light-emitting layer
Show answer and explanation
Answer: C. The band gap of the semiconductor used in the light-emitting layer
An LED emits light with energy near the band gap when electrons and holes recombine; wavelength λ(nm) ≈ 1240/Eg(eV). InGaN at about 2.7 eV emits blue at about 460 nm, and AlGaInP at about 1.9 eV emits red at about 650 nm. Adjusting the active-layer composition tunes the band gap—more indium in InGaN shifts it to green. White LEDs combine a blue LED with a yellow phosphor.
FAQ
Where does the number 1240 come from?
It is hc/q = 1.23984×10⁻⁶ eV·m, Planck's constant times the speed of light divided by the elementary charge, converted to nm. It expresses the fact that photon energy (eV) times wavelength (nm) is always about 1240.
Why doesn't silicon emit light?
Silicon has an indirect band gap: the electron momentum differs between the bottom of the conduction band and the top of the valence band, so emitting light requires help from a phonon (a lattice vibration). That process is unlikely, and non-radiative recombination that produces heat happens first, so light emission is very weak. Absorption can also proceed with phonon assistance, so silicon still works well for photodetectors and image sensors.
Can I calculate the wavelength of a white LED?
Most white LEDs use light from a blue LED (about 450–460 nm) to excite a yellow phosphor, and the mix looks white. This formula covers the blue LED part; the phosphor emission works by a different mechanism.