optical bandgap is not real bandgap while electrical is the real bandgap
Hello Monika. I agree that the wikipedia article entitled "Bandgap" is not very clear.
To understand the difference, you need access to an E-k diagram showing the dispersion relation of the energy of the allowed electronic states as a function of the momentum k of the electron in the first Brillouin zone. Easy to find in a dedicated textbook or on the internet.
The quantity k is the wave number of the wave associated with the particle. Different curves denote different energy bands.
To simplify things, let us consider semiconductors.
The electrical bandgap is the energy between the highest maximum of the highest fully occupied band (valence band = VB), generally denoted E_v and the lowest minimum of the lowest empty band (conduction band =CB), generally denoted E_c.
• If the CB minimum and VB maximum occur for the same value of k (in this case generally at k=0), then the energy difference E_g=E_c-E_v can be accommodated by the absorption (for the creation of an e-h pair) or emission (for the recombination of e-h pairs) of a photon. No phonon is required in these transitions. The semiconductor is called a direct bandgap semiconductor (like e.g. GaAs and various III-V materials used in optoelectronic devices), with same optical and electrical bandgap.
• However, if the CB minimum and the VB maximum don't occur for the same k value, a photon with a higher energy will be required to send an electron from the VB to the BC at quasi-constant k, and then one or several phonons (quantum of lattice vibration) with high k and low ∆E will be required to dissipate the excess energy of the "excited" electron and let it reach the CB minimum.
In this case, the optical bandgap is the lowest photon energy that will be able to create an electron-hole pair. This will generally occur for a non-zero momentum value, that depends on the band structure of the considered semiconductor.
With this in mind, it is clear that the optical bandgap of an indirect bandgap semiconductor is larger than its electrical bandgap.
Of course, both optical and electrical bandgaps are real well-defined quantities!
In all these transitions, both the total energy and the total momentum (of electron + hole + any involved photon or phonons) must be conserved. Photons can have an energy amounting to 1 or several eV, but a very small momentum, whereas phonons have a smaller energy (< a few 10s of meV) but may have a large momentum as compared to the maximum momentum in the 1st Brillouin zone.