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Author: Ali Kosari Mehr
Bands having irregular shapes are formed in energy-momentum space by overlapping electronic orbits of constituents in a solid. These bands represent the allowed energy levels for electrons in various momentums. The highest band (in terms of energy) occupied by electrons is called valence band, and the lowest band empty is called conduction band. As opposed to metals in which either the two highest bands overlap or the valence band is occupied in part, there is a forbidden energy gap between the valence band and the conduction band in semiconductors/dielectrics – this gap is called bandgap. Precisely, bandgap or electronic bandgap is the distance between the highest part of the valence band and the lowest part of the conduction band in the energy axis. The bandgap of dielectrics is large enough that the valence band is filled with electrons and the conduction band is completely empty while the bandgap of semiconductors is such that some thermal electrons can be lifted from the valence band to the conduction band.
Bandgaps are divided into two categories: direct and indirect. Provided that the lowest part of the conduction band matches the highest part of the valence band in the same momentum, the bandgap is direct. If not, the bandgap is indirect. Indirect bandgaps only weakly interact with light (and photons being quantized light waves) since a transition from the valence band to the conduction band cannot be completed only by photons of the respective energy and since this transition also requires phonons (being quantized sound waves caused by the vibration of constituents in solids) so that the momentum of the electron can also change.
Optical bandgap is the energy threshold at which photons are absorbed in semiconductors. Accordingly, a photon with this energy causes the formation of an excited electron in the conduction band and a hole in the valence band. The resulting electron and hole form bound states owing to Coulomb’s force, as a result of which their energy is lower than an unbound state – for further information, please refer to the concept of excitons. Therefore, the electron-hole recombination in this case releases lower energy than the electronic bandgap in which the energy of the electron-hole bound state has not been included; the difference between the optical bandgap and the electronic bandgap is often in the range between 200 meV and 400 meV.
Ultraviolet-visible spectroscopy is a means whereby a sample is illuminated by photons in the whole gamut of the ultraviolet-visible spectrum, resulting in the parameters absorbance (i.e., -log transmittance) and reflectance being determined in this spectrum. Transmittance is the fraction of incident electromagnetic power that is transmitted through a sample, and reflectance is the fraction of incident electromagnetic power that is reflected at the boundary. In thin films and coatings, the absorption coefficient (α) can be calculated by the following expression from film thickness, transmittance, and reflectance:
α=(1/d)ln[[(1-R2)/(2T)]+[(1-R)4/(4T2)]1/2+R2] (1)
Furthermore, the optical bandgap can be calculated by the following expression:
αhν=A(hν-Egopt)r (2)
where h, v, A, and Egopt are Planck’s constant, frequency, a parameter depending on the transition probability, and optical bandgap, respectively. Moreover, r is a number characterizing the transition process; r=1/2 for allowed direct transitions and r=2 for allowed indirect transitions. For determining the value of r (or the direct/indirect nature of bandgaps), (αhν)1/r is plotted against (hv) by the expression 1 – Tauc plot. Next, one investigates what input values for r in the expression 2 result in a better fit of the plot obtained by the expression 1. By this means, the value of r is determined in the expression 2. Given the value of r, the optical bandgap can be determined at α=0 where the Tauc plot should exhibit linear behavior.
