The analysis of the Band Structure of the compound Bi2S3 ( from my memoir of Magister)

The set of curves En (k) constitutes the dispersion curves of the electrons in the crystal: also called band structure.

In the same way that it suffices to know V (r) on the unit cell of the crystal, the curves En(k) have the same symmetry as the reciprocal space and it suffices to describe En(k) in the first Brillouin zone.

The band structure represents all the energy states accessible to electrons. By filling the states with all the electrons of the crystal, we arrive at a configuration where at T = 0 K, the states of the conduction band are not populated; on the other hand, the states of the valence band are populated by electrons.

The energy bands give the possible energies of an electron as a function of the wave vector. These bands are represented in reciprocal space, and for simplicity, only the directions of higher symmetries in the first Brillouin zone are treated. The following figure represents the first Brillouin zone of Bi2S3.

The valence bands and the conduction bands are separated by a prohibited band or gap. The energy gap is defined as the difference between the maximum of the valence band MBV and the minimum of the conduction band MBC.

The band structure of Bismuth Sulfide is shown in the following figure.

The energy bands illustrate a layered crystal structure, and in the region of the valence band, there is a substantial dispersion of bands in the branches ГS and ГY. There is also a small dispersion in the SR direction which shows a weak interaction between the layers. The bands along ГZ and ГX show localized electronic states.

Because of the complementarity between the direct space and the reciprocal space, the highly dispersive electronic bands in the reciprocal space correspond to the highly delocalized electronic states in the direct space; while the less dispersive (flat) bands correspond to localized electronic states.

The band structure shows two gaps. One direct with a value of 1.42 eV between the points Гv - Гc and the other indirect with a value of 1.32 eV between the points Zv- Гc. For a practical objective, and particularly for photovoltaic applications, Bismuth Sulfide is considered as a direct gap material since the latter is dominant/

The following table presents the calculated gap value as well as other theoretical and experimental results.

Knowing that the DFT-GGA underestimates the gap by up to 50%, it is estimated that the real value can reach more than 2.5 eV; and since the gap decreases with temperature and with dimensions, we can imagine that it will decrease until the experimental values.

We also note that the determined gap of the band structure is the fundamental gap; while the measured gap is an optical gap which is less than the first but which is close to it in the case of semiconductors.

From the table, we see that the calculated gap is in agreement with the theoretical results available.

Reference: https://aljazairi1970.blogspot.com/2020/03/my-memoir-of-magister-in-french.html