Why is the size dependence of the band gap similar in MX₂ ( M = Mo , W and X=S,Se )?

The size dependence of the electronic structure of semiconductor nanocrystals have been investigated for many years, with band gap changes being seen only upto 20 Å in some instances, while it is upto 200 Å in other instances. This variation has been found to be associated with the excitonic Bohr radius of the semiconductor involved. In contrast, when one considers van der Waals materials like the transition metal dichalcogenides, one finds that the number of layers upto which band gap variations are found in Mo and W-based based transition metal dichalcogenides MX₂ (where M=Mo and W; X=S and Se) are very similar. We have mapped the electronic structure of these materials onto a tight binding model and derived a low energy effective hamiltonian by integrating out the high energy bands. We find that the model derived for a trilayer is able to capture the electronic structure upto 10-layers and beyond as revealed by a comparison with the ab-initio results. Within this model, the origin of the similar size dependence across materials has been addressed.