Note that the energy bandgap primarily relates to the electronic properties of materials. Solubility typically pertains to the ability of a substance to dissolve in a solvent. The two effects are not directly linked. Solubility is determined by intermolecular interactions, temperature, pressure, and solute/solvent nature.
However, we can observe certain indirect relationships. Doping modifies the electrical properties. Thus, polarity, ionization, complexation, and electrostatic interactions are changed.
It is necessary to consider the specific material and solvent involved to understand the solubility behavior accurately