How can I determine the minimum of the conduction band of a semiconductor using DFT/VASP, or any other option?
Firstly due to the bandgap problem in density functional theory, the bandgap is underestimated (difference between valence band maximum and conduction band minimum). However, for semi-conductors, hybrid functionals such as HSE06 do a good job of predicting the bandgap. However, such calculations would be computationally expensive. To find the conduction band minimum you can run an HSE06 calculation with reasonable Brillion zone sampling. You have to run some convergence tests to determine the sampling since it is the system and supercell size-dependent. Once you run the calculation you can find the density of states (DOSCAR in case you are using VASP). From here you can plot the density of states and find both the valence band maxima and conduction band minima.
Bonus tip: If you are using VASP. You can use python toolkits like Pymatgen to help you with a lot of analysis techniques.