Question

Why are elemental dopants for Silicon or Germanium usually chosen from group $XIII$ or group $XV$?

Answer 1

When a pure semiconductor is mixed with a small amount of impurities i.e., elements having number of valence electrons slightly different, the concentration of charge carries changes.

The process of addition of impurity is called doping. The size of the dopant atom should be equivalent to the size of $Si$ or $Ge$. So that the symmetry of pure $Si$ or $Ge$ does not disturb, and dopants can contribute the charge carrier on forming covalent bonds with $Si$ or Germanium atoms.

As the silicon and germanium belongs to $XIV$ group so similar size of atom will be $XIII$ and $XV$ group of modern periodic table.