Confused on relation between bandgaps and pn-junction operation

The band gap energy is related to the maximum built-in voltage corresponding to the maximum band bending in an abrupt pn-junction: $$V_{bi}≲ frac{E_g}{q}$$ This can be easily seen by drawing a band diagram in equilibrium. In the n-type region, the Fermi level is close to the conduction band edge and in the p-type region the Fermi level it is close to the valence band edge. The higher the doping, the closer the Fermi level comes to the respective band edge. The (ideal) saturation current of a diode decreases exponentially with the band gap energy $$J_s∝n_i^2∝exp{(-frac{E_g}{kT})}$$ For the dominant generation current in a Si diode it decreases as $$J_s∝n_i∝exp{(-frac{E_g}{2kT})}$$ thus the band gap energy determines the magnitude of the pn-junction current. Under forward bias the current increases exponentially because the applied voltage decreases the built-in potential barrier $V_{bi}$.