i have some confusion existing between the bandgap and the contact potential!!!

Is there any relationship between the band gap and the contact potential?

(Eg and V0)

i found that vo is given by vo=(kT/q)*ln((Na*Nd)/(ni*ni))

Hence the vo depends on doping concentrations

But ni*ni=Nc*Nv*e^(-(Eg/kT)

hence there is a concentration(intrinsic ni) dependency of Eg

hence vo and Eg are directly proportional ,one will get from the above relation !

is there any mistakes in the above problem!!!!