Barrier potential of a p-n junction diode does not depend on -

The barrier potential is set up by the immobile ions in the depletion region, formed by diffusion of carriers across the junction.
It depends on the semiconductor material (Si about 0.7 V, Ge about 0.3 V), the doping density (more doping gives a higher barrier) and the temperature (it decreases about 2 mV per degree rise).
It is changed by the applied bias: forward bias lowers the effective barrier and reverse bias raises it.
It does not depend on the design of the diode, i.e. its geometrical shape or construction.
So the barrier potential does not depend on diode design.