The zener diode is heavilier doped than a normal diode , that is why its depletion region is smaller than the delpetion region of a normal diode and that is why the electric field is very strong.Zener diodes are able to undergo Zener breakdown by pulling charge carriers out of pairs of electrons(chemical bond between atoms).Why don't normal normal diodes do the same thing?Energy is required for a bond to be broken.Work is the transformation of energy from one kind to another and the transfer of energy from one object to another . Because the potentials of a zener diode and a normal diode are the same , the electric field of those two can produce the same work on an electron.
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The reason for this is quantum tunneling . Because the depetion region is smaller electrons can "borrow" energy and overcome the barrier potential of the chemical bond.