||Superconductivity above the upper
critical field of Niobium cylinders with as-grown (i.e. rough),
chemically, and electrolytically polished surfaces is investigated.
Using the popular criterion for the surface critical field, i.e. the
onset of a screening in the low-frequency (10 Hz) susceptibility chi,
we find the ratio R between the surface critical field and the upper
critical field to be larger than the classical Ginzburg-Landau based
value of 1.69 and to increase from 1.8 to 2.0 by electropolishing the
surface. This increase is tentatively related to a reduction of the
correlation length near the surface due to impurities released from a
dirty layer to the Nb due the polishing processes. We have also
determined from the ac susceptibility the ac-conductivity, and find
that the surface critical field marks only the appearance of a strongly
increasing Ohmic conductance, accompanied by a rather small screening
component. At a lower field, where the Ohmic conductivity tends to
diverge, we observe a sharp increase of the screening component, i.e.
of a long-range coherent surface superconductivity. The appearance of
phase coherence at this coherent surface critical field is corroborated
by the onset of a (inductively measured) finite critical surface
current. The field and temperature variation of this current can be
well explained in terms of the critical state model by J. Fink (Phys.
Rev. Lett. 14, 853 (1965)). For the smoothest surface, also the
magnitude of the surface critical current is close to the prediction of
this GL-based model. Upon surface roughening, the current density is
reduced, and along with the decrease of the ratio R this indicates a
larger instability of surface superconductivity. Our analyses of the
frequency variation of both conductivity components reveal the validity
of dynamical scaling near the surface coherent field. The critical
exponents suggest that the coherence is achieved by a percolation
between superconducting patches nucleated at the surface critical field.