Shapiro steps in Josephson junctions with alternating critical current density

We treat theoretically Shapiro steps in tunnel Josephson junctions with spatially alternating critical current density. Explicit analytical formulas for the width of the first integer (normal) and half-integer (anomalous) Shapiro steps are derived for short junctions. We develop a coarse-graining approach, which describes Shapiro steps in the voltage-current curves of the asymmetric grain boundaries in yttrium barium copper oxide thin films and different superconductor-ferromagnet-superconductor Josephson-type heterostructures.