Josephson plasma resonance in Bi 2 Sr 2 CaCu 2 O 8+y with spatially dependent interlayer phase coherence

We study the Josephson plasma resonance (JPR) in ${\mathrm{Bi}}_{2}{\mathrm{Sr}}_{2}{\mathrm{CaCu}}_{2}{\mathrm{O}}_{8+y}$ (BSCCO) with spatially dependent interlayer phase coherence (IPC). The half-irradiated BSCCO (HI-BSCCO), in which columnar defects are introduced only in a half of the sample, shows several resonance peaks, which are not simple superposition of the peaks in irradiated and pristine parts. JPR in HI-BSCCO changes its character from irradiated to pristine type at a crossover frequency $({\ensuremath{\omega}}_{\mathrm{cr}}).$ We demonstrate that the one-dimensional linearized sine-Gordon equation, which takes into account the spatial dependence of IPC, can reproduce most of the experimental findings including the presence of ${\ensuremath{\omega}}_{\mathrm{cr}}.$