Thermally induced dimensional crossover in single-crystal ${\mathrm{Bi}}_2$${\mathrm{Sr}}_2$${\mathrm{CaCu}}_2$${\mathrm{O}}_{\mathit{x}}$

The linear ac magnetic response of three ${\mathrm{Bi}}_2$ ${\mathrm{Sr}}_2$ ${\mathrm{CaCu}}_2$ ${\mathrm{O}}_{\mathit{x}}$ single crystals with various doping states was investigated with a miniature two-coil mutual inductance technique in magnetic fields parallel to the c axis. The dissipation peak appeared right below ${\mathit{T}}_{\mathit{c}}$, and a drop in dissipation was observed at a frequency-independent temperature ${\mathit{T}}_{\mathit{D}}$ together with an increase in shielding signal, upon field cooling the sample. The observed frequency-independent feature can be understood as a consequence of the thermally induced dimensional crossover.