Proximity effect on the superconducting transition temperature of ${\mathrm{YBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{7\mathrm{-}\mathit{x}}$/${\mathrm{PrBa}}_2$${\mathrm{Cu}}_3$${\mathrm{O}}_{7\mathrm{-}\mathit{x}}$ superlattices

The superconducting transition temperature ${\mathit{T}}_{\mathit{c}}$ of ${\mathrm{YBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_{7\mathrm{-}\mathit{x}}$/${\mathrm{PrBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_{7\mathrm{-}\mathit{x}}$ superlattices is studied theoretically with use of the Gor’kov equation. We show that the proximity effect in which the pair amplitude of the superconducting ${\mathrm{YBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_{7\mathrm{-}\mathit{x}}$ layer leaks to the semiconducting ${\mathrm{PrBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_{7\mathrm{-}\mathit{x}}$ layer plays a dominant role in explaining the depression of ${\mathit{T}}_{\mathit{c}}$ in these superlattices with a minor modification from the charge-carrier-transfer effect across the Y and Pr interface. By adjusting the value of the hopping matrix between the Cu-O plane, our numerical result can be quantitatively compared with recent experimental measurement of ${\mathit{T}}_{\mathit{c}}$ in ${\mathrm{YBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_{7\mathrm{-}\mathit{x}}$/${\mathrm{PrBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_{7\mathrm{-}\mathit{x}}$ superlattices.