Polaron effects and charge carrier mobility in high-$T_c$ cuprate superconductors (HTSCs) have been investigated theoretically. The appropriate Boltzmann transport equations under relaxation time approximation were used to calculate the mobility of polaronic charge carriers and bosonic Cooper pairs above and below the pseudogap (PG) temperature $T^\ast$. It is shown that the scattering of polaronic charge carriers and bosonic Cooper pairs at acoustic and optical phonons are responsible for the charge carrier mobility above and below the PG temperature. We show that the energy scales of the binding energies of large polarons and polaronic Cooper pairs can be identified by PG cross-over temperature on the cuprate phase diagram.

The extended Ambegaokar–Baratoff model was applied to calculate the critical current ($I_c$) of a superconductor/insulator/superconductor (S/I/S) Josephson junction with multiple tunnelling channels. We analysed the sensitivity of the critical current, multiplied by normal-state resistance (i.e.,$I_c$$R_n$ product), to the symmetries of the order parameter in the Bardeen-Cooper-Schrieffer (BCS)-type gap formula. In addition, we showed that the temperature dependence of the product $I_c$$R_n$ ($T$),which corresponds to multigap symmetry, exhibitsa shoulder-like feature and has a pronounced kink at $T$ ${\simeq}$10K for the SmFeAsO$_{0.9}$F$_{0.1}$ superconductor.