A possible mechanism of the lifetime effects of electron pairs at the $SrTiO_3/LaTiO_3$ interface

The lifetime effects of the electron pairs are supposed to be responsible for the atypical behavior of the single electron transistors manufactured at the strontium titanate/ lanthanum aluminate (STO/LAO) interface. In addition, as shown in Richter's experiments, the energy gap $\Delta$ should persist above the superconducting transition temperature $T_c$. In order to explain the experiments, the present paper attempts to associate the phonon-electron interactions to the mechanism of the formation and decay of the electron pairs. Moreover, through employing the boson fermion model that includes two superconducting mechanisms, we reproduce the step-like $\Delta-T_c$ relations with the energy gap $\Delta$ remaining finite above $T_c$. The results are roughly consistent with the experiments. The newly-introduced models and formalism may be helpful in describing superconductors with lifetime effects and a finite energy gap $\Delta$ above $T_c$.