The effect of gap fluctuations on interacting and non-interacting polarization for nano-superconducting grains in electron- and hole-doped cuprates

The behavior of non-interacting and interacting polarization under influence of fluctuations of the superconducting gap with d-wave symmetry and under consideration of the gap dependence on nano- grain size is obtained in terms of the frequency, temperature and the size at zero and finite temperatures for rectangular cuprate nano-superconducting grains. By using Eliashberg equations and applying the relations of the fermionic dispersion for the hole-doped and electron-doped cuprates, we numerically compute the real part of size-dependent polarization for both types of cuprates. We show that the peak of real part of polarization moves to higher frequency by including the additional fluctuating part of gap (or the nano-size effect). Also, we obtain the temperatures for different frequencies, in which the effect of gap fluctuations fades. In the case of size-dependent gap, there is a critical frequency; for frequencies lower (higher) than the critical frequency, the nano-effect weakens (improves) the superconducting state. Moreover, it is concluded that the real part of polarization for hole- doped cuprates in terms of the grain size has more significant amount in comparison with electron-doped ones.