  We have developed the recent investigations on the second-order phase
transition in the holographic superconductor using the probe limit for a
nonlinear Maxwell field strength coupled to a massless scalar field. By
analytic methods, based on the variational Sturm- Liouville minimization
technique, we study the effects of the space-time dimension and the
non-linearity parameter on the critical temperature and the scalar condensation
of the dual operators on the boundary. Further, as a motivated result, we
analytically deduce the DC conductivity in the low and zero temperatures
regime. Especially in the zero temperature limit and in two dimensional toy
model, we thoroughly compute the conductivity analytically. Our work clarifies
more features of the holographic superconductors both in different space
dimensions and on the effect of the non-linearity in Maxwell's strength field.
