Holographic Superconductors with Logarithmic Nonlinear Electrodynamics in an External Magnetic Field

Based on the matching method, we explore the effects of adding an external magnetic field on the $s$-wave holographic superconductor when the gauge field is in the form of the logarithmic nonlinear source. First, we obtain the critical temperature as well as the condensation operator in the presence of logarithmic nonlinear electrodynamics and understand that they depend on the nonlinear parameter $b$. We show that the critical temperature decreases with increasing $b$, which implies that the nonlinear gauge field makes the condensation harder. Then, we turn on the magnetic field in the bulk and find the critical magnetic field, $B_c$, in terms of the temperature, which also depends on the nonlinear parameter $b$. We observe that for temperature smaller than the critical temperature, $T<T_c$, the critical magnetic field increases with increasing $b$ and goes to zero as $T\rightarrow T_c$, independent of the nonlinear parameter $b$. In the limiting case where $ b\rightarrow0 $, all results restore those of the holographic superconductor with magnetic field in Maxwell theory.