Investigation of anisotropic upper critical magnetic field in two-band model for the copper-based superconductors $\mathbf {YBa_{2}Cu_{4}O_{8}}$ and $\mathbf {Bi_{2}Sr_{2}CaCu_{2}O_{8}}$

Abstract

This research work focuses on the theoretical investigation of upper critical magnetic field ($H_{C2}$), lower critical magnetic field ($H_{C1}$), Ginzburg-Landau coherence length ($\xi _{GL}$) and Ginzburg-Landau penetration depth ($\lambda _{GL}$) in two-band model for the copper-based superconductors $\hbox {YBa}_{2}\hbox {Cu}_{4}\hbox {O}_{8}$ and $\hbox {Bi}_{2}\hbox {Sr}_{2}\hbox {CaCu}_{2}\hbox {O}_{8}$. By employing the phenomenological Ginzburg-Landau (GL) free energy density functional theory in two-band model in the presence of two superconducting order parameters, we obtained the mathematical expressions for the temperature dependence of the upper critical magnetic field ($H_{C2}(T)$) and lower critical magnetic field ($H_{C1}(T)$); the angular dependence of upper critical magnetic field, $H_{C2}(\theta )$; the temperature dependence of GL coherence length, $\xi _{GL}(T)$ and GL penetration depth, $\lambda _{GL}(T)$ for the superconductors $\hbox {YBa}_{2}\hbox {Cu}_{4}\hbox {O}_{8}$ and $\hbox {Bi}_{2}\hbox {Sr}_{2}\hbox {CaCu}_{2}\hbox {O}_{8}$. By using the experimental values in the obtained expressions, the phase diagrams for the temperature dependence of upper critical magnetic fields parallel, $H^{\Vert c}_{C2}(T)$ and perpendicular, $H_{C2}^{\perp c}(T)$ to the crystallographic c-axis are plotted for $\hbox {YBa}_{2}\hbox {Cu}_{4}\hbox {O}_{8}$ and $\hbox {Bi}_{2}\hbox {Sr}_{2}\hbox {CaCu}_{2}\hbox {O}_{8}$. Similarly, using the obtained expressions the phase diagrams for the angular dependence of upper critical magnetic field, $H_{C2}(\theta )$ are plotted for $\hbox {YBa}_{2}\hbox {Cu}_{4}\hbox {O}_{8}$ and $\hbox {Bi}_{2}\hbox {Sr}_{2}\hbox {CaCu}_{2}\hbox {O}_{8}$. We demonstrated that, the upper critical magnetic field along the ab-plane is greater than along the crystallographic c-axis for both $\hbox {YBa}_{2}\hbox {Cu}_{4}\hbox {O}_{8}$ and $\hbox {Bi}_{2}\hbox {Sr}_{2}\hbox {CaCu}_{2}\hbox {O}_{8}$. We have also plotted the phase diagrams for the temperature dependence of lower critical magnetic fields parallel, $H^{\Vert c}_{C1}(T)$ and perpendicular, $H_{C1}^{\perp c}(T)$ to the crystallographic c-axis for the two copper-based superconductors. Moreover, the phase diagrams for the temperature dependence of GL coherence lengths in the ab-plane, $\xi ^{ab}_{GL}(T)$ and along the c-axis, $\xi _{GL}^{c}(T)$ are plotted. Furthermore, we have plotted the phase diagrams for the temperature dependence of GL penetration depths in the ab-plane, $\lambda ^{ab}_{GL}(T)$ and along the c-axis, $\lambda _{GL}^{c}(T)$. Finally, the phase diagrams for the GL characteristic parameters, $\kappa ^{c}_{GL}$ and $\kappa ^{ab}_{GL}$ are plotted and the large values indicate that, $\hbox {YBa}_{2}\hbox {Cu}_{4}\hbox {O}_{8}$ and $\hbox {Bi}_{2}\hbox {Sr}_{2}\hbox {CaCu}_{2}\hbox {O}_{8}$ are extreme type-II copper-based superconductors. The results we obtained in this research work are in broad agreement with previous experimental findings.

Graphic abstract