Eur. Phys. J. B (2023) 96: 136
https://doi.org/10.1140/epjb/s10051-023-00605-z

Regular Article – Solid State and Materials

Investigation of anisotropic upper critical magnetic field in two-band model for the copper-based superconductors ${\mathbf{YBa}}_{\mathbf{2}}{\mathbf{Cu}}_{\mathbf{4}}{\mathbf{O}}_{\mathbf{8}}$ and ${\mathbf{Bi}}_{\mathbf{2}}{\mathbf{Sr}}_{\mathbf{2}}{\mathbf{CaCu}}_{\mathbf{2}}{\mathbf{O}}_{\mathbf{8}}$

¹ Department of Physics, College of Natural Science, Adigrat University, Adigrat, Ethiopia
² Department of Physics, College of Natural Science, Bahir Dar University, Bahir Dar, Ethiopia
³ Department of Physics, College of Natural Science, Aksum University, Aksum, Ethiopia

^a tsadik.k@gmail.com

Received: 13 September 2023
Accepted: 8 October 2023
Published online: 24 October 2023

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 (${\xi }_{\mathit{GL}}$) and Ginzburg-Landau penetration depth (${\lambda }_{\mathit{GL}}$) in two-band model for the copper-based superconductors ${\text{YBa}}_2{\text{Cu}}_4{\text{O}}_8$ and ${\text{Bi}}_2{\text{Sr}}_2{\text{CaCu}}_2{\text{O}}_8$. 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, ${\xi }_{\mathit{GL}}(T)$ and GL penetration depth, ${\lambda }_{\mathit{GL}}(T)$ for the superconductors ${\text{YBa}}_2{\text{Cu}}_4{\text{O}}_8$ and ${\text{Bi}}_2{\text{Sr}}_2{\text{CaCu}}_2{\text{O}}_8$. By using the experimental values in the obtained expressions, the phase diagrams for the temperature dependence of upper critical magnetic fields parallel, $H_{C2}^{\Vert c}(T)$ and perpendicular, $H_{C2}^{\perp c}(T)$ to the crystallographic c-axis are plotted for ${\text{YBa}}_2{\text{Cu}}_4{\text{O}}_8$ and ${\text{Bi}}_2{\text{Sr}}_2{\text{CaCu}}_2{\text{O}}_8$. Similarly, using the obtained expressions the phase diagrams for the angular dependence of upper critical magnetic field, $H_{C2}(\theta )$ are plotted for ${\text{YBa}}_2{\text{Cu}}_4{\text{O}}_8$ and ${\text{Bi}}_2{\text{Sr}}_2{\text{CaCu}}_2{\text{O}}_8$. We demonstrated that, the upper critical magnetic field along the ab-plane is greater than along the crystallographic c-axis for both ${\text{YBa}}_2{\text{Cu}}_4{\text{O}}_8$ and ${\text{Bi}}_2{\text{Sr}}_2{\text{CaCu}}_2{\text{O}}_8$. We have also plotted the phase diagrams for the temperature dependence of lower critical magnetic fields parallel, $H_{C1}^{\Vert c}(T)$ 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, ${\xi }_{\mathit{GL}}^{\mathit{ab}}(T)$ and along the c-axis, ${\xi }_{\mathit{GL}}^c(T)$ are plotted. Furthermore, we have plotted the phase diagrams for the temperature dependence of GL penetration depths in the ab-plane, ${\lambda }_{\mathit{GL}}^{\mathit{ab}}(T)$ and along the c-axis, ${\lambda }_{\mathit{GL}}^c(T)$. Finally, the phase diagrams for the GL characteristic parameters, ${\kappa }_{\mathit{GL}}^c$ and ${\kappa }_{\mathit{GL}}^{\mathit{ab}}$ are plotted and the large values indicate that, ${\text{YBa}}_2{\text{Cu}}_4{\text{O}}_8$ and ${\text{Bi}}_2{\text{Sr}}_2{\text{CaCu}}_2{\text{O}}_8$ are extreme type-II copper-based superconductors. The results we obtained in this research work are in broad agreement with previous experimental findings.