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

This research work focuses on the theoretical investigation of upper critical magnetic field ( $$H_{C2}$$ H C 2 ), lower critical magnetic field ( $$H_{C1}$$ H C 1 ), Ginzburg-Landau coherence length ( $$\xi _{GL}$$ ξ GL ) and Ginzburg-Landau penetration depth ( $$\lambda _{GL}$$ λ GL ) in two-band model for the copper-based superconductors $$\hbox {YBa}_{2}\hbox {Cu}_{4}\hbox {O}_{8}$$ YBa 2 Cu 4 O 8 and $$\hbox {Bi}_{2}\hbox {Sr}_{2}\hbox {CaCu}_{2}\hbox {O}_{8}$$ Bi 2 Sr 2 CaCu 2 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 ( $$H_{C2}(T)$$ H C 2 ( T ) ) and lower critical magnetic field ( $$H_{C1}(T)$$ H C 1 ( T ) ); the angular dependence of upper critical magnetic field, $$H_{C2}(\theta )$$ H C 2 ( θ ) ; the temperature dependence of GL coherence length, $$\xi _{GL}(T)$$ ξ GL ( T ) and GL penetration depth, $$\lambda _{GL}(T)$$ λ GL ( T ) for the superconductors $$\hbox {YBa}_{2}\hbox {Cu}_{4}\hbox {O}_{8}$$ YBa 2 Cu 4 O 8 and $$\hbox {Bi}_{2}\hbox {Sr}_{2}\hbox {CaCu}_{2}\hbox {O}_{8}$$ Bi 2 Sr 2 CaCu 2 O 8 . By using the experimental values in the obtained expressions, the phase diagrams for the temperature dependence of upper critical magnetic fields parallel, $$H^{\Vert c}_{C2}(T)$$ H C 2 ‖ c ( T ) and perpendicular, $$H_{C2}^{\perp c}(T)$$ H C 2 ⊥ c ( T ) to the crystallographic c-axis are plotted for $$\hbox {YBa}_{2}\hbox {Cu}_{4}\hbox {O}_{8}$$ YBa 2 Cu 4 O 8 and $$\hbox {Bi}_{2}\hbox {Sr}_{2}\hbox {CaCu}_{2}\hbox {O}_{8}$$ Bi 2 Sr 2 CaCu 2 O 8 . Similarly, using the obtained expressions the phase diagrams for the angular dependence of upper critical magnetic field, $$H_{C2}(\theta )$$ H C 2 ( θ ) are plotted for $$\hbox {YBa}_{2}\hbox {Cu}_{4}\hbox {O}_{8}$$ YBa 2 Cu 4 O 8 and $$\hbox {Bi}_{2}\hbox {Sr}_{2}\hbox {CaCu}_{2}\hbox {O}_{8}$$ Bi 2 Sr 2 CaCu 2 O 8 . We demonstrated that, the upper critical magnetic field along the ab-plane is greater than along the crystallographic c-axis for both $$\hbox {YBa}_{2}\hbox {Cu}_{4}\hbox {O}_{8}$$ YBa 2 Cu 4 O 8 and $$\hbox {Bi}_{2}\hbox {Sr}_{2}\hbox {CaCu}_{2}\hbox {O}_{8}$$ Bi 2 Sr 2 CaCu 2 O 8 . We have also plotted the phase diagrams for the temperature dependence of lower critical magnetic fields parallel, $$H^{\Vert c}_{C1}(T)$$ H C 1 ‖ c ( T ) and perpendicular, $$H_{C1}^{\perp c}(T)$$ H C 1 ⊥ 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, $$\xi ^{ab}_{GL}(T)$$ ξ GL ab ( T ) and along the c-axis, $$\xi _{GL}^{c}(T)$$ ξ GL c ( T ) are plotted. Furthermore, we have plotted the phase diagrams for the temperature dependence of GL penetration depths in the ab-plane, $$\lambda ^{ab}_{GL}(T)$$ λ GL ab ( T ) and along the c-axis, $$\lambda _{GL}^{c}(T)$$ λ GL c ( T ) . Finally, the phase diagrams for the GL characteristic parameters, $$\kappa ^{c}_{GL}$$ κ GL c and $$\kappa ^{ab}_{GL}$$ κ GL ab are plotted and the large values indicate that, $$\hbox {YBa}_{2}\hbox {Cu}_{4}\hbox {O}_{8}$$ YBa 2 Cu 4 O 8 and $$\hbox {Bi}_{2}\hbox {Sr}_{2}\hbox {CaCu}_{2}\hbox {O}_{8}$$ Bi 2 Sr 2 CaCu 2 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