Phenomenological Theory of Superconductivity and Magnetism in Ho${}_{1-x}$Dy${}_x$Ni${}_2$B${}_2$C

The coexistence of the superconductivity and magnetism in the Ho${}_{1-x}$Dy${}_x$Ni${}_2$B${}_2$C is studied by using Ginzburg-Landau theory. This alloy shows the coexistence and complex interplay of superconducting and magnetic order. We propose a phenomenological model which includes two magnetic and two superconducting order parameters accounting for the multi-band structure of this material. We describe phenomenologically the magnetic fluctuations and order and demonstrate that they lead to anomalous behavior of the upper critical field. The doping dependence of $T_c$ in Ho${}_{1-x}$Dy${}_x$Ni${}_2$B${}_2$C showing a reentrance behavior are analyzed yielding a very good agreement with experimental data.