Coexistence of superconductivity and magnetism in spin-fermion model of ferrimagnetic spinel in an external magnetic field

Abstract

A two-sublattice spin-fermion model of ferrimagnetic spinel, with spin-$1/2$ itinerant electrons at the sublattice $A$ site and spin-$s$ localized electrons at the sublattice $B$ site is considered. The exchange between itinerant and localized electrons is antiferromanetic. As a result the external magnetic field, applied along the magnetization of the localized electrons, compensates the Zeeman splitting due to the spin-fermion exchange and magnon-fermion interaction induces spin anti-parallel p-wave superconductivity which coexists with magnetism. We have obtained five characteristic values of the applied field (in units of energy) $H_{cr1}<H_3<H_0<H_4<H_{cr2}$. At $H_0$ the external magnetic field compensates the Zeeman splitting. When $H_{cr1}<H<H_{cr2}$ the spin antiparallel p-wave superconductivity with $T_{1u}$ configuration coexists with magnetism. The superconductor to normal magnet transition at finite temperature is second order when $H$ runs the interval $(H_3,H_4)$. It is an abrupt transition when $H_{cr1}<H<H_3$ or $H_4<H<H_{cr2}$. This is proved calculating the temperature dependence of the gap for three different values of the external magnetic field $H_{cr1}<H<H_3$, $H_4<H<H_{cr2}$ and $H=H_0$. In the first two cases the abrupt fall to zero of the gap at superconducting critical temperature shows that the superconductor to normal magnet transition is first order. The Hubbard term (Coulomb repulsion), in a weak coupling regime, does not affect significantly the magnon induced superconductivity. Relying on the above results one can formulate a recipe for preparing a superconductor from ferrimagnetic spinel: i) hydrostatic pressure above the critical value of insulator-metal transition. ii) external magnetic field along the sublattice magnetization with higher amplitude.