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

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