Triplet superconductivity in ferromagnets due to magnon exchange

We consider the superconducting pairing induced by spin-wave exchange in a ferromagnet with both conduction and localized electrons, the latter being described as spins. We use the microscopic Eliashberg theory to describe the pairing of conducting electrons and the random phase approximation approach to treat the localized spins assuming an exchange coupling between the conducting electrons and spins. In the framework of a nonrelativistic Hamiltonian, we found that the spin-wave exchange results in equal spin electron pairing described by the two components of the order parameter, ${\mathrm{\Delta }}^{\uparrow }$ (both spins up) and ${\mathrm{\Delta }}^{\downarrow }$ (both spins down). Due to the conservation of total spin projection on the axis of the spontaneous ferromagnetic moment, the spin-wave exchange at low temperatures includes an emission of magnons and an absorption of thermal magnons by the conduction electrons. The absorption and emission processes depend differently on the temperature, with the absorption being progressively suppressed as the temperature drops. As a result, the superconducting pairing exists only if the electron–spin-wave exchange parameter $g$ exceeds some critical value $g_c$. At $g>g_c$, pairing vanishes if the temperature drops below the lowest point $T_{cl}$ or increases above the upper critical point $T_{ch}\approx T_m$ (the Curie temperature) where the spin waves cease to exist. This behavior inherent to the spin-carrying glue is in an obvious disagreement with the results of the conventional BCS approach, which assumes that the effective electron-electron attraction is simply proportional to the static magnetic susceptibility.

Figure 1

The phase diagram of the ferromagnet in the plane of temperature $T_{cl}$ and the strength $g$ of the exchange coupling of localized spins and conducting electrons. As the temperature increases from $T=0$, the superconducting triplet pairing establishes itself due to the exchange of electrons with spin waves at the second-order transition line $T_{cl}(g,\kappa )$. The descending line $g\left(T_{cl}\right)$ is a direct consequence of the spin conservation at the magnon exchange, which is effective only in the presence of thermal magnons. Note that this drop in $g$ vs $T_{c1}$ becomes more pronounced as the anisotropy parameter and thus the gap in the spin-wave spectrum increase. Warming up the superconductor results in the destruction of spin waves and the Cooper pairs as the temperature approaches the Curie temperature $T_m$. This change from triplet superconducting phase to normal phase is signaled by a first-order phase transition; its exact position may be found if magnetic fluctuations are accounted for.

Figure 2

The frequency dependence of the order parameter ${\mathrm{\Delta }}^{\downarrow }$ in the ferromagnet with magnetic anisotropy $\kappa =0.1$. The dependence at $\kappa <\omega <2+\kappa$ reflects the spectrum of magnons for $\epsilon \left(\theta \right)=\kappa +1-cos\theta$, where $\theta$ is the polar angle in the $x,y$ plane.

Figure 3

The frequency dependence of the order parameter ${\mathrm{\Delta }}^{\uparrow }$ in the ferromagnet with magnetic anisotropy $\kappa =0.1$. The dependence at $-2-\kappa <\omega <\kappa <-\kappa$ reflects the spectrum of magnons $\epsilon \left(\mathbf{q}\right)=\kappa +1-cos\theta$.