Spin transport in an electrically-driven magnon gas near Bose-Einstein condensation: a Hartree-Fock-Keldysh theory

Abstract

An easy-plane ferromagnetic insulator in a uniform external magnetic field and in contact with a phonon bath and a normal metal bath is studied theoretically in the presence of dc spin current injection via spin Hall effect in the metal. The Keldysh path integral formalism is used to model the magnon gas driven into nonequilibrium steady state by mismatched bath temperatures and/or electrical spin injection, and we analyze the magnon system in the normal (uncondensed) state, but close to field- and electrically-driven instability to Bose-Einstein condensation (BEC), within the self-consistent Hartree-Fock approximation. We show that the BEC instability in the electrically-driven magnon system is signaled by a sign change in the imaginary part of the poles for long-wavelength magnon modes and by the divergence of the nonequilibrium magnon distribution function. In the presence of two bath temperatures, we find that the correlation length of the superfluid order parameter fluctuations exhibits nontrivial finite temperature crossover behavior that is richer than the standard crossover behavior obtained for the vacuum-superfluid transition in an equilibrium dilute Bose gas. We study the consequences of these thermal crossovers on the magnon spin conductivity and obtain an inverse square-root divergence in the spin conductivity in the vicinity of the electrically-induced BEC instability. A spintronics device capable of testing our spin transport predictions is discussed.