The RKKY interaction across a tunnelling junction out of equilibrium in relation to the spin current

The Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between two magnetic spin impurities across a tunnelling junction is studied when the system is driven out of equilibrium through biasing the junction. The nonequilibrium situation is handled with the Keldysh time-loop perturbation formalism in conjunction with appropriate coupling methods for tunnelling systems due to Caroli and Feuchtwang. We find that the presence of a nonequilibrium bias across the junction leads to an interference of several fundamental oscillations, such that in this tunnelling geometry, it is possible to tune the interaction between ferromagnetic and antiferromagnetic coupling at a fixed impurity configuration, simply by changing the bias across the junction. Furthermore, it is shown that the range of the RKKY interaction is altered out of equilibrium such that in particular the interaction energy between two slabs of spin scales extensively with the thickness of the slabs in the presence of an applied bias. The spin current has been calculated for this system and it is shown that it is proportional to the long-range interaction. Other aspects of the relationship of the current to the spin geometry are explored.