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Mirror symmetry and exchange of magnetic impurities mediated by electrons of Rashba spin–orbit interaction in a four-terminal Landauer setup

The two-dimensional paramagnetic-impurity-embedded electron gas with Rashba spin–orbit interaction in a four-terminal Landauer setup is studied. The mean-field-assisted Landauer–Keldysh formalism is employed to investigate the electron and impurity magnetizations (spin polarizations). A mirror symmetry is identified to characterize both electron and impurity magnetizations when the impurities are symmetrically (with respect to this mirror) positioned and when pinning fields are absent. In the equilibrium Landauer setup where electrodes remain at the same chemical potentials, the adopted formalism is justified by recovering the conventional (without spin–orbit interactions) Ruderman–Kittel–Kasuya–Yosida (RKKY) exchange by applying a pinning field to the impurity. In the same setup, when further the Rashba spin–orbit interaction is turned on, the exchange between two impurities with one of the spins being pinned is comprehended as a consequence of the interplay between spin precession and the exchange oscillation. We find that in such an equilibrium system, at most two components of the spins can show up. For biased (non-equilibrium) setup, on the other hand, three components of the impurity spins can all be non-vanishing, which is distinguishable from the equilibrium case.