Quasiclassical theory of magnetoelectric effects in superconducting heterostructures in the presence of spin-orbit coupling

The quasiclassical theory in terms of equations for the Green's functions (Eilenberger equations) is generalized in order to allow for quantitative description of the magnetoelectric effects and proximity-induced triplet correlations in the presence of spin-orbit coupling in hybrid superconducting systems. The formalism is valid under the condition that the spin-orbit coupling is weak with respect to the Fermi energy, but exceeds the superconducting energy scale considerably. On the basis of the derived formalism it is shown that the triplet correlations in the spin-orbit coupled normal metal can be induced by proximity to a singlet superconductor without any exchange or external magnetic field. They contain an odd-frequency even-momentum component, which is stable against disorder. The value of the proximity-induced triplet correlations is on the order of ${\mathrm{\Delta }}_{\mathrm{so}}/{\varepsilon }_F$, which is absent in the framework of the standard quasiclassical approximation, but can be described by our theory. The spin polarization, induced by the Josephson current flowing through the superconductor/Rashba metal/superconductor junction, is also calculated.

Figure 1

(a) Sketch of the 2D NSO/S interface under consideration. Both materials are 2D, the spin-orbit coupling $\alpha \ne 0$ only to the left of the interface, while $\alpha =0$ to the right, and there is also an intrinsic singlet superconducting pairing there. (b) Another possible realization of the proximity effect between NSO and a superconductor. The difference from panel (a) is that the interface is between the different parts of the same 2D material; therefore the value of $\alpha$ is the same for both sides of the interface. NSO by itself is not superconducting, but the superconductivity is realized in the left part by the proximity to the singlet superconductor, which is placed on top of the NSO material.

Figure 2

(a) Sketch of the 2D S/NSO/S junction under consideration. (b) Alternative realization of the S/NSO/S Josephson setup.