Theory of the spin-galvanic effect and the anomalous phase shiftφ0in superconductors and Josephson junctions with intrinsic spin-orbit coupling

Abstract

Due to the spin-orbit coupling (SOC) an electric current flowing in a normal metal or semiconductor can induce a bulk magnetic moment. This effect is known as the Edelstein (EE) or magneto-electric effect. Similarly, in a bulk superconductor a phase gradient may create a finite spin density. The inverse effect, also known as the spin-galvanic effect, corresponds to the creation of a supercurrent by an equilibrium spin polarization. Here, by exploiting the analogy between a linear-in-momentum SOC and a background SU(2) gauge field, we develop a quasiclassical transport theory to deal with magneto-electric effects in superconducting structures. For bulk superconductors this approach allows us to easily reproduce and generalize a number of previously known results. For Josephson junctions we establish a direct connection between the inverse EE and the appearance of an anomalous phase-shift ϕ0 in the current-phase relation. In particular we show that ϕ0 is proportional to the equilibrium spin-current in the weak link. We also argue that our results are valid generically, beyond the particular case of linear-in-momentum SOC. The magneto-electric effects discussed in this study may find applications in the emerging field of coherent spintronics with superconductors.

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

“…For Josephson junction the inverse magneto-electric effect is a cause of the anomalous phase shift ϕ 0 , which modifies the current-phase relation according to j = j c sin(ϕ − ϕ 0 ). This is the so called ϕ 0 -junction, and its interpretation in terms of the inverse magneto-electric effect was reported in 46 . It was actively studied recently in half-metal junctions, noncoplanar ferromagnetic junctions, ferromagnetic Josephson junctions with spin orbit interaction or TI surface states [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48] .…”

“…More recently, it was reported in several works on the basis of the lattice numerical calculation 56 , a gauge-covariant analytical approach 46 , and, at last, on the basis of exact Gor'kov technique, that triplet superconductivity can be arXiv:1611.01737v2 [cond-mat.supr-con] 23 Sep 2017 generated in Rashba metals by proximity to a singlet superconductor 57 . However, the Gor'kov equations are of very limited use for inhomogeneous problems.…”

“…So, it is desirable to generalize the quasiclassical technique in order to be able to describe the SOC-induced triplet correlations and the magneto-electric and spin-galvanic effect, which are also beyond the framework of quasiclassical approximation. A way to such a generalization has already been proposed in the framework of gauge-covariant Green's functions approach 46 . But the results for the proximityinduced triplet correlations in the ballistic limit seem to be not fully coinciding with the results of exact Gor'kov's approach 57 .…”

Anomalous current in diffusive ferromagnetic Josephson junctions

“…Below we derive analytical expressions for the anomalous and usual Josephson current components in generic trilayer S/F/S structures shown in Fig. 1 using Usadel equations (12) and (13) with boundary conditions (19) and current (20). We calculate the amplitudes of anomalous currents for the structures shown in Fig.…”

“…This effect is referred to as the anomalous Josephson effect (AJE) and takes place only in systems with a broken time-reversal symmetry. The AJE has been predicted in junctions which combine conventional superconductors with magnetism and spinorbital interaction [15][16][17][18][19][20][21][22][23], between unconventional superconductors [24], and between topologically nontrivial superconducting leads [25]. In the presence of magnetic flux piercing the normal interlayer superconducting proximity currents are generated, which naturally leads to a phase shift of the CPR [26,27].…”