Theory of diffusive φ ₀ Josephson junctions in the presence of spin-orbit coupling

Abstract

We present a full microscopic theory based on the SU(2) covariant formulation of the quasiclassical formalism to describe the Josephson current through an extended superconductor-normal metalsuperconductor (SNS) diffusive junction with an intrinsic spin-orbit coupling (SOC) in the presence of a spin-splitting field h. We demonstrate that the ground state of the junction corresponds to a finite intrinsic phase difference 0 < ϕ0 < 2π between the superconductor electrodes provided that both, h and the SOC-induced SU(2) Lorentz force are finite. In the particular case of a Rashba SOC we present analytic and numerical results for ϕ0 as a function of the strengths of the spin fields, the length of the junction, the temperature and the properties of SN interfaces.The dc Josephson effect establishes that the supercurrent flowing between two superconductors connected by a weak link (normal metal, ferromagnet or semiconductor) is given by I J = I c sin ϕ . Here ϕ is the phase difference between the superconducting electrodes and I c the critical current, i.e. the maximum supercurrent that can flow through the junction. The ground state of such a junction corresponds to a zero current state and vanishing phase difference and for that reason it is denoted as a 0-junction. In analogy, one can define a ϕ 0 -junction with a more general current phase relation described byThe ground state corresponds to a finite phase difference ϕ 0 across the junction. Examples of non-zero junctions are superconductor-ferromagnet-superconductor (SFS) junction of certain thickness with a ground state at ϕ 0 = π, predicted in 1982 [1] and first detected in 2001 by Ryazanov et al. [2]. Besides 0 and π junctions currently there is no experimental evidence of a ϕ 0 -junction with 0 < ϕ 0 < π for single junctions [3][4][5]. It was, however, theoretically suggested by A. Buzdin [6] that if the weak link is made of a non-centrosymmetric magnetic metal such a junction is possible. This prediction has been originally formulated in terms of the Ginzburg-Landau (G-L) theory written in the presence of a Rashba-like SOC and an exchange field. ϕ 0 -junctions have been also analyzed in different types of ballistic junctions with the Rashba SOC [7]. The universality of this result in the presence of disorder or a generic SOC is however questionable, and no further conclusions can be drawn from previous works.In this letter we address this question and present a complete theoretical description of the Josephson effect through a weak link with arbitrary linear in momentum SOC and disorder. First, by using the analogy of SOC to a SU(2) gauge field and simple symmetry arguments we set the conditions for the ϕ 0 -junction to exist in a ballistic system, and demonstrate the connection between ϕ 0 -junction behavior and the Edelstein effect in superconductors [8]. In a second part we use the SU(2) covariant Eilenberger equation [9] to show that SOC-induced SU(2) Lorentz force is responsible for the intrinsic ϕ 0 for an arbitrary degree of disorder. Finally we analyze ...

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