Boundary conditions for spin-active interfaces with arbitrary transparency
Hybrid interfaces between magnets and superconductors need boundary conditions that connect the quasiclassical Green's functions on each side. Earlier widely used expressions cover low transparency, arbitrary transparency without spin dependence, or spin-active interfaces only in restricted limits. This paper derives boundary conditions for spin-active interfaces with arbitrary transparency, where at least one of the two materials is in the diffusive regime.
The result is compact. The boundary condition is written as a matrix current nR·JR = Σn [(1 + Xn QR)^-1 Xn, QR], with Xn = iπν h·στ3 + (πν)^2 Wn† (1 + iπν Q̃L h·στ3)^-1 Q̃L Wn. Here ν is the density of states of the right material, h the effective interfacial exchange field, and WnWn† contains the spin-dependent transmission rates of the nth mode. The left electrode need not be diffusive: for a clean left side, Q̃L is replaced by the quasiclassical Eilenberger Green function of the isolated left material.
The action formalism turns interface scattering into a boundary term
The derivation starts from a total action S = SL + SR + ST, where SL and SR are bulk actions of the left and right materials and ST is the boundary action. The right material is assumed diffusive, so its bulk action is a functional of the matrix field QR subject to the nonlinear constraint QR^2 = 1. Varying the action and requiring δS = 0 for perturbations that respect this constraint gives the Usadel equation in the bulk and the boundary condition at the interface.
The boundary action generalizes an earlier expression to include spin-dependent transmission coefficients and an interfacial exchange field. The transmission matrix is parameterized as Wn = tn + un Pn·σ, so the transmission rates are |tn + un|^2 for spins parallel to Pn and |tn − un|^2 for spins antiparallel. This is what makes the resulting boundary condition able to describe normal transport, magnetoresistance transport and spin precession through the boundary within one framework. Because the construction uses the Keldysh-Nambu-spin formalism, the equations hold in both the normal and the superconducting state.
Known boundary conditions reappear as limiting cases
The paper checks its expressions against earlier literature. In the low-transparency limit, the boundary conditions reduce to known results for spin-active interfaces. In the limit of spin-independent barriers, they agree with the Nazarov and Tanaka-Nazarov boundary conditions for any transparency. The authors therefore present their result as a generalization of earlier expressions to generic spin-dependent transparencies.
The derivation also extends the Tanaka-Nazarov construction to a junction where one material is clean and the other diffusive. That extension is what allows the formalism to describe the proximity effect of an unconventional superconductor on a dirty metal through a spin-active interface, which is the setting used later for the p-wave and triplet cases.
Zero-bias conductance depends on the d-vector orientation
The boundary conditions are applied to the anomalous proximity effect of a p-wave superconductor through a spin-active interface. The paper reports an analytic expression for the zero-bias conductance in junctions with a p-wave superconductor and a spin-filter, and shows how it depends on the relative orientation of the exchange field and the d-vector.
The surface Green's function modified by the interfacial exchange field is derived separately for a d-vector perpendicular to h and for a d-vector parallel to h. In the perpendicular case, C commutes with h·στ3 and the low-energy expression for X reduces to a form proportional to C. In the parallel case, C and h·σ commute with σz, so the problem decomposes into two-by-two blocks and the low-energy result for X contains both a h·στ3 term and a τ1 term. These two orientations are therefore not interchangeable when reading the conductance result.
Anomalous Josephson currents need different spin orientations in the two junctions
The paper contrasts two Josephson junctions through a magnetic interface: two spin-split superconductors, and a spin-split superconductor with a triplet superconductor. For the two spin-split superconductors, anomalous Josephson currents appear when the magnetizations on the two sides and the one in the barrier together span the whole spin space, expressed by the constraint P·(h1 × h2) ≠ 0. The anomalous current is maximized when the three vectors are mutually perpendicular.
For the spin-split superconductor with a triplet superconductor, the mutually orthogonal orientation gives no anomalous Josephson current, because P·(h × d) is time-reversal even and cannot correspond to a time-reversal odd equilibrium current. If the relative orientation between h and d is not perpendicular, anomalous Josephson currents are allowed through higher-order combinations of h, d and P. The paper states that the conditions on the relative orientations of the spins are distinctly different for the two junction types, in agreement with time-reversal symmetry.
A diode effect requires nonzero and unequal spin transparencies
The paper shows that anomalous Josephson currents are accompanied by a diode effect only if both transparencies for spin up and down are nonzero and distinct. In other words, the barrier must have nonzero filtering but must not be a perfect filter: the minority spin needs a nonzero transmission probability that differs from that of the majority spin.
For the triplet superconductor with a spin-split superconductor, the diode effect is allowed only in orientations where d, h and P are not orthogonal, and it is due to a higher-order invariant. The paper reports that the diode efficiency vanishes in high-symmetry orientations but can be nonzero otherwise, and may be positive or negative depending on the d-vector orientation. It remains much smaller than for the junction with spin-split superconductors, reflecting that non-reciprocal effects in the spin-split superconductor / ferromagnetic insulator / triplet superconductor junction require higher-order invariants.
The method is broader than the junctions used to illustrate it
The paper notes that although it focuses on junctions with spin-split superconductors, unconventional superconductors and spin-active interfaces with a net polarization, the proposed method is not restricted to these junction types. It can be applied to a large range of junctions, including those that on the clean side have any type of unconventional magnetism, valley degree of freedom or multiple bands, via appropriate modification of the surface Green's function and the transmission matrices W and W†.
The construction through the action formalism also allows derivation of the corresponding full counting statistics for charge currents, thermal currents and spin currents. The authors state that this makes it possible to study effects beyond the stationary boundary conditions examined in the paper, such as noise in charge currents, spin currents and thermal currents. The boundary conditions themselves are derived for interfaces where at least one material is diffusive; the clean-side extension is handled by replacing Q̃L with the Eilenberger Green function of the isolated left material.
The orientation constraint is the key difference between the two junctions
The most decision-relevant comparison in this paper is between the two Josephson junction types. For two spin-split superconductors, the anomalous current condition is P·(h1 × h2) ≠ 0, and the effect is maximized when the three vectors are mutually perpendicular. For a spin-split superconductor with a triplet superconductor, that mutually orthogonal orientation produces no anomalous current, and a nonzero effect requires h and d not to be perpendicular, through higher-order combinations of h, d and P.
The second threshold is the barrier quality. In both cases, a diode effect appears only when both spin transparencies are nonzero and distinct, so a perfect spin filter is not enough. The paper also reports that the diode efficiency in the triplet case is generally much smaller than in the spin-split superconductor case. Readers comparing candidate junctions should therefore check the orientation constraint first, then the spin-filter quality, rather than treating anomalous Josephson current and diode effect as one signature.
Sources used in this answer
General boundary conditions for the quasiclassical kinetic equations in the presence of magnetism and unconventional superconductivity
Derives compact boundary conditions for spin-active interfaces with arbitrary transparency and an interfacial exchange field, valid in normal and superconducting states, and applies them to the anomalous proximity effect and to Josephson coupling between triplet and spin-split superconductors.