Abstract
A Josephson junction made of a generic magnetic material sandwiched between two conventional superconductors is studied in the ballistic semiclassic limit. The spectrum of Andreev bound states is obtained from the single valuedness of a particle-hole spinor over closed orbits generated by electron-hole reflections at the interfaces between superconducting and normal materials. The semiclassical quantization condition is shown to depend only on the angle mismatch between initial and final spin directions along such closed trajectories. For the demonstration, an Andreev-Wilson loop in the composite position-particle-hole-spin space is constructed and shown to depend on only two parameters, namely, a magnetic phase shift and a local precession axis for the spin. The details of the Andreev-Wilson loop can be extracted via measuring the spin-resolved density of states. A Josephson junction can thus be viewed as an analog computer of closed-path-ordered exponentials.
Highlights
A Josephson junction made of a generic magnetic material sandwiched between two conventional superconductors is studied in the ballistic semi-classic limit
For example, how the Josephson current is affected by the spin-dependent interactions and finite temperature, or whether a finite magnetic moment can be created in the junction one needs to extend the formalism and take into account all energies in the spectrum and the electronic distribution functions. For this we introduce the quasiclassical Green’s function, solve Eilenberger equation for the superconductor/normal metal/superconductor (S/N/S) junction and explore how the vectors ne,h and the phase Φ, associated with the AW-loop, manifest in physical observables
When the expectation values of the electron se (x) = ψ† (x) σψ (x) and hole sh (x) = χ† (x) σχ (x) spin vectors are propagated around the Andreev-Wilson loops based at the point x, they rotate by the angle Φ about the directions of ne (x) and nh (x), respectively
Summary
A Josephson junction made of a generic magnetic material sandwiched between two conventional superconductors is studied in the ballistic semi-classic limit. Assuming the left (L) and right (R) interfaces with transmission probabilities TL,R = 1 − RL,R and reflection coefficients rL,R with |rL,R|2 = RL,R one obtains for a single channel junction such generic spin-dependent field.
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