Semiclassical Quantization of Spinning Quasiparticles in Ballistic Josephson Junctions

Abstract

A Josephson junction made of a generic magnetic material sandwiched between two conventional superconductors is studied in the ballistic semi-classic 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. In the last years the spin-orbit and spin-splitting effects in superconducting heterostructures [1,2] are receiving a great deal of attention in the context of an emerging superconducting spintronics [3,4] and in connection with possible realizations of Majorana bound states in nanowires [5]. A Josephson junction with a magnetoactive normal bridge exemplify a prototype structure hosting such kind of spin interactions. The physics of superconductor/normal metal/superconductor (S/N/S) ballistic Josephson junctions is mainly determined by the so called Andreev bound states (ABS) localized in the Nregion. These states, which carry a significant fraction of the Josephson supercurrent [6,7], have been extensively studied in ballistic superconducting point contacts [8][9][10].Theoretically the quantization of states trapped in some classically allowed region can be understood from the Bohr-Sommerfeld quantization rule [11,12] which requires the phase accumulated along a closed classical trajectory to be a multiple of 2π. In a ballistic S/N/S junction the trapping in the N-region occurs due to Andreev reflections with conversion of the incident electron to the reflected hole and vice versa at the S/N interfaces [13]. Each Andreev reflection brings a phase shift θ (E) = arccos (E/∆), where E < ∆ is the energy measured with respect to the Fermi level [14]. The classical loop trajectory is now defined in the space composed of the position and particle-hole subspaces. In the position subspace the electron and the reflected hole accumulate the phase equal to 2EL/v, where L is the distance between the S electrodes and v is the component of the velocity perpendicular to the junction plane. From the two Andreev reflections (shifts in the particle-hole subspace) the phase acquires the contribution 2θ (E) ± ϕ, depending on the propagating direction, where ϕ is the phase difference between the two S-electrodes (see Fig.1), [15]. Hence the quantization condition for ABS reads: 2E n L/|v| − 2θ (E n ) + sgn(v)ϕ = 2nπ. The spin-orbit coupling (SOC) and spin-splitting (exchange or Zeeman), possibly textured fields in a magnetic material, generate precession of the electron and hole spins, which should modify the properties of ABS. How the semiclassical condition is modified in th...

Anomalous current in diffusive ferromagnetic Josephson junctions

“…The general symmetries of the BdG Hamiltonian are well known [46]. In the quasiclassical limit, which is equivalent to the Andreev approximation [47], transport properties are determined by particles living exactly at the Fermi surface. In the BdG Hamiltonian this corresponds to the ξ = 0 case.…”

Andreev spectrum of a Josephson junction with spin-split superconductors

“…(10) can be generalized to the case of a spin-active weak link using the formalism developed in Ref. [37].…”

Spectral properties and quantum phase transitions in superconducting junctions with a ferromagnetic link

“…Eq. 14 determines the spectrum of Andreev bound states (ABS) [9,10]. In a short junction, L ξ 0 , where ξ 0 ≡ v F /∆ is the superconducting coherence length, one obtains σ = ±∆ cos σΦ + ϕ 2 .…”

“…Ballistic SFS junctions have been widely explored in the past, mainly in two limiting cases. One of them is the semiclassical limit, in which the Fermi energy, µ, is assummed to be much larger than any other energy involved in the system, including the superconducting gap, ∆, and the Zeeman splitting, h [6][7][8][9][10]. In this limit, one can directly apply the Bohr-Sommerfeld semiclassical quantization condition [11] and demonstrate that, in the absence of interface barriers, the spectrum consists of two double-degenerate Andreev bound states with opposite energies.…”