Necessary conditions for spin-resolved Josephson diode effect across strongly spin-polarized magnetic materials

TL;DR

The work addresses nonreciprocal Josephson transport (the Josephson diode effect) in junctions that include strongly spin-polarized ferromagnets between spin-singlet superconductors. It introduces quantum geometric phases $\Delta\varphi'$ arising from noncoplanar spin textures, which enter the current-phase relation alongside the superconducting phase difference $\Delta\chi$ and enable diode behavior. The authors enumerate four necessary conditions—nonzero $\Delta\varphi'$, spin-band DOS asymmetry, participation of both spin channels, and higher harmonic content with $\Delta\varphi' \neq k\pi/2$—and provide a minimal phenomenological model to illustrate how these ingredients produce significant charge diodes and, in some regimes, perfect spin diodes. This framework offers a practical route to design and understand nonreciprocal superconducting devices using strongly spin-polarized materials, including ferromagnetic trilayers and conical magnets, by exploiting geometric-phase physics in the CPRs.

Abstract

We present a set of necessary conditions for the appearance of charge and spin Josephson diode effects across strongly spin-polarized inhomogeneous magnetic materials (FM) placed between two spin-singlet superconductors. Noncoplanarity of the FM's spin texture gives rise to quantum geometric phases, $Δ\varphi'$, that enter the Josephson current-phase relation (CPR) similarly to the superconducting phase difference, resulting in charge and spin Josephson diode effects. Our study shows that such effects appear if the CPR possesses no phase-inversion center, achieved under the following conditions. First, noncoplanarity of the spin texture is necessary to break the spatial inversion symmetry. Second, both spin bands have to contribute to the transport, i.e., the effect is absent in half-metallic junctions. Third, different band-specific densities of states are required, and this condition is ensured by the strong spin polarization of the FM. Finally, higher harmonics in the CPR are necessary, i.e., the effect is absent in the tunneling limit. However, even in this case, the CPR must not have a phase-inversion center, which is ensured by the restriction of the quantum geometric phase to values $Δ\varphi'\neq kπ/2, k\in\mathbb{Z}$. We formulate a minimal phenomenological model that incorporates all these points, qualitatively illustrating our theory.

Figures (3)

• Figure 1: Two configurations of strongly spin-polarized magnetic materials that exhibit quantum geometric phases. Panel (a) shows a ferromagnetic trilayer where the exchange field of the central metallic layer (FM), $\Vec{J}$, sets a global quantization axis and the quantum geometric phase $\Delta\varphi$ is determined by the relative azimuthal angle between the exchange fields of the satellite ferromagnetic layers, $\vec{J}_L$ and $\vec{J}_R$. Panel (b) shows a conical magnet (CM) that in the so-called adiabatic approximation (see text) can be viewed as a ferromagnet with the opposite spin geometric phase $\Delta\varphi_s$ of the two spin bands.
• Figure 2: The Josephson energy (blue), the charge current (violet), and the spin current (orange) computed from the minimal model of Eq. \ref{['eqn:Ej_minimal']} for (a) a strongly spin-polarized junction in the tunneling limit $I_{1,1}=0$, (b) a weakly spin-polarized highly transmissive junction $I_{1,0}=I_{0,1}$ and $I_{1,1}\neq 0$, and (c) a strongly spin-polarized highly transmissive junction, $I_{1,0}\neq I_{0,1}$ and $I_{1,1}\neq 0$. In all panels, $\Delta\varphi'\neq k\pi/2$.
• Figure 3: The charge ($\eta_\mathrm{ch}$; dashed violet) and spin ($\eta_\mathrm{sp}$; dotted orange) diode efficiencies as functions of the geometric phase $\Delta\varphi'$ in a highly transmissive junction, $I_{1,1}\neq 0$, involving a strongly spin-polarized magnetic material, $I_{1,0}\neq I_{0,1}$.