Theory of quantum-geometric charge and spin Josephson diode effects in strongly spin-polarized hybrid structures with noncoplanar spin textures

TL;DR

The paper addresses nonreciprocal Josephson transport in superconducting hybrids formed by strongly spin-polarized ferromagnets coupled through spin-active insulating interfaces. It develops a modified quasiclassical Usadel formalism with two decoupled spin bands, Riccati coherence parametrization, and S-matrix boundary conditions to capture quantum-geometric effects from noncoplanar magnetization textures. The main findings show charge diode efficiencies up to about 33% and a perfect spin diode (100% efficiency) when a noncoplanar spin texture exists and spin-band densities of states differ; a harmonic analysis reveals that crossed equal-spin pair transmission drives the diode behavior. The results establish design principles for diode-like superconducting spintronic devices and propose measurement schemes in SQUID geometries, with potential realization in Ni/Co-based ferromagnets and ferromagnetic insulators like GdN/EuS, facilitating spin-switching of equal-spin supercurrents in hybrid structures.

Abstract

We present a systematic study of the spin-resolved Josephson diode effect (JDE) in strongly spin-polarized ferromagnets (sFM) coupled to singlet superconductors (SC) via ferromagnetic insulating interfaces (FI). All metallic parts are described in the framework of the quasiclassical Usadel Green's function theory applicable to diffusive systems. The interfaces are characterized by an S-matrix obtained for a model potential with exchange vectors pointing in an arbitrary direction with respect to the magnetization in the sFM. Our theory predicts a large charge Josephson diode effect with an efficiency exceeding $33\%$ and a perfect spin diode effect with $100\%$ efficiency. To achieve these the following conditions are necessary: (i) a noncoplanar profile of the three magnetization vectors in the system and (ii) different densities of states of spin-$\uparrow$ and spin-$\downarrow$ bands in the sFM achieved by a strong spin polarization. The former gives rise to the quantum-geometric phase, $Δ\varphi$, that enters the theory in a very similar manner as the superconducting phase difference across the junction, $Δχ$. We perform a harmonic analysis of the Josephson current in both variables and find symmetries between Fourier coefficients allowing an interpretation in terms of transfer processes of multiple equal-spin Cooper pairs across the two ferromagnetic spin bands. We point out the importance of crossed pair transmission processes. Finally, we study a spin-switching effect of an equal-spin supercurrent by reversing the magnetic flux in a SQUID device incorporating the mentioned junction and propose a way for measuring it.

Figures (20)

• Figure 1: Schematic representation of the two different pairing mechanism for spin-triplet correlations. The left panel shows pairing which leads to mixed-spin triplet correlations (with spin projection $s_z = 0$) with finite center-of-mass momentum $Q \neq 0$. The right panel shows the equal-spin triplet correlations (with spin projection $s_z = \pm 1$), which are generated in a strongly spin-polarized ferromagnet and do not acquire a finite center-of-mass momentum, i.e. $Q=0$. Only the latter ones contribute to the transport across the metallic ferromagnet in our model.
• Figure 2: Sketch of the system under study where two superconductors (SC; orange) are brought into contact with a strongly spin-polarized ferromagnet (sFM; green) via two ferromagnetic insulating layers (FI1/FI2; grey). At the outer interfaces the system is connected to superconducting reservoirs characterized by the respective pair potential $\Delta_{1/2} = \abs{\Delta_{1/2}} e^{i\chi_{1/2}}$, where 1/2 corresponds to the left/right interface.$^1$
• Figure 3: (a): Schematic representation of the parabolic band model. The SC (left; orange) has a spin-degenerate Fermi surface, whereas the sFM (right) has a separate Fermi surface for each spin band (blue $\uparrow$, red $\downarrow$). Two possible Fermi surface geometries are (b) $k_F^\downarrow < k_F < k_F^\uparrow$ and (c) $k_F^\downarrow < k_F^\uparrow < k_F$.
• Figure 4: A schematic representation of the potentials characterizing the SC/sFM interface in our model. In the superconductor (SC) it assumed that no bias potentials and exchange fields are present. In the ferromagnetic insulator both spin bands are shifted above the Fermi energy and are therefore insulating whereas in the strongly spin-polarized ferromagnet the bands are splitted but metallic. Since the splitting in both magnetic materials is of the order of $E_F$ the Fermi surfaces are split, see Fig. \ref{['fig:sketch_parabolas']}. The potentials in the two regions for each spin direction are given by $V^\mathrm{B/sFM}_{\uparrow(\downarrow)} = V^\mathrm{B/sFM} \mp J^\mathrm{B/sFM}/2$, where it is ensured that $V^\mathrm{B}_{\uparrow(\downarrow)} > E_F$ and $V^\mathrm{sFM}_{\uparrow(\downarrow)} < E_F$.
• Figure 5: Spin-resolved current-phase relation (CPR) $I_{{\uparrow\uparrow}}(\Delta \chi)$, $I_{{\downarrow\downarrow}}(\Delta\chi)$, and charge current $I_\mathrm{ch} (\Delta\chi )$ for (a) a coplanar arrangement of $(\bm{m}_1,\bm{m}_2,\bm{M})$, i.e., $\Delta\varphi = 0$, and (b) a noncoplanar arrangement, here $\Delta\varphi/\pi = 0.44$. (c) spin currents $I_{\mathrm{sp}}(\Delta\chi)$ corresponding to panel (a) and (b) for the coplanar case (dashed line) and the noncoplanar case (solid line). (d) absolute value of $I_{\mathrm{ch}}$ from panel (b), with the sign denoted by red ($+$) and blue ($-$) color; dots represent values calculated via QCGFs and used for the Fourier decomposition of the CPR. Parameters are appropriate for a symmetric junction with $V_\mathrm{B} = 1.3 E_F$, $V_\mathrm{sFM} = 0$, $J_\mathrm{sFM} = J_\mathrm{B} = 0.4 E_F$, $d_\uparrow = 0.6 \lambda_F / 2\pi$ and $d_\downarrow = 0.8 \lambda_F / 2\pi$. The left and the right interface differ only in $\varphi$.