Purely even harmonic Josephson current due to crossed pair transmission across strongly spin-polarized materials

TL;DR

This work shows that in a Josephson junction formed by a strongly spin-polarized ferromagnet coupled to two superconductors via a spin-active left barrier and a non-spin-active right barrier, the supercurrent is exclusively carried by equal-spin triplet pairs and is dominated by the second harmonic, sin(2Δχ). The authors identify crossed pair transmission as the microscopic mechanism: two singlet Cooper pairs in the superconductors effectively convert to two equal-spin triplet pairs in the ferromagnet, requiring higher-order Andreev processes that transfer net charge 4e. The analysis combines a diagrammatic ballistic treatment with self-consistent diffusive simulations, showing that lower-order processes cancel due to spin selection while fourth-order processes yield a robust even-harmonic CPR; the effect scales as sin^2α with misalignment α between the left barrier and the sFM magnetization and vanishes for collinear configurations. These results have implications for superconducting spintronics, enabling long-range, diode-like Josephson behavior in strongly spin-polarized hybrids and informing experimental designs using ferromagnetic bilayers and spin-active interfaces.

Abstract

We revisit the problem of the second harmonic generation in the current-phase relation across ferromagnetic bilayers placed between BCS superconductors. In particular, we consider a strongly spin-polarized metallic ferromagnet coupled to two superconducting leads via thin spin-active (left) and non-spin-active (right) insulating layers. The system is examined in the framework of the quasiclassical Green$^\prime$s function formalism, both in the ballistic (Eilenberger) and the diffusive (Usadel) limit. Strong spin polarization allows for neglecting short-range mixed-spin correlations, and the Josephson supercurrent across the ferromagnet is fully mediated by long-range equal-spin triplet correlations. Using a diagrammatic technique for ballistic propagators introduced in Refs. [1-3], we describe the relevant Andreev processes responsible for the effective conversion of two spin-singlet Cooper pairs in the superconductor into two $\uparrow\uparrow$ and $\downarrow\downarrow$ pairs in the ferromagnet. Contrary to the naive picture of direct conversion, we show that the lowest order process involves four Cooper pairs in the superconductor, among which three are incoming, and one is outgoing, giving rise to net charge transport of 4e across the non-spin-active interface. The self-consistent numerical treatment of the diffusive junction, typically more relevant in experiments, confirms this picture quantitatively.

Figures (10)

• Figure 1: (a) Sketch of the system under study. A strongly spin-polarized metallic ferromagnet (sFM; green) is connected to two BCS superconductors (SC; orange) by a thin ferromagnetic insulator on the left side (FI; grey) and a thin non spin-active insulator on the right side (I; blue). The misalignment between the barrier's exchange field $\textbf{J}_\mathrm{FI}$ and the exchange field in the sFM $\textbf{J}_\mathrm{sFM}$, is characterized by spherical angles ($\alpha,\varphi$). (b) A schematic representation of an effective process that gives rise to the second harmonic in the CPR. Two equal-spin triplet pairs with opposite spin coming from the sFM are transmitted through the insulating barrier into the superconductor as two spin-singlet Cooper pairs. As we show in Sec. \ref{['sec:ballistic:results']}, this is a naive representation showing just the effective process; the detailed Andreev scattering process is more complicated.
• Figure 2: Sketches of the Andreev processes up to the second order in the transmission amplitude following from Eq. \ref{['eq:Gamma_1']} and explicitly shown in Eqs. \ref{['eq:contribution_first']} and \ref{['eq:contribution_third']}. The singlet Cooper pairs (coherence amplitudes $\gamma_{{\mathrm{S}}},\tilde{\gamma}_{{\mathrm{S}}}$) in the SC and the triplet pair correlations (coherence amplitudes $\gamma_{\eta\eta},\tilde{\gamma}_{\eta\eta}$) in the sFM are denoted by circles containing the corresponding spins. Red and blue colors denote spin-$\uparrow$ and spin-$\downarrow$, respectively. Here $\gamma_i$ describes the conversion process of holelike to particlelike Andreev amplitudes whereas $\tilde{\gamma}_i$ describes the conversion of particlelike to holelike Andreev amplitudes. The black line in the middle of all panels represents the sFM/SC barrier which does not allow for spin-flip processes. The colored arrows represent the trajectory of a particle (solid line) or hole (dashed line). The black arrows at the edge of the singlet Cooper pairs or triplet pairs show the direction of the center-of-mass momentum. The green dot in panels (b), (e) and (f) indicate where a spin-flip process would be needed in order for the diagram to contribute to the charge current. All of the above diagrams also appear with spins exchanged, i.e., the red and blue color are exchanged.
• Figure 3: A schematic sketch of the lowest order processes contributing to the Josephson current across the junction following from Eq. \ref{['eq:current_contribution']}. The notation is the same as for Fig. \ref{['fig:lower_order_contributions']}. This process leads to a net charge transfer of $4e$ and consequently a second harmonic contribution in the Josephson CPR.
• Figure 4: (a) The current-phase relations for the spin-resolved and the total charge current for $\alpha = \pi/2$ and the same insulators' widths $d_L = d_R =0.6\lambda_F/2\pi$. Panel (b) shows the Fourier components corresponding to the CPR shown in panel (a). Panel (c) displays the critical charge current $I_\mathrm{ch}^\mathrm{crit}$ as a function of the tilt of the left barrier's exchange field $\alpha$.
• Figure 5: (a) The critical current as a function of the barrier width for the spin-$\downarrow$ channel of the spin-active (left) insulator, which modulates the spin-mixing angles. The blue curve denotes the fully self-consistent solution, whereas the red line denotes the approximate scaling from Eq. \ref{['eq:approx_sma']}. Panel (b) shows the functional dependence of the constituents of Eq. \ref{['eq:approx_sma']} on the barrier width of the spin-$\downarrow$ channel. We do not show $t^\mathrm{L}_\uparrow$, as it is almost constant as a function of $d_\mathrm{L}^\downarrow$. Other parameters are the same as for Fig. \ref{['fig:CPR_FT_crit_curr_alpha']}(a).