Phase-induced switching of ferromagnetic insulators in Josephson spin valves

Abstract

We study the Josephson effect in junctions composed of two ferromagnetic insulator/diffusive superconductor bilayers separated by an insulating barrier. By computing the free energy of the system, we identify two distinct contributions: (i) The work performed by a current source to create a supercurrent through the junction, and (ii) an antiferromagnetic coupling between ferromagnetic insulators, mediated by the superconducting condensate across the insulating barrier. The competition between these contributions allows for switching between parallel and antiparallel configurations of the magnetizations of the ferromagnetic insulators. We explicitly show that the switching occurs at finite temperatures and for superconducting phase differences satisfying $π/2 < φ< 3π/2$. Importantly, this effect can be realized in ferromagnetic insulators with sufficiently large easy-plane anisotropy energy. Using realistic junction parameters, we demonstrate that the switching can be controlled by phase bias and triggered by half-flux-quantum voltage pulses or external magnetic field pulses on the microsecond timescale. These results provide a route towards controllable Josephson-based superconducting memory devices based on EuS/Al heterostructures.

Figures (5)

• Figure 1: (a): Sketch of the Josephson junction considered in this work. It consists of five layers, FI/S/I/S/FI. Each superconducting (S) layer has a width $d_S$ and each ferromagnetic insulator (FI) layer produces an exchange field $\boldsymbol{b}_{L,R} a\delta(x \pm d_S)$ in each superconductor. The $\boldsymbol{b}_{L,R}$ of the FIs are aligned along unit vectors $\boldsymbol{m}_{L,R}$. A (time-dependent) voltage bias $V(t)$ is applied to the superconducting elements fixing the phase difference between them to $\phi(t) = \phi(-\infty) + \int_{-\infty}^{t} 2eV(t')dt'$. (b,c): Sketch of the difference in tunneling energies between Josephson junctions with the FIs in the parallel (P) and antiparallel (AP) configurations, $F^{P}_{JJ} - F^{AP}_{JJ}$, as a function of the phase difference (b) and of the system temperature (c).
• Figure 2: Zero-field phase diagrams in the $(\phi, T)$ plane illustrating the AP-P phase transition for different exchange fields $h$. The horizontal dashed lines indicate $T_c(h)$, the temperature above which superconductivity is suppressed by the exchange field. The shaded areas correspond to the parallel phase, while the unshaded regions below the dashed lines represent antiparallel alignment of the magnetic moments.
• Figure 3: Phase diagram for our device with $h/T_{c0}=0.2\pi$ at absolute temperature $T=0.05 T_{c0}$ as a function of the superconducting phase $\phi$ and the external magnetic field $B$ applied along $\boldsymbol{z}$ axis.
• Figure 4: Switching between AP and P configurations induced by half-flux-quantum (HFQ) voltage pulses (a,b) and by a magnetic field pulse (c,d). The system parameters correspond to the EuS/Al/AlO$_x$/Al/EuS junction discussed in the main text. Two voltage pulses are applied at $\tfrac{t_0 \gamma \pi T_c}{2e^2 R_B M_0} = 10$ and $\tfrac{t_1 \gamma \pi T_c}{2e^2 R_B M_0} = 60$, each with duration $\delta t = \tfrac{2e^2 R_B M_0}{\gamma \pi T_c} \approx 5.5\ \mu\text{s}$ and amplitude $V_p = \tfrac{\gamma \pi T_c}{4 e^3 R_B M_0} \times \tfrac{\pi}{\delta t} \approx 34\ \mu\text{V}$. The phase difference is computed from the second Josephson relation, $\dot{\phi}(t)/(2e) = V(t)$, with $\phi(-\infty) = 0$, yielding $\phi(t) = \int_{-\infty}^t 2eV(t)\,dt$. The magnetic field pulse is applied along the $\boldsymbol{y}$ axis at $\tfrac{t_0 \gamma \pi T_c}{2e^2 R_B M_0} = 10$, with duration $\delta t = 30 \times \tfrac{2e^2 R_B M_0}{\gamma \pi T_c}$ and amplitude $B_p = 0.5 \times \tfrac{\pi T_c}{2e^2 R_B M_0} \approx 0.54\ \mu\text{T}$.
• Figure 5: Critical current dependencies on the spin-splitting field $h$ and temperature $T$ for parallel (a) and antiparallel (b) configurations of the FI magnetic moments according to Eqs. (\ref{['eq:Ic_P']}) and (\ref{['eq:Ic_AP']}).