Gate-Tunable Superconducting Spin Valve in a van der Waals Ferromagnet/Superconductor/Ferromagnet Trilayer

TL;DR

This work addresses how to electrically control superconducting spin-valve behavior in a vdW trilayer F1/S/F2 by tuning the ferromagnets' chemical potentials μ_{F1} and μ_{F2} with gate voltages. Using a minimal tight-binding model and a 12×12 Nambu-spin-layer Green's function, the authors solve the Gor'kov equations self-consistently to obtain the order parameter Δ and spectral properties, revealing gate-tunability among standard, inverse, and triplet SVE regimes. They show that the effective exchange field h_{eff} in the S layer, set by hybridization with the F spectra, dictates the SVE type, and that non-BCS Δ(T) behavior—such as reentrant superconductivity and bistability—emerges under gating even for parallel magnetizations. These results establish vdW F/S/F trilayers as a flexible platform for superconducting spintronics, enabling electrically reconfigurable proximity-induced phenomena and exotic superconducting states with potential for device applications.

Abstract

We theoretically demonstrate a gate-tunable superconducting spin valve effect (SVE) in a van der Waals (vdW) heterostructure composed of a monolayer superconductor (S) sandwiched between two ferromagnetic (F) monolayers (F/S/F). By electrostatically gating the ferromagnetic layers to modulate their chemical potentials, the system can be continuously tuned between the standard, inverse and triplet (non-monotonic) SVE regimes within the same device. This tunability originates from the gate-controlled hybridization between the superconducting and ferromagnetic electronic spectra, which determines the effective exchange field induced in the S-layer. Furthermore, we reveal that gating enables exotic, non-BCS temperature dependencies of the superconducting order parameter, including reentrant superconductivity, bistable states, first-order phase transitions, and the emergence of superconductivity at finite temperatures. Our results establish vdW F/S/F trilayers as a versatile and highly controllable platform for superconducting spintronics, where external gate voltages can selectively activate different spin-valve functionalities and unconventional superconducting states.

Figures (8)

• Figure 1: $\mathbf{F_1/S/F_2}$ van der Waals spin valve. The heterostructure comprises a monolayer superconductor (S) encapsulated between two ferromagnetic monolayers ($\rm F_1$ and $\rm F_2$). Schematic representation includes top and bottom gate voltages $V_{1,2}$ applied to the respective $\rm F_{1,2}$ layers.
• Figure 2: Phase diagram for standard and inverse SVE. The diagram delineates the regions in the $(\mu_{F_1}, \mu_{F_2})$ parameter space where the standard (St) and inverse (Inv) SVEs are realized. Parameters used: $t_F = 1.25 t_S$, $t_{FS} = 0.017 t_s$, $\mu_S = 0.333 t_S$, and $h = 0.167 t_S$.
• Figure 3: $T_c$ and electronic spectra of the $\mathbf{F_1/S/F_2}$ spin valve. (a) Superconducting critical temperature $T_c$ versus the on-site energy $\mu_{F_2}$ of the bottom ferromagnetic layer for parallel (blue) and antiparallel (orange) magnetization configurations. Inset: expanded view of the right region of suppressed superconductivity for the parallel configuration. (b)–(e) Spin-resolved electronic spectral density for spin-up (yellow) and spin-down (red) states as a function of quasiparticle energy $\varepsilon$ and $\zeta = -2(\cos p_y a + \cos p_z a)$, calculated at the $\mu_{F_2}$ values marked by points b–e in panel (a). Spectral branches associated with the F$_1$ and F$_2$ layers are labeled accordingly. Here, $\Delta_0$ denotes the superconducting order parameter at $T=0$. Parameters: $T = 0.214 \Delta_0$, $\Delta_0 = 0.016 t_S$, $t_F = 1.25 t_S$, $t_{FS} = 0.017 t_S$, $\mu_S = 0.333 t_S$, $\mu_{F_1} = 0.55 t_S$, $h = 0.167 t_S$. Specific values: (b),(d) $\mu_{F_2} = 0.183 t_S$; (c),(e) $\mu_{F_2} = 0.333 t_S$.
• Figure 4: Non-monotonic angular dependence of the superconducting order parameter. (a) Phase diagram indicating regions of monotonic (M) and non-monotonic (N) $\Delta(\theta)$ behavior. (b,c) Examples of non-monotonic $\Delta(\theta)$ dependencies corresponding to the red points in panel (a): (b) $\mu_{F_1} = 0.625 t_S$, $\mu_{F_2} = 0.247 t_S$; (c) $\mu_{F_1} = 0.5 t_S$, $\mu_{F_2} = 0.583 t_S$. Other parameters are the same as in Fig. (\ref{['fig:TcMu']}).
• Figure 5: Singlet and triplet superconducting correlations as functions of $\bm {(\mu_{F_1}, \mu_{F_2})}$. (a) Singlet correlations $d_0 (\omega_0)$ at the first Matsubara frequency, (b) triplet correlations $d_x(\omega_0)$, (c) triplet correlations $-i d_y (\omega_0)$, (d) triplet correlations $-i d_z (\omega_0)$. $\theta=\pi/2$. Other parameters are the same as in Fig. (\ref{['fig:TcMu']}).