Designing lattice spin models and magnon gaps with supercurrents

Abstract

Electric control over magnetic interactions at the level of individual spins is relevant for a variety of quantum applications, such as qubits, memory and sensor functionality. We show here that spin lattices and magnon gaps can be controlled with a supercurrent. Remarkably, a spin-polarized supercurrent makes the interaction between magnetic adatoms placed on the surface of a superconductor depend not only on their relative distance, but also on their absolute position in space. This property permits electric control over the interaction not only between two individual spins, but over an entire spin lattice, allowing for tunable non-collinear ground states and a practical arena to study the properties of different spin Hamiltonians. Moreover, we show that a supercurrent controls the magnon gap in antiferromagnetic and altermagnetic insulators. These results provide an accessible way to realize electrically controlled spin switching and magnon gaps without dissipative currents.

Figures (11)

• Figure 1: Two spins $\boldsymbol{S}_1$ and $\boldsymbol{S}_2$, here represented as large arrows in the $xy$--plane, are placed on the top of a 1D superconductor with equal-spin Cooper pairs. The circles represent the lattice points, and the lattice constant is $a$. The interaction between the two spins, and hence their ground state configuration, can be tuned by a supercurrent. The positions of the spins are $r_1,r_2$ and the relative coordinate is $\tau = r_2-r_1$. (a) shows the ground state configuration for two spins separated by a distance $\tau/a=1$ and $\tau/a=3$ when the supercurrent is zero. When a charge current $Q^{\sigma}/Q_c\approx0.87$ is turned on in (b), the ground state changes for $\tau/a=3$. In (c), a spin current $Q^{\sigma}/Q_c\approx0.87\sigma$ flows through the superconductor. This alters the ground state compared to (a) and (b), and the spins are allowed to be non-collinear. Moreover, the spin current creates a dependence on the center-of-mass coordinate of the spins, unlike conventional spin-spin interactions that only depend on the relative coordinate. The blue arrows represent the ground state of the spins when they are moved 8 lattice sites to the right. The chemical potential is set to $\mu=-t$. The supercurrent is normalized on the critical supercurrent $aQ_c=0.0505$, which is found by solving the gap equation.
• Figure 2: (a-d) The ground state configuration in the $xy$-plane of two spins $\boldsymbol{S}_1$ and $\boldsymbol{S}_2$ whose interaction is mediated by a superconductor with equal-spin Cooper pairs. Spins of the same color are separated by $\tau/a$. (a) There are no supercurrents in the superconductor, ${Q}^{\sigma}=0$. The ground states at $\tau/a=3$ and $\tau/a=6$ are degenerate in the $xz$-plane with negligible $y$-components (not shown here). (b) A charge current $Q^{\sigma}/Q_c\approx0.87$ flows through the superconductor. In both (a) and (b), there is no center-of-mass-dependence $(r_1)$. (c) The position of $\boldsymbol{S}_1$ is given by $r_1/a$, and the vertical gray lines separate spin pairs with different $r_1/a$. A spin supercurrent $Q^{\sigma}/Q_c\approx0.87\sigma$ flows through the superconductor and makes the RKKY interaction depend on the center-of-mass coordinate of the two spins. (d) Spin supercurrent dependence of the magnetic ground state for $r_1/a=8$. (e) Spin supercurrent dependence for a trimer with spin positions $r_1/a=4$, $r_2/a=6$ and $r_3/a=8.$ The chemical potential is $\mu=-t$.
• Figure 3: (a) A bilayer consisting of an antiferromagnetic insulator (AFM) and a triplet superconductor (tSC) with equal-spin Cooper pairs relative to the Néel order. (b) The supercurrent-induced contribution $\mathcal{D}$ to the magnon gap in the AFM in (a). The parameters used are $T=0.2T_c$ and $\mu=-t$, which gives $aQ_c\approx 0.039$. (c) A bilayer consisting of an AFM and a spin-split singlet superconductor (sSC). The spin splitting is induced by proximity to a ferromagnetic insulator (FM). The critical momentum is $aQ_c(T/T_c=0.2)\approx 0.043$ and $aQ_c(T/T_c=0.7)\approx 0.033$. (d) The supercurrent-induced contribution $\mathcal{D}^s$ to the magnon gap in the AFM in (b). The spin splitting is $h=0.01t$ and the chemical potential is $\mu=-t$. (e) shows the corresponding minima of the quasiparticle energy dispersion.
• Figure S1: The coupling constants in units of $\Lambda^2/t$ for a superconductor with equal-spin Cooper pairs. The chemical potential is $\mu=-t$ and the magnitude of the superconducting OP is $\Delta^\sigma=0.1t$. A spin current $Q^\sigma/Q_c \approx 0.87\sigma$, or equivalently $aQ^{\sigma}=2\pi\sigma\cdot 7/1000$, flows through the superconductor. The coupling constants shown here were used to calculate the ground state configuration in Fig. 2(c) in the main text.
• Figure S2: The ground state configuration in the $xy$-plane of two spins $\boldsymbol{S}_1$ and $\boldsymbol{S}_2$ whose interaction is mediated by a superconductor with equal-spin Cooper pairs. The parameters used are the same as in Fig. 2(c) in the main text, except (a) $\mu/t=+1$, (b) $Q^\sigma/Q_c\approx 0.37 \sigma$ and (c) $Q^\sigma/Q_c\approx 0.62 \sigma$.