Hybrid collective excitations in topological superconductor/ferromagnetic insulator heterostructures

Abstract

We develop a linear response theory for the dynamical proximity effect in topological superconductor/ferromagnetic insulator (TS/FI) hybrids. Our approach integrates the nonequilibrium quasiclassical Keldysh-Usadel formalism for the TS with the Landau-Lifshitz-Gilbert equation for the FI's magnetization dynamics. This framework reveals a proximity-induced coupling between magnons and superconducting collective modes. Crucially, we find that spin-momentum locking in the TS surface state drives a hybridization between magnons and the superconducting Nambu-Goldstone (phase) mode, giving rise to composite magnon-Nambu-Goldstone excitations. We analyze the coupling strength's dependence on key parameters both analytically and numerically. In contrast, we demonstrate that the Higgs (amplitude) mode does not couple to magnons at linear order and is thus excluded from the hybrid excitation spectrum. The hybridization between magnons and the superconducting phase mode provides a mechanism for the interconversion of spin signals and the spinless signals carried by collective superconducting excitations, thereby giving new impetus to the development of superconducting spintronics.

Figures (6)

• Figure 1: Schematic of the TS/FI heterostructure. (a) Side view of the system, illustrating the interface exchange coupling $J_{ex}$ between the superconducting and magnetic subsystems. Normal-state dispersion relation of the 2D conductive surface state of the TI, showing spin-momentum locking is depicted at the top right corner. The equilibrium magnetization of the FI $\bm m_0 = \hat{x}$ and a magnon excitation $\delta \bm m$ of the FI induce in the TS an equilibrium effective exchange field $\bm h_x$ and a magnon-induced dynamic contribution $\delta \bm h_y$, respectively. The dynamic exchange field $\delta \bm h_y$ triggers a time-dependent response of the superconducting order parameter in the TS, resulting in a hybridized magnon–Nambu–Goldstone (NG) excitation. (b) Top view of the FI layer. (c) Top view of the TS layer. (d) The basic modes of the non-interacting system (i.e. $J_{ex} =0$). (e) Schematic plot of the magnon -NG mode hybridization in the system and (f) anticrossing strength as a function of the temperature.
• Figure 2: Real and imaginary parts of the Higgs mode susceptibility $\chi_H(\omega, k)$ as a function of the excitation frequency at $D k^2/\Delta_0 = 0.25$ and $T=0.1 T_c$ (orange lines) in comparison to the corresponding results taken from Ref. Nosov2024 (black points).
• Figure 3: The spectrum (a) and decay rate (b) of superconducting phase mode at various plasma frequencies calculated at $\Gamma = 0.018 \Delta_0$ and $T=0.1 T_c$, which is shown as a horizontal dashed line. Color dashed lines in panel (a) are results of analytical calculation making use of Eq. (\ref{['NG_analytical']}).
• Figure 4: The spectrum of superconducting phase mode at various temperatures: (a) - $\hbar\omega_p = 1.79 \Delta_0$ and $\Gamma = 0.0018 \Delta_0$, (b) - $\hbar\omega_p = 1.79\Delta_0$ and $\Gamma = 0.018 \Delta_0$, (c) - $\hbar\omega_p = 4.28 \Delta_0$ and $\Gamma = 0.0018 \Delta_0$, (d) - $\hbar\omega_p = 4.28 \Delta_0$ and $\Gamma = 0.018 \Delta_0$.
• Figure 5: Hybridization between NG and magnon modes. Left column: $\hbar\omega_p = 4.28 \Delta_0$; right column - $\hbar\omega_p = 42.8\Delta_0$. Top row: excitations propagating along the $x$-axis; bottom row: along the $y$-axis. The anticrossing strength $\delta \omega_a$ is defined in Eq. \ref{['domega_a']}. The parameters used for the calculations are as follows. Superconducting subsystem: $\Gamma = 0.018 \Delta_0$, $T=0.1 T_c$, $\xi =3.5 nm$Onar2016 and $\sigma = 3 \cdot 10^{14} c^{-1}$ (normal state conductivity). Magnetic subsystem: $\gamma = 1.76 \cdot 10^7 G^{-1} s^{-1}$, $\omega_0 = \gamma K = 10^{-17} erg \approx 0.018 \Delta_0$, $D_m=5 \cdot 10^{-29} erg \cdot cm^2$Xiao2010 and $4 \pi M_s \approx 2 \cdot 10^3 Oe$kajiwara2010. The exchange field is set to $h_0 \approx 4.16 \Delta_0$. Insets in panels (a) and (b) display the anticrossing strength $\delta \omega_a$ as a function of the temperature.