Hybrid collective excitations in topological superconductor/ferromagnetic insulator heterostructures

This paper develops a linear response theory for topological superconductor/ferromagnetic insulator hybrids, revealing that spin-momentum locking drives a hybridization between magnons and the superconducting phase mode to create composite excitations, while the amplitude mode remains decoupled, offering a new mechanism for spin-signal interconversion in superconducting spintronics.

The Gist

Imagine you have two very different neighbors living next to each other. One is a Superconductor, a material where electricity flows with zero resistance, like a perfectly smooth, frictionless ice rink. The other is a Ferromagnetic Insulator, a material that acts like a giant, rigid magnet but doesn't conduct electricity, like a solid block of iron.

Usually, these two neighbors ignore each other. But in this paper, the scientists discover that if you put them in a special "topological" neighborhood, they start a fascinating dance where they can't help but influence each other's movements.

Here is the breakdown of what's happening, using simple analogies:

1. The Special Neighborhood: Spin-Momentum Locking

In a normal superconductor, electrons are like a crowd of people walking in any direction. But in this Topological Superconductor (TS), the electrons are special. They have a rule called "Spin-Momentum Locking."

Think of it like a dance floor where every dancer is forced to hold hands with their partner in a specific way: if you walk forward, you must spin clockwise; if you walk backward, you must spin counter-clockwise. You cannot walk without spinning. This "lock" is the secret ingredient that makes the magic happen.

2. The Two Dancers: Magnons and the "Phase" Wave

Inside this system, there are two types of "waves" or vibrations:

• The Magnon (The Magnetic Wave): In the magnetic neighbor (the Ferromagnetic Insulator), the atoms act like tiny compass needles. When they wiggle together, it creates a wave called a magnon. Imagine a stadium wave where everyone stands up and sits down in sequence.
• The Nambu-Goldstone Mode (The Phase Wave): In the superconductor, the electrons are all dancing in perfect unison. Sometimes, the timing of their dance shifts slightly. This shift is called the Phase Mode (or Nambu-Goldstone mode). It's like the whole crowd suddenly deciding to clap a split-second earlier or later than before.

3. The Big Discovery: They Start Dancing Together

The paper shows that because of the "Spin-Momentum Locking" rule, these two waves can't stay separate.

• The Connection: When the magnetic neighbor wiggles (the Magnon), it pushes on the electrons in the superconductor. Because of the locking rule, this push doesn't just make the electrons move; it forces them to change the timing of their dance (the Phase Mode).
• The Hybrid: This creates a new, hybrid creature: a Magnon-Phase Hybrid. It's like a "Frankenstein" wave that is part magnetic wiggle and part superconducting timing shift. They are now so linked that you can't have one without the other.

4. The "Higgs" Mode: The Wallflower

Superconductors have another type of vibration called the Higgs Mode (or Amplitude Mode). If the Phase Mode is about when the electrons dance, the Higgs Mode is about how hard they dance (the strength of the pairing).

The scientists found that the magnetic neighbor cannot get the Higgs Mode to dance. No matter how much the magnet wiggles, the Higgs Mode stays still. It's like a wallflower at a party who refuses to dance with the magnetic neighbor. The paper proves mathematically that this interaction is impossible in this specific setup.

5. Why Does This Matter? (The Superpower)

Why should we care about this dance?

• Translating Languages: This hybrid dance acts as a translator. It can take a "spin signal" (information carried by magnetic waves) and turn it into a "charge signal" (information carried by superconducting waves), and vice versa.
• Superconducting Spintronics: This is a new field of electronics that uses both spin and superconductivity. Because this hybrid wave moves with almost no energy loss (thanks to the superconductor), it could lead to ultra-fast, ultra-efficient computer chips that don't get hot.

Summary

The paper describes a scenario where a magnetic material and a special superconductor are forced to interact. Due to a unique rule where electron movement is tied to their spin, the magnetic waves (magnons) and the superconducting timing waves (Phase modes) merge into a single, hybrid wave. However, the superconducting strength waves (Higgs mode) remain unaffected. This discovery opens the door to new technologies that can convert magnetic information into electrical signals with incredible efficiency.

Technical Summary

1. Problem Statement

The paper addresses the dynamical coupling between magnetic excitations (magnons) and superconducting collective modes in hybrid structures consisting of a Topological Superconductor (TS) and a Ferromagnetic Insulator (FI).

• Context: While static proximity effects in Superconductor/Ferromagnet (S/F) systems are well-studied (e.g., spin-valve effects, triplet conversion), the dynamical coupling between magnons and superconducting collective modes (specifically the Nambu-Goldstone phase mode and the Higgs amplitude mode) remains largely unexplored.
• Specific Gap: In 3D superconductors, the phase mode is typically gapped by Coulomb interactions (hybridizing with plasmons). However, in 2D systems, the phase mode remains gapless. The authors investigate whether this gapless 2D phase mode can hybridize with magnons in a TS/FI heterostructure, specifically leveraging the unique spin-momentum locking of the TS surface state.
• Key Question: Does the spin-momentum locking in a TS enable a linear coupling between magnons and the superconducting order parameter (OP) collective modes, leading to hybrid excitations?

2. Methodology

The authors develop a linear response theory based on the non-equilibrium quasiclassical Keldysh–Usadel formalism.

• System Model: A 2D conductive surface state of a TS (modeled as a helical metal with full spin-momentum locking) interfaced with a thin-film FI.
• Hamiltonian: The TS electrons are described by a Hamiltonian including spin-orbit coupling (SOC), chemical potential, and an exchange field ($h$) induced by the FI via the proximity effect. The ground state is a "helical state" with a spatially inhomogeneous OP phase ($\Delta(r) = \Delta e^{iqr}$) to cancel spontaneous currents.
• Dynamics:
  • Superconducting Side: Described by the Usadel equation for the quasiclassical Green's function, expanded to first order in perturbations (OP fluctuations $\delta\Delta$, exchange field fluctuations $\delta h$, and electric potential $\delta\phi$).
  • Magnetic Side: Described by the Landau-Lifshitz-Gilbert (LLG) equation, including a spin-torque term arising from the back-action of the TS spin polarization on the FI magnetization.
  • Electrostatics: The Poisson equation is solved self-consistently to account for charge oscillations (plasmon coupling) in the 2D system.
• Coupling Mechanism: The theory integrates the linear response of the OP to the magnon-induced exchange field and the response of the FI magnetization to the AC supercurrent (via the inverse magnetoelectric effect).

3. Key Contributions

1. Theoretical Framework: Established a comprehensive linear response theory for dynamical proximity effects in TS/FI hybrids, combining Keldysh-Usadel formalism with LLG dynamics.
2. Identification of Hybridization: Demonstrated that spin-momentum locking is the crucial ingredient that enables a linear coupling between magnons and the superconducting Nambu-Goldstone (NG) phase mode.
3. Exclusion of Higgs Mode: Proved analytically that the Higgs (amplitude) mode does not couple to magnons at the linear order in this specific helical ground state, distinguishing it from the phase mode.
4. Anisotropy: Revealed that the coupling strength is highly anisotropic, depending on the propagation direction of the magnon relative to the equilibrium magnetization.

4. Key Results

A. Hybridization of Magnons and NG Mode

• Mechanism:
  1. A magnon ($\delta m$) induces a dynamic exchange field ($\delta h$) in the TS.
  2. Due to spin-momentum locking, $\delta h$ linearly excites the phase of the superconducting order parameter ($\delta \Delta_p$), but not the amplitude.
  3. The oscillating phase mode generates an AC supercurrent.
  4. Via the direct magnetoelectric effect (spin-momentum locking), this current induces an AC spin polarization ($s$) in the TS.
  5. This spin polarization exerts a torque on the FI magnetization (via the LLG equation), exciting magnons.
• Result: This feedback loop creates composite magnon-NG excitations.
• Spectrum: The dispersion relations of the uncoupled magnon and NG modes exhibit an anticrossing (hybridization gap) when their frequencies intersect.
• Coupling Strength ($\delta \omega_a$):
  • Scales linearly with the interface exchange coupling constant ($J_{ex}$) and the equilibrium exchange field ($h_0$).
  • Scales with the square root of the magnon gap frequency ($\omega_0$) at low temperatures.
  • Is anisotropic: Maximal when the magnon wavevector is parallel to the equilibrium magnetization ($\mathbf{k} \parallel \hat{x}$) and zero when perpendicular ($\mathbf{k} \parallel \hat{y}$).

B. Absence of Higgs-Magnon Coupling

• Microscopic calculations (Appendix B) show that the coefficient $F^a_{\omega_r, k}$, representing the linear response of the amplitude mode to the magnon, is exactly zero for a TS in the helical ground state.
• Consequently, the Higgs mode remains decoupled from magnons in the linear regime and does not form hybrid excitations.

C. Numerical and Analytical Validation

• Numerical solutions of the coupled matrix equations confirm the analytical predictions.
• The anticrossing strength is significant and experimentally accessible, particularly in the limit of high superconducting plasma frequency ($\tilde{\omega}_p \gg 1$), which is typical for 2D systems.
• The effect vanishes as $T \to T_c$, confirming its purely superconducting origin.

5. Significance and Implications

• Fundamental Physics: The work uncovers a new class of dynamical proximity effects where magnetic and superconducting collective modes hybridize. It highlights the unique role of topological surface states (spin-momentum locking) in mediating this interaction, which is absent in conventional s-wave superconductors without SOC.
• Superconducting Spintronics: The hybridization provides a novel mechanism for the interconversion of spin signals and spinless collective superconducting excitations.
  • Magnons can be converted into phase oscillations (and vice versa).
  • This could lead to new devices for spin-charge conversion, magnon-based superconducting logic, or low-dissipation spin transport.
• Experimental Probes: The predicted anticrossing in the excitation spectrum offers a clear signature for experimental detection (e.g., via inelastic neutron scattering or microwave spectroscopy) of the elusive NG mode in 2D superconductors and its coupling to magnetism.

In summary, the paper establishes that in TS/FI heterostructures, the interplay of spin-momentum locking and dynamical proximity effects leads to the formation of hybrid magnon-Nambu-Goldstone modes, while explicitly excluding the Higgs mode from this interaction, opening new avenues for controlling spin dynamics in superconducting systems.