Hybrid magnon -- Nambu-Goldstone excitations in topological superconductor/ferromagnetic insulator thin-film heterostructures

This paper predicts a novel dynamical proximity effect in topological superconductor/ferromagnetic insulator heterostructures where spin-momentum locking facilitates the coupling of ferromagnetic magnons and superconducting Nambu-Goldstone modes into hybrid excitations, enabling the interconversion of spin and spinless signals for advanced superconducting spintronics.

The Gist

Imagine a world where two very different neighbors live right next to each other: one is a superconductor (a material that conducts electricity with zero resistance and has a special "dance" of electrons), and the other is a magnetic insulator (a material that is magnetic but doesn't conduct electricity).

Usually, these two neighbors ignore each other's internal rhythms. But this paper discovers a secret handshake between them that creates a brand-new type of energy wave.

Here is the story of that discovery, broken down into simple concepts.

1. The Two Neighbors

• The Magnetic Neighbor (The Ferromagnetic Insulator): Think of this layer as a field of tiny compass needles all pointing in the same direction. When these needles wobble together in a wave, it's called a magnon. It's like a "spin wave" or a ripple of magnetism moving through the field.
• The Superconducting Neighbor (The Topological Superconductor): This layer is special. Inside it, electrons are locked in a very strict dance. In this specific type of superconductor, an electron's spin (its magnetic direction) is permanently glued to its momentum (which way it's moving). If it moves North, it spins Up. If it moves East, it spins Right. This is called Spin-Momentum Locking.

2. The Secret Handshake (The Proximity Effect)

When these two layers touch, they start to influence each other. This is called the "proximity effect."

In the past, scientists knew that if the magnetic neighbor wobbled, it could create some "triplet" pairs in the superconductor (like a specific type of dance partner). But this paper found something much more exciting: The magnetic wobble actually shakes the rhythm of the superconductor itself.

3. The "Ghost" Wave (The Nambu-Goldstone Mode)

Inside the superconductor, there is a special, invisible wave called the Nambu-Goldstone (NG) mode.

• Analogy: Imagine a giant trampoline. If you push down on one spot, a wave travels across it. In a normal trampoline, the wave is just the fabric moving up and down. In this superconductor, the "fabric" is the quantum phase of the electrons. The NG mode is a ripple in this phase.
• Usually, this ripple is "soft" and easy to move, but it doesn't carry a magnetic charge, so it's hard to see or talk to.

4. The Hybrid Monster (Magnon-NG Excitation)

Here is the magic trick: Because the electrons in the superconductor are "locked" (Spin-Momentum Locking), the magnetic wobble from the neighbor directly pushes on the superconductor's rhythm.

• The Result: The magnetic wave (Magnon) and the superconducting rhythm wave (NG mode) stop being separate. They merge into a hybrid creature.
• The Metaphor: Imagine a drummer (the magnet) and a singer (the superconductor). Usually, they play different songs. But because of the "lock," the drummer's beat forces the singer to change their melody instantly. Now, they are singing and drumming in perfect, inseparable unison. You can't have the beat without the melody, and you can't have the melody without the beat. They have become a Magnon-NG hybrid.

5. The Direction Matters (Anisotropy)

This connection is very picky about direction.

• If the magnetic wave moves parallel to the magnetic field, the handshake is strong, and they merge perfectly.
• If the magnetic wave moves perpendicular (sideways), the handshake breaks, and they ignore each other.
• Analogy: It's like a Velcro strip. It sticks strongly if you pull it in one direction, but if you slide it sideways, it just slides right off.

6. Why Does This Matter? (The Super-Spintronics)

Why should we care about this hybrid wave?

• The Problem: In modern electronics, we use spin (magnetism) to carry information (like in hard drives). But in superconductors, we use charge (electricity) or phase (the rhythm). These two languages don't usually speak to each other.
• The Solution: This hybrid wave acts as a translator.
  • You can send a magnetic signal (spin) into the magnetic layer.
  • It instantly turns into a superconducting rhythm wave (spinless signal).
  • This allows us to convert magnetic data into superconducting data without losing energy.

The Big Picture

This paper predicts a new way to build computers. By stacking these two special materials, we can create a bridge that converts magnetic signals into superconducting signals and back again. This could lead to a new generation of ultra-fast, ultra-efficient computers (Superconducting Spintronics) that use the best of both worlds: the speed of superconductors and the storage power of magnets.

In short: The authors found a way to make a magnetic wave and a superconducting wave hold hands and dance together, creating a new type of energy that could revolutionize how we process information.

Technical Summary

1. Problem Statement

The paper addresses a gap in the understanding of dynamic proximity effects in superconductor/ferromagnet (S/F) hybrid structures. While static proximity effects (e.g., the induction of odd-frequency triplet pairing) are well-established, the dynamic interplay between magnetic excitations (magnons) and collective superconducting modes remains less explored.

Specifically, the authors investigate the coupling between:

• Magnons: Elementary magnetic excitations in a Ferromagnetic Insulator (FI).
• Nambu-Goldstone (NG) Mode: The gapless phase mode of the superconducting order parameter.

Previous studies have shown that in conventional S/F systems, magnons can couple to triplet Cooper pairs to form "magnon-cooparons," but a direct linear coupling between magnons and the superconducting phase (NG) mode has not been demonstrated. The authors hypothesize that this coupling becomes possible in Topological Superconductor (TS) systems due to unique spin-momentum locking properties.

2. Methodology

The authors employ a microscopic theoretical framework combining several advanced techniques:

• System Model: A bilayer heterostructure consisting of a thin film of a Ferromagnetic Insulator (FI) placed on top of a Topological Superconductor (TS). The TS is modeled as having a 2D helical surface state with full spin-momentum locking (similar to topological insulators).
• Hamiltonian: The electron system in the TS is described by a Hamiltonian including kinetic energy, chemical potential, a superconducting order parameter ($\Delta$), and an exchange field ($h$) induced by the FI via interfacial exchange coupling ($J_{ex}$).
• Ground State: The equilibrium state is a helical superconducting state where the order parameter has a spatially modulated phase ($\Delta(r) = \Delta e^{iqr}$) due to the exchange field, breaking time-reversal symmetry.
• Dynamics:
  • Magnon Dynamics: Described by the Landau-Lifshitz-Gilbert (LLG) equation, including a spin-torque term arising from the back-action of the TS on the FI.
  • Superconducting Response: Analyzed using the Keldysh-Green's function formalism within the Usadel equations. This allows for the calculation of linear response functions for the order parameter and current.
• Coupling Mechanism: The authors derive the mutual linear response:
  1. How a magnon-induced dynamic exchange field ($\delta h$) perturbs the superconducting order parameter.
  2. How the superconducting phase mode (NG) generates an AC current, which via the direct magnetoelectric effect (spin-momentum locking), creates a spin polarization that exerts a torque on the FI magnetization.

3. Key Contributions

The paper makes several novel theoretical contributions:

• Discovery of Magnon-NG Hybridization: The authors predict a new type of composite excitation where magnons in the FI and the Nambu-Goldstone phase mode in the TS hybridize. This is distinct from the previously known magnon-cooparon (magnon + triplet pairs).
• Role of Spin-Momentum Locking: The study identifies full spin-momentum locking in the TS surface state as the critical ingredient enabling this coupling. In conventional superconductors without strong spin-orbit coupling, the order parameter does not couple linearly to magnons. In the TS, the mixed singlet-triplet nature of the pairing allows for a linear response of the order parameter to the magnon field.
• Suppression of Higgs Mode Coupling: Microscopic calculations reveal that while the NG mode couples strongly to magnons, the amplitude (Higgs) mode does not ($F^a_{\omega,k} = 0$). This is a unique feature of the helical ground state in TS/FI systems.
• Anisotropic Coupling: The strength of the hybridization is highly anisotropic. It depends on the angle between the magnon wave vector ($\mathbf{k}$) and the equilibrium magnetization ($\mathbf{m}_0$). The coupling is maximal when $\mathbf{k} \parallel \mathbf{m}_0$ and vanishes when $\mathbf{k} \perp \mathbf{m}_0$.

4. Results

The theoretical calculations yield the following specific results:

• Dispersion Relations:
  • In the absence of coupling ($J_{ex}=0$), the NG mode and magnon modes are distinct. The NG mode exhibits a dispersion crossover from $\omega \propto \sqrt{k}$ to $\omega \propto k$.
  • When coupling is turned on ($J_{ex} \neq 0$), the modes undergo anticrossing (avoided crossing) near their intersection points. This creates two new hybrid branches: an upper branch ($\omega_{up}$) and a lower branch ($\omega_{dn}$).
• Damping Characteristics:
  • The decay rates (damping) of the hybrid modes are calculated. Far from the anticrossing, the damping resembles that of the uncoupled modes.
  • Near the anticrossing point, the lower branch exhibits a pronounced peak in damping, indicating strong mixing and enhanced dissipation due to the energy transfer between the magnetic and superconducting sectors.
• Parameter Dependence: The splitting of the hybrid modes ($\delta \omega$) scales linearly with the interface exchange coupling and the equilibrium exchange field ($h_0$). The effect is robust for realistic parameters (e.g., YIG as the FI and FeTe$_{0.5}$Se$_{0.5}$ as the TS).
• Frequency Range: The hybridization occurs in the gigahertz frequency range (approx. 10 GHz), making it potentially accessible to experimental microwave spectroscopy.

5. Significance

The findings have profound implications for both fundamental physics and applied technology:

• Superconducting Spintronics: The paper proposes a new mechanism for the interconversion of spin signals and spinless collective excitations.
  • Magnons (spin signals) can be converted into NG mode oscillations (spinless).
  • Conversely, NG mode excitations can generate spin currents (via the magnetoelectric effect) to manipulate magnetization.
  • This offers a pathway for controlling magnetic dynamics using superconducting phase modes and vice versa, without the need for charge currents.
• Experimental Probing: The predicted anisotropic anticrossing and specific damping signatures provide a clear "fingerprint" for experimentalists to detect the elusive Nambu-Goldstone mode in superconductors, which is typically difficult to observe directly.
• New Physics in Topological Systems: It highlights the unique dynamical properties of topological superconductors, where the interplay of topology, superconductivity, and magnetism leads to emergent quasiparticles not found in conventional materials.

In summary, this work establishes a theoretical foundation for a new class of hybrid quasiparticles in TS/FI heterostructures, bridging the fields of magnonics and superconductivity through the unique properties of topological surface states.