Ferromagnetic resonance modulation in topological materials with bulk--boundary coexistence

This paper extends ferromagnetic resonance (FMR) modulation theory to topological materials with coexisting bulk and boundary states, applying it to a $d$-wave superconductor to reveal distinct excitation peaks and temperature-dependent decay behaviors that demonstrate the comparable contributions of both states to the FMR response.

The Gist

Imagine you are trying to understand how a crowd of people moves in a large, complex building. Usually, you might study the people walking through the main halls (the bulk) or the people huddled in the quiet corners (the boundary). But what happens when the building is designed so that the main hall and the corner are actually the same space, and people are moving in both places at the exact same time?

This is the puzzle physicists are trying to solve with topological materials. These are special materials where the "inside" and the "surface" behave in a unique, intertwined way.

Here is a simple breakdown of what this paper does, using everyday analogies:

1. The Problem: A Broken Ruler

Scientists have a tool called Ferromagnetic Resonance (FMR). Think of this like a tuning fork. If you strike a magnet (the tuning fork), it vibrates at a specific pitch. If you place it next to another material, that material changes the pitch and how quickly the vibration dies out (the "damping").

By listening to these changes, scientists can "hear" what the electrons in the material are doing.

However, existing theories were like rulers that only measured straight lines. They worked great for materials that were either all "inside" or all "surface." But when the inside and surface mix together (which happens in topological materials), the old rulers broke. They couldn't tell if the signal was coming from the bulk or the boundary.

2. The Solution: A New, Flexible Ruler

The authors of this paper built a new mathematical framework (a new ruler) that can measure both the "inside" and the "surface" of a material at the same time.

They created a model where a magnet (the tuning fork) sits on top of a semi-infinite material (like a very deep ocean). Their math naturally accounts for the waves crashing on the surface and the currents deep below, treating them as equal partners in the dance.

3. The Experiment: The Superconductor Dance Floor

To test their new ruler, they looked at a specific type of material: a d-wave superconductor (a material that conducts electricity with zero resistance).

Imagine this superconductor as a dance floor.

• The Bulk (Inside): Most dancers are moving in a chaotic, energetic crowd.
• The Boundary (Edge): There is a special VIP lane along the edge where a few dancers move in a very specific, synchronized way (these are called Andreev Bound States).

In this specific material, the VIP lane and the main crowd exist at the same energy level. They are dancing together.

4. The Discovery: Two Distinct Beats

When they applied their new theory to this "dance floor," they found two unique "beats" (peaks in the signal) that proved the inside and outside were interacting:

• Beat #1: The "Edge-to-Edge" Echo (Low Energy)

  • The Analogy: Imagine two dancers in the VIP lane bumping into each other. Because they are so close and synchronized, they create a very loud, sharp echo right at the start of the music (near zero energy).
  • The Science: This is a sharp peak caused by electrons jumping between the edge states. It's a clear signature that the "VIP lane" exists.

• Beat #2: The "Edge-to-Bulk" Jump (Higher Energy)

  • The Analogy: Now imagine a dancer from the VIP lane suddenly jumping into the main crowd, or a dancer from the crowd jumping into the VIP lane. This requires a bit more energy, like a big leap.
  • The Science: This creates a second peak at a higher energy level (the superconducting gap). It shows the connection between the surface and the deep interior.

5. Why It Matters: Listening to the Material

The paper also looked at how temperature changes the music.

• At very low temperatures: The signal fades away slowly, like a drumbeat that keeps going (a "power-law decay"). This tells us the edge states are very stable.
• At medium temperatures: The signal drops off quickly, like a lightbulb being switched off (an "exponential decay").

The Big Takeaway:
This research is like giving scientists a new pair of ears. Before, they might have heard a muffled sound and guessed what was happening. Now, with this new theory, they can clearly distinguish between the "inside" and the "outside" of these complex materials.

This is crucial because it helps us identify topological materials (which could be the key to future quantum computers) without needing to build expensive, microscopic machines to look inside. We can just "listen" to their magnetic vibrations and know exactly what's going on.

In short: The authors built a universal translator that lets us understand how the "inside" and "outside" of exotic materials talk to each other, revealing hidden secrets about how these materials conduct electricity and magnetism.

Technical Summary

1. Problem Statement

Ferromagnetic Resonance (FMR) modulation is a powerful technique for probing spin dynamics in adjacent fermionic systems (FS) interfaced with a ferromagnetic insulator (FI). While existing microscopic theories successfully describe FMR modulation in translationally invariant bulk systems or isolated boundary states, they fail to address topological materials where bulk and boundary states coexist at the same energy.

In gapless topological phases (e.g., nodal topological superconductors and topological semimetals), bulk nodal quasiparticles and topologically protected boundary modes (such as Andreev Bound States) coexist. Previous formulations could not treat these two sectors on an equal footing, making it difficult to disentangle their respective contributions to the FMR signal (specifically the resonance field shift $\Delta H$ and the Gilbert damping enhancement $\delta \alpha_G$). The authors aim to develop a unified microscopic framework to analyze FMR modulation in such bulk–boundary coexisting systems.

2. Methodology

The authors developed a microscopic theory for an FI/FS bilayer system where the FS is a semi-infinite fermionic system.

• Hamiltonian Formulation: The system consists of a 2D Heisenberg model for the FI and a semi-infinite tight-binding model for the FS. The interaction is modeled via an interfacial exchange coupling ($H_{int}$), which includes randomness to account for realistic interface disorder.
• Surface Green's Function Approach: The core of the methodology is the calculation of the surface dynamical spin susceptibility ($\chi^R_{q\parallel}$) of the semi-infinite FS.
  • Instead of treating bulk and edge separately, the authors utilize the surface Green's function ($g^R_{q\parallel,0}$) derived from a semi-infinite system.
  • They employ a Möbius transformation technique to solve the Dyson equation for the surface Green's function in closed form. This involves diagonalizing a $2N \times 2N$ matrix derived from the bulk Hamiltonian parameters.
  • This approach naturally incorporates both bulk and boundary states without artificial separation, as the surface Green's function inherently contains contributions from all states accessible at the surface.
• Magnon Self-Energy: The FMR modulation parameters ($\Delta H$ and $\delta \alpha_G$) are derived from the magnon self-energy ($\Sigma^R_k$), which is expressed as a convolution of the interfacial coupling strength and the surface spin susceptibility.
• Specific Application: The theory is applied to the (110) surface of a $d$-wave superconductor. This system is chosen because it hosts zero-energy surface Andreev Bound States (ABS) (edge states) that coexist with bulk nodal quasiparticles. The authors compare this with the (010) surface, which lacks these specific edge states.

3. Key Contributions

• Unified Framework: The paper establishes a general, broadly applicable framework for calculating FMR modulation in systems with bulk–boundary coexistence. It moves beyond effective models to a rigorous microscopic treatment using surface Green's functions.
• Disentanglement of Excitations: The theory allows for the clear identification of distinct excitation channels:
  1. Edge-to-Edge: Excitations within the boundary states.
  2. Edge-to-Bulk: Excitations from boundary states to bulk states.
• Quantitative Prediction for $d$-wave Superconductors: The authors provide specific predictions for the frequency and temperature dependence of the Gilbert damping constant ($\delta \alpha_G$) on the (110) surface of a $d$-wave superconductor.

4. Key Results

The application of the theory to the (110) surface of a $d$-wave superconductor yields two characteristic features in the frequency dependence of $\delta \alpha_G$ at $T=0$:

1. Edge-to-Edge Peak (Zero Energy):

   • A pronounced peak appears near $\hbar\omega \sim 0$.
   • Origin: Quasiparticle excitations within the zero-energy surface Andreev Bound States (ABS).
   • Scaling: The peak is followed by a power-law decay of $\delta \alpha_G \propto \omega^{-3}$. This scaling arises from the Lorentzian density of states of the ABS broadened by impurity scattering.
   • Temperature Dependence: At low temperatures, $\delta \alpha_G$ decays as $T^{-1}$, governed by the Fermi distribution function.

2. Edge-to-Bulk Peak (Gap Energy):

   • An additional peak emerges near the superconducting gap energy ($\hbar\omega \sim \Delta_0$).
   • Origin: Quasiparticle excitations from the edge states to the bulk continuum.
   • Temperature Dependence: At intermediate temperatures, this contribution leads to an exponential decay in the low-frequency limit.

Comparison with (010) Surface:
On the (010) surface (which lacks the zero-energy ABS), $\delta \alpha_G$ shows a smooth $\omega^2$ dependence at low frequencies and lacks the distinct zero-energy peak, confirming that the unique features on the (110) surface are direct signatures of the coexisting edge states.

Absence of Coherence Peak:
Unlike $s$-wave superconductors, the $d$-wave case does not exhibit a coherence peak near the transition temperature ($T_c$) due to the suppression of the density of states divergence by the bulk nodal quasiparticles.

5. Significance and Implications

• New Probe for Topological States: The study demonstrates that FMR modulation can serve as a spin-sensitive probe for detecting Andreev Bound States and topological boundary modes, complementing traditional tunneling spectroscopy (which detects zero-bias conductance peaks).
• Experimental Feasibility: The predicted edge-to-edge and edge-to-bulk peaks fall within experimentally accessible frequency ranges (GHz), particularly for superconductors with low $T_c$ (e.g., heavy-fermion systems), where the energy scales match the GHz regime.
• General Applicability: The theoretical framework is not limited to $d$-wave superconductors. It is applicable to a wide class of topological materials, including nodal topological superconductors and topological semimetals, providing a tool to quantitatively analyze the interplay between bulk and boundary dynamics in spintronics.
• Disorder and Interface Effects: The paper highlights the role of interface disorder and impurity scattering in broadening the ABS features, suggesting that realistic interface modeling is crucial for interpreting experimental FMR data in topological heterostructures.

In summary, this work bridges the gap between microscopic spin dynamics theory and the unique electronic structure of topological materials, offering a robust method to detect and characterize the coexistence of bulk and boundary states through magnetic resonance techniques.