Chirality-resolved spectroscopy of Caroli-de Gennes-Matricon states in multiband FeTe$_{1-x}$Se$_{x}$ superconductors

Using terahertz Faraday magneto-optical spectroscopy, researchers directly resolved the helicity and band origin of quantized Caroli-de Gennes-Matricon states in multiband FeTe$_{1-x}$Se$_x$ superconductors, enabling the independent determination of quasiparticle lifetimes and other key parameters while providing dynamical evidence for multiband vortex-core excitations.

The Gist

Imagine a superconductor as a perfectly smooth, frictionless highway where electricity flows without losing any energy. But sometimes, if you push this highway too hard with a magnetic field, tiny whirlpools (called "vortices") form in the flow. Inside the center of these whirlpools, the smooth flow breaks down, and the electrons get trapped in a special, swirling dance.

This paper is about taking a high-speed, color-coded camera to watch that dance and figure out exactly who is dancing and how they are moving.

The Dance Floor: The "CdGM" States

In the middle of these magnetic whirlpools, electrons get stuck in specific energy levels, like steps on a staircase. Physicists call these steps Caroli–de Gennes–Matricon (CdGM) states.

Think of these steps like a spiral staircase inside a tornado. The electrons can only stand on specific steps, and they have to spin in a specific direction to stay there.

• The Problem: In most materials, these steps are so close together and the electrons move so chaotically that you can't tell them apart. It's like trying to count individual raindrops in a heavy storm.
• The Solution: The researchers used a special material called FeTe$_{1-x}$Se$_x$ (a mix of iron, tellurium, and selenium). This material is special because the "steps" are far apart, and the electrons move cleanly enough that the steps are distinct.

The Camera: Terahertz Light and "Handedness"

To see these steps, the scientists used Terahertz light (a type of invisible light between microwaves and infrared). But they didn't just shine a flashlight; they used a very specific trick involving polarization.

Imagine light as a spinning top. It can spin clockwise (right-handed) or counter-clockwise (left-handed).

• The Analogy: Think of the electrons in the whirlpool as dancers. Some dancers (the "electron-like" ones) only like to spin counter-clockwise. Others (the "hole-like" ones) only like to spin clockwise.
• The Magic: When the scientists shone counter-clockwise spinning light, it made the counter-clockwise dancers jump up a step. When they shone clockwise spinning light, it made the clockwise dancers jump.

Because the light and the dancers have to match their "handedness" (chirality) to interact, the scientists could tell exactly which type of electron was doing what. It's like having a lock that only opens with a left-handed key, allowing them to count the left-handed dancers separately from the right-handed ones.

What They Found

By watching how the light twisted as it passed through the material (a phenomenon called Faraday rotation), they discovered:

1. Two Different Groups: They confirmed that there are indeed two distinct groups of dancers (electron bands and hole bands) inside the whirlpools, and they respond to light differently.
2. Measuring the Dance: They could measure how long the dancers stayed on a step before falling off (their "lifetime"), how heavy they felt (their "mass"), and how big the whirlpool was (the "coherence length").
3. Changing the Mix: They tested different versions of the material by changing the ratio of Tellurium to Selenium. They found that changing this mix is like changing the music on the dance floor: it changes how many dancers are on the floor and how long they can keep dancing.
   • In one mix, the "electron" dancers were the main crowd.
   • In another mix, the "hole" dancers were more balanced with the electrons.

Why It Matters

Before this, scientists could only see the "static" picture of these whirlpools (like a frozen photo). This paper is the first to use light to see the dynamic movement and the specific "handedness" of the particles inside.

They proved that Terahertz magneto-optics is a powerful new tool. It's like upgrading from a black-and-white photo to a 3D, slow-motion, color-coded video that lets you see the individual steps of the quantum dance inside a superconductor. This helps us understand how these materials work, which is a crucial step toward building better superconductors for the future.

Technical Summary

Technical Summary: Chirality-Resolved Spectroscopy of Caroli–de Gennes–Matricon States in Multiband FeTe$_{1-x}$Se$_x$ Superconductors

Problem and Motivation
In type-II superconductors within the Abrikosov phase, vortex cores govern low-energy dissipation and electrodynamics. While the classic Bardeen–Stephen model treats vortex cores as normal-metal regions (valid in the dirty limit), clean superconductors ($l > \xi_0$) host discrete bound states known as Caroli–de Gennes–Matricon (CdGM) states. In conventional superconductors, the large coherence length and small gap-to-Fermi-energy ratio ($\Delta/E_F$) cause these levels to be thermally and lifetime-broadened, preventing spectroscopic resolution. Although high-$T_c$ cuprates possess short coherence lengths, their nodal order parameters create a quasi-continuous spectrum that obscures discrete states.

Iron-based superconductors, specifically FeTe$_{1-x}$Se$_x$, offer a favorable alternative due to their short coherence lengths, large $\Delta/E_F$ ratio, and nodeless $s_{\pm}$ order parameter. However, while scanning tunneling spectroscopy (STS) has resolved static local densities of states in these materials, it cannot access the helicity (handedness) of vortex-core quasiparticles imposed by the magnetic field. The authors aim to resolve the helicity and band origin of these quasiparticles dynamically.

Methodology
The study employs terahertz (THz) Faraday magneto-optical spectroscopy on epitaxial FeTe$_{1-x}$Se$_x$ thin films (specifically $x=0.15$ and $x=0.38$) with critical temperatures ($T_c$) around 13 K. The experimental setup utilizes circularly polarized THz radiation generated by a Hg-arc lamp and a Martin-Puplett interferometer, passing through a superconducting magnet.

The core physical principle relies on strict angular-momentum selection rules for optical transitions between CdGM levels. Because photons carry angular momentum $\hbar$, transitions between occupied and unoccupied CdGM states are polarization-selective:

• Electron-like carriers absorb left-handed circularly polarized radiation (transitions from $\mu = -1/2$ to $\mu = +1/2$).
• Hole-like carriers absorb right-handed circularly polarized radiation (transitions from $\mu = +1/2$ to $\mu = -1/2$).

This polarization selectivity induces a difference in the refractive indices for left- and right-handed light ($N_+$ and $N_-$), resulting in Faraday rotation ($\theta_F$). The rotation angle is directly proportional to the difference in optical response between the two helicities, allowing the authors to probe the vortex-core dynamics and distinguish between electron and hole band contributions.

Key Results
The authors successfully resolved long-lived CdGM resonances with opposite circular polarizations for electron- and hole-like bands. Key findings include:

1. Band-Resolved Dynamics: The Faraday rotation spectra exhibit distinct signs and frequency dependencies for the two samples. For $x=0.38$, the rotation is predominantly positive, while for $x=0.15$, it changes sign across the frequency range. This confirms that both electron and hole CdGM states contribute to the response, with their relative weights shifting with isovalent substitution $x$.
2. Parameter Extraction: By fitting the magnetic field and temperature dependence of the Faraday rotation to a theoretical model involving frequency-dependent vortex mass and optical conductivity, the authors independently determined:
   • Quasiparticle Lifetimes ($\tau$): Ranging from 0.41 ps to 0.80 ps depending on the band and composition.
   • Vortex Masses: The effective vortex mass per unit length was derived, showing distinct values for electron and hole bands.
   • Coherence Lengths ($\xi_0$) and Upper Critical Fields ($B_{c2}$): Estimated for each band. For $x=0.38$, both condensates have similar coherence lengths (~1.7–2.0 nm). For $x=0.15$, the electron condensate has a longer coherence length (2.7 nm) compared to the hole condensate (1.7 nm).
   • Carrier Densities: The sum of carrier densities in the vortex core remains relatively constant, but the ratio of electron to hole density shifts with $x$.
3. Systematic Evolution: The study reveals a systematic redistribution of superfluid weight between electron and hole bands as a function of the Te/Se substitution ($x$). The "chemical pressure" from substitution modifies the relative sizes of the Fermi pockets, altering carrier densities and lifetimes.
4. Regime Verification: The derived parameters confirm that these films operate in the "moderately clean limit" ($\omega_0 \tau \gtrsim 1$), where well-defined resonances exist, and the mean free path is comparable to or larger than the coherence length.

Significance and Claims
The paper establishes terahertz magneto-optics as a direct probe of helical vortex-core excitations. The primary significance lies in the ability to:

• Resolve Helicity: Directly access the chirality of vortex-core quasiparticles, a capability not available in static STS measurements.
• Dynamical Evidence: Provide dynamical evidence for multiband CdGM states in iron-based superconductors, distinguishing the contributions of electron and hole bands to the vortex mass and superfluid density.
• Independent Characterization: Enable the independent determination of coherence lengths and upper critical fields for individual bands within a multiband superconductor without relying on assumptions about interband coupling.

The authors conclude that this magneto-optical approach provides a route to chirality-resolved spectroscopy of vortex matter in complex superconductors, applicable to any type-II superconductor with sufficiently short coherence lengths, including heavy-fermion and engineered heterostructures. The work does not claim to resolve interband coupling effects, noting that the analysis was performed within the framework of non-interacting condensates.