Anomalous terahertz nonlinearity in disordered s-wave superconductor close to the superconductor-insulator transition

This study reveals that in strongly disordered NbN thin films near the superconductor-insulator transition, an anomalous third-harmonic generation signal persists above the critical temperature due to quantum path interference between unpaired electrons and Cooper pairs, which subsequently couples strongly to the driven Higgs mode below the transition.

The Gist

The Big Picture: Listening to the "Heartbeat" of Superconductors

Imagine a superconductor as a giant, synchronized dance floor. When it gets cold enough, all the electrons (the dancers) pair up and move in perfect unison. In physics, this synchronized movement creates a specific "heartbeat" or vibration called the Higgs mode. Scientists use a special kind of light (Terahertz light) to tap on this dance floor and listen for that heartbeat.

Usually, if the dance floor is messy or the dancers are tripping over each other (disorder), the heartbeat gets quiet or disappears. However, this paper discovered something surprising: when the dance floor is extremely messy, right on the edge of stopping the dance entirely, a strange new sound appears that shouldn't be there at all.

The Experiment: Four Different Dance Floors

The researchers studied thin films of a material called Niobium Nitride (NbN). They made four versions of these films, each with a different level of "messiness" (disorder):

1. Clean: Very organized, dancers move smoothly.
2. Moderately Messy: Some bumps in the road.
3. Very Messy: The dancers are struggling to stay in sync.
4. Chaotic: So messy that the dancing stops completely (it becomes an insulator).

They shined a low-frequency "tap" (0.42 THz light) on these films and listened for a "triple tap" (the third harmonic, or THG) that happens when the material reacts non-linearly.

The Surprise: A Ghostly Signal Above Freezing

The Expectation:
In the clean and moderately messy films, the "triple tap" signal only appeared when the material was superconducting (cold). Once it warmed up and the superconductivity stopped, the signal vanished completely. This is normal.

The Discovery:
In the Very Messy film (near the point where superconductivity dies), they found something weird:

• Above the freezing point (Normal State): Even when the material wasn't superconducting, a faint "triple tap" signal persisted. It was like hearing a faint echo of the dance music even after the dancers had gone home.
• Below the freezing point (Superconducting State): When they cooled it down, the signal didn't just get louder; it got chaotic. The single clear "beat" split into multiple overlapping beats, creating a complex, wobbly sound.

Ruling Out the Suspects

The scientists had to figure out why this weird signal existed in the messy film when it was warm.

Suspect 1: "Ghost Dancers" (Superconducting Fluctuations)

• The Theory: Maybe tiny, invisible islands of superconductivity were still floating around in the warm material, creating the signal.
• The Test: They applied a strong magnetic field (like a giant magnet) to the warm, messy film. This should crush any tiny superconducting islands.
• The Result: The signal did not change. The magnetic field killed the superconductivity, but the "ghostly" signal remained.
• Conclusion: The signal is not caused by superconducting fluctuations. It is an intrinsic property of the messy material itself, likely caused by how the disorder changes the way electrons move and scatter.

Suspect 2: The "Echo" (Reflections)

• The Theory: Maybe the signal was just light bouncing around inside the machine.
• The Test: They checked the timing and intensity.
• The Result: The signal was too strong and happened at the wrong times to be a simple echo.

The "Multi-Peak" Mystery: A Choir of Islands

When the messy film was cooled down and became superconducting, the signal became a mess of multiple peaks.

• The Analogy: Imagine a choir. In a clean film, everyone sings the same note perfectly (one clear peak). In the messy film, the dancers have formed small, isolated groups (islands).
  • Group A is singing a note based on their local rhythm.
  • Group B is singing a slightly different note.
  • The "normal" electrons (the ones not dancing) are also making a sound.
• The Interference: Because these groups are slightly out of sync and singing different notes, their sounds crash into each other. This creates a "beating" effect (like two slightly out-of-tune guitar strings played together) and splits the sound into multiple peaks.
• The Cause: The disorder created a patchwork quilt of superconducting islands. The signal is the result of the "Higgs mode" (the superconducting heartbeat) interfering with the "normal state" signal within these tiny islands.

The Gold Comparison

To prove that this "messy material signal" wasn't unique to superconductors, they tested thin films of Gold (which never superconducts).

• They made Gold films with different levels of messiness.
• They found the exact same pattern: a weak signal that gets stronger as the material gets messier, peaks at a certain level of messiness, and then fades away if it gets too messy.
• This confirmed that the "ghostly signal" is a universal feature of disordered metals, not a secret superconducting trick.

Summary of Findings

1. Disorder creates a new signal: In extremely messy superconductors, a strange non-linear signal appears even when the material is warm (not superconducting).
2. It's not superconductivity: This warm signal is caused by the disorder itself, not by hidden superconducting islands.
3. The Higgs mode is still king: When the material gets cold, the superconducting "Higgs mode" takes over and makes the signal much stronger.
4. Messiness creates complexity: The chaotic, multi-peaked sound in the cold, messy state is a fingerprint of the material being a patchwork of tiny superconducting islands that are all singing slightly different songs.

In short, the paper shows that making a superconductor messy doesn't just break it; it reveals a hidden, complex layer of physics where disorder and superconductivity interact in surprising ways.

Technical Summary

Technical Summary: Anomalous Terahertz Nonlinearity in Disordered s-wave Superconductor Close to the Superconductor-Insulator Transition

Problem Statement
The detection of the Higgs mode (amplitude mode) in superconductors via nonlinear terahertz (THz) spectroscopy is a central topic in condensed matter physics. While disorder is known to enhance the coupling between the THz field and the Higgs mode in moderately disordered systems, the behavior of nonlinear responses in the strongly disordered regime near the superconductor-insulator transition (SIT) remains unclear. Specifically, it is unknown how the emergence of superconducting granularity (islands of pairing separated by weak regions) and the persistence of a pseudogap above the critical temperature ($T_c$) affect the nonlinear THz response. Previous studies have suggested that normal-state nonlinearity in disordered NbN might arise from superconducting fluctuations, but this hypothesis lacks definitive experimental verification. Furthermore, the fate of the Higgs mode and its nonlinear signature in highly inhomogeneous superconductors has not been experimentally investigated.

Methodology
The authors conducted systematic third-harmonic generation (THG) studies on NbN thin films with four distinct disorder levels, characterized by their Ioffe-Regel parameters ($k_F l$), ranging from clean ($k_F l \sim 7.2$) to strongly disordered near the SIT ($k_F l \sim 2.5$). The experiments utilized intense 0.42 THz pump pulses to drive the samples, recording the resulting 1.26 THz (3$\omega$) radiation.

• Temperature Dependence: THG intensity and spectral shapes were measured across a wide temperature range, from below $T_c$ up to 300 K.
• Magnetic Field Tuning: To distinguish between superconducting fluctuations and intrinsic normal-state mechanisms, magnetic fields (up to 9 T) were applied perpendicular to the sample surface to suppress global superconducting coherence while preserving localized pairing in the strongly disordered sample.
• Comparative Control: Non-superconducting gold (Au) films with varying disorder levels ($k_F l$ from 7.4 to 92.6) were measured to isolate disorder-induced nonlinearity from superconducting contributions.
• Spectral Analysis: Time-domain waveforms were analyzed via Fast Fourier Transform (FFT) to identify multi-peak structures and beating features. Optical conductivity data were fitted using both Mattis-Bardeen and Larkin-Ovchinnikov (LO) models to quantify mesoscopic inhomogeneity.

Key Results

1. Anomalous Normal-State THG: In the strongly disordered sample ($k_F l \sim 2.5$) near the SIT, a weak but detectable THG signal persists well above $T_c$, extending up to $\sim 10 T_c$. This signal is absent in cleaner superconducting samples and non-superconducting counterparts with low disorder.
2. Magnetic Field Independence: Crucially, the normal-state THG signal in the strongly disordered sample remains unchanged even under high magnetic fields (up to 9 T) that fully suppress global superconductivity. This observation rules out superconducting fluctuations as the origin of the normal-state nonlinearity.
3. Disorder-Induced Mechanism: Comparative measurements on Au films reveal a "dome-shaped" dependence of THG intensity on disorder ($k_F l$), peaking around $k_F l \sim 60-70$. This suggests a universal disorder-enhanced nonlinearity in the normal state, likely arising from disorder-induced modifications to the electronic band structure (anharmonicity) or a disparity between energy and momentum relaxation times ($\tau_E / \tau_M$), rather than superconducting correlations.
4. Higgs Mode Dominance Below $T_c$: Upon cooling below $T_c$, the THG intensity in the strongly disordered sample increases sharply, indicating that the driven Higgs mode remains the dominant contribution to the nonlinear response, similar to cleaner samples.
5. Multi-Peak Spectral Structure: Unlike the single-peak THG spectra observed in cleaner samples, the strongly disordered sample exhibits a broadened, multi-peak structure below $T_c$. This feature disappears when superconductivity is suppressed by magnetic fields or heating. The authors attribute this to quantum path interference between distinct channels: the Higgs mode response from Cooper pairs within emergent superconducting islands and the normal-state nonlinearity from unpaired electrons.

Significance and Claims
The paper claims to provide the first experimental investigation of the nonlinear THz response of the Higgs mode near the SIT. The primary significance lies in:

• Distinguishing Origins of Nonlinearity: The work definitively separates disorder-induced normal-state nonlinearity from superconducting fluctuation effects, demonstrating that the former is robust against magnetic fields and intrinsic to the disordered electronic structure.
• Probing Mesoscopic Inhomogeneity: The authors propose that the interference-induced multi-peak structure in the THG spectrum serves as a sensitive probe for mesoscopic superconducting inhomogeneity (granularity) in strongly disordered systems.
• Robustness of the Higgs Mode: The findings confirm that despite strong disorder and the emergence of granularity, the Higgs mode remains a well-defined collective excitation that dominates the nonlinear response below $T_c$.
• Universal Disorder Effect: The observation of a dome-shaped disorder dependence in both NbN and Au suggests that disorder-enhanced THG is a universal phenomenon in metals, independent of superconductivity.

The authors conclude that their work advances the understanding of how disorder tunes nonlinear responses in superconductors and offers a new perspective on the interplay between normal-state electrons and the superconducting Higgs mode in inhomogeneous systems.