Signatures of Dynes superconductivity in the THz response of ALD-grown NbN thin films

This study demonstrates that the terahertz optical conductivity of atomic layer deposition-grown NbN thin films deviates from standard BCS predictions and is accurately described by Dynes electrodynamics, revealing a small, temperature-independent pair-breaking rate and thickness-dependent evolution of the energy gap and superfluid density.

The Gist

The Big Picture: A Superconductor with a "Leaky" Gap

Imagine a superconductor (a material that conducts electricity with zero resistance) as a high-security fortress.

In a perfect, textbook superconductor (described by the old BCS theory), there is a massive, impenetrable moat called the "Energy Gap."

• Below the gap: No energy is allowed to cross. It's like a wall of force. Electrons are paired up (Cooper pairs) and dance in perfect harmony. They can't be broken apart by small nudges.
• Above the gap: If you hit the fortress with enough energy (like a big hammer), you can break the pairs, and electricity starts flowing with resistance again.

The Problem:
For decades, scientists have noticed that in real-world, messy superconductors (like the thin films in this study), the moat isn't perfectly dry. There are little puddles of water inside the moat. Even with very low energy, some electrons get through. The old "perfect fortress" theory couldn't explain this.

The Solution (The Dynes Model):
Scientists use a formula called the Dynes model to describe this. It suggests the moat isn't a solid wall, but a foggy, leaky barrier. There's a constant, tiny "pair-breaking rate" (let's call it Gamma, $\Gamma$) that acts like a slow leak, letting some energy slip through even when it shouldn't.

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What Did These Scientists Do?

The researchers grew very thin films of a material called Niobium Nitride (NbN) using a high-tech process called Atomic Layer Deposition (ALD). Think of ALD as building a wall one atom at a time, giving them incredible control over how thick the wall is (from 4.5 nanometers to 20 nanometers).

They then shined Terahertz (THz) light on these films.

• The Analogy: Imagine THz light as a specific type of "sonar" or "ping" that bounces off the electrons. By listening to how the sound changes, they can map out the shape of the energy gap.

The Key Discovery: The "Half-Step"

When they looked at the thickest film (20 nm), they found something the old theories missed:

1. The Old Theory (BCS): Predicted that absorption (energy being taken in) would stay at zero until the energy hit the full size of the gap ($2\Delta$). Then, boom, it would jump up.
2. The Reality (Dynes): They saw a step appear much earlier, at exactly half the gap size ($\Delta$).

The Metaphor:
Imagine you are trying to climb a ladder to get over a wall.

• BCS Theory: Says you can't climb at all until you reach the very top rung.
• Dynes Reality: Says there is a small, hidden step halfway up the wall. You can start climbing (absorbing energy) much earlier than expected.

This "step" is the smoking gun for the Dynes model. It proves that the "fog" (the pair-breaking rate) is real and measurable using light.

Why Was This Hard to See Before?

Usually, scientists measure these things using tunneling (squeezing electrons through a barrier). But tunneling is like looking at the fortress through a tiny keyhole; it only sees a very small, local spot. If the wall is uneven, the keyhole might miss the big picture.

This study used THz light, which is like shining a floodlight over the whole fortress. It averages out the whole surface.

• The Surprise: When they looked at the whole film, the "leak" (Gamma) was constant. It didn't change with temperature.
• The Contrast: Previous tunneling studies suggested the leak got much bigger as things got hotter. The scientists realized that the "leak" they see with light is different from the "leak" seen through a keyhole. It suggests the "leak" is a fundamental property of the material's bulk, not just a surface glitch.

What About the Thin Films?

They tested films of different thicknesses (4.5 nm to 20 nm).

• Thicker films: Behaved more like the "perfect" superconductor, but still had that tiny, constant leak.
• Thinner films: As the films got thinner, the superconducting properties got weaker (the critical temperature dropped), but the "leak" (Gamma) stayed surprisingly small and steady.

This tells us that even when the material is very disordered and thin, the mechanism causing the "leak" is stable and doesn't depend heavily on how hot or cold the room is.

Why Does This Matter?

1. Better Models: It confirms that for many modern superconductors, the old "perfect" math doesn't work. We need the "leaky" Dynes math to predict how they behave.
2. Quantum Computers: Superconductors are the heart of quantum computers. If you are building a quantum circuit, you need to know exactly how much energy leaks out. If you think the gap is perfect but it's actually "foggy," your computer might make errors.
3. New Materials: This study shows that even with advanced manufacturing (ALD), these "leaks" exist. Understanding them helps engineers design better materials for sensors and quantum devices.

Summary in One Sentence

The scientists used a special "floodlight" (THz light) to prove that superconducting films aren't perfect fortresses with dry moats, but rather have a constant, tiny "fog" (the Dynes effect) that lets energy slip through at half the expected energy level, a discovery that helps us build better quantum technology.

Technical Summary

1. Problem Statement

In conventional $s$-wave superconductors, the Bardeen-Cooper-Schrieffer (BCS) theory predicts a complete energy gap ($2\Delta$) in the density of states (DOS) at zero temperature, implying no absorption of photons with energy $E < 2\Delta$. However, in disordered and ultra-thin superconductors, tunneling spectroscopy experiments frequently reveal a non-zero DOS at zero bias (in-gap states) even at temperatures far below the critical temperature ($T_c$).

To describe this, the phenomenological Dynes formula is often used in tunneling studies, introducing a pair-breaking scattering rate ($\Gamma$) that smears the gap. While the Dynes model successfully fits tunneling data, its implications for optical conductivity (specifically the frequency-dependent complex conductivity $\hat{\sigma}(f)$) have rarely been discussed or directly observed. Specifically, the Dynes model predicts a distinct step-like absorption feature at half the spectral gap ($\Delta$), a feature not captured by standard BCS electrodynamics (Mattis-Bardeen theory). The authors aim to experimentally verify these optical signatures in Niobium Nitride (NbN) thin films.

2. Methodology

• Sample Preparation: The study utilized NbN thin films with thicknesses ranging from 4.5 nm to 20 nm, grown on R-plane sapphire substrates using Atomic Layer Deposition (ALD). This growth technique allows for precise control over film thickness and stoichiometry.
• Transport Characterization: DC transport measurements (van der Pauw geometry) were performed to determine the critical temperature ($T_c$), normal-state resistivity ($\rho_0$), and sheet resistance.
• Terahertz (THz) Spectroscopy: The optical response was probed using two complementary techniques over a frequency range of 0.3 to 2.1 THz (covering energies both below and above $2\Delta$):
  1. THz Time-Domain Spectroscopy (TDS): Provided broadband complex conductivity ($\hat{\sigma} = \sigma_1 + i\sigma_2$) directly.
  2. THz Frequency-Domain Spectroscopy (FDS): Utilized tunable backward wave oscillators (BWO) to measure transmission spectra, analyzing Fabry-Pérot (FP) resonances to extract temperature-dependent optical properties with high precision.
• Modeling: The experimental data were fitted using:
  • The Mattis-Bardeen model (generalized by Zimmermann for dirty limits, $\Gamma=0$).
  • The Dynes electrodynamics model (incorporating a finite pair-breaking rate $\Gamma$), which modifies the DOS and predicts specific absorption steps.

3. Key Contributions

• Direct Optical Observation: This work provides the first direct optical observation of the step-like absorption feature at $\Delta$ predicted by the Dynes model in the real part of the optical conductivity ($\sigma_1$).
• Validation of Dynes Electrodynamics: The authors demonstrate that standard BCS/Mattis-Bardeen theory fails to describe the THz response of these disordered NbN films, whereas the Dynes formalism provides a near-perfect fit.
• Temperature Independence of $\Gamma$: A crucial finding is that the pair-breaking rate $\Gamma$ is temperature-independent in the THz regime, contrasting with previous tunneling studies where $\Gamma$ appeared to increase significantly at higher temperatures.
• Thickness Dependence: The study systematically tracks the evolution of superconducting parameters ($2\Delta_0$, superfluid density $n_s$, and $\Gamma$) as a function of film thickness in the ALD-grown regime.

4. Key Results

• Optical Conductivity Signatures:
  • For the 20 nm film, $\sigma_1(f)$ at low temperatures ($T=3.1$ K) shows a distinct onset of absorption at $E \approx \Delta$ (half the gap), which is absent in BCS predictions.
  • The Dynes model, with a constant pair-breaking rate of $\Gamma \approx 0.036 \Delta_0$, perfectly reproduces the experimental data, including the step feature and the sub-gap absorption.
• Fabry-Pérot Resonances:
  • FDS measurements of FP modes confirmed the TDS results. The transmission spectra showed deviations from BCS predictions near the gap edge, consistent with the presence of in-gap states.
  • Both TDS and FDS yielded consistent values for $\Gamma$, confirming the robustness of the finding.
• Pair-Breaking Rate ($\Gamma$):
  • $\Gamma$ was found to be small (a few percent of the gap) and temperature-independent across the measured range for all five samples.
  • This contrasts with tunneling data from Chockalingam et al. (on similar NbN films), where $\Gamma$ increased up to 50% of $\Delta_0$ at higher temperatures. The authors suggest this discrepancy arises from differences in probing depth (bulk vs. surface) and spatial averaging.
• Superconducting Properties vs. Thickness:
  • As film thickness decreased, $T_c$ and the superfluid density ($n_s$) were suppressed, while normal-state resistivity increased.
  • The coupling ratio $2\Delta_0 / k_B T_c$ remained high ($\approx 4.0$) for thicker films, indicating strong electron-phonon coupling.
  • The suppression of superconductivity in thinner films is attributed to increased electron-electron repulsion (fermionic regime) rather than the formation of a pseudo-gap state (bosonic regime).

5. Significance

• Theoretical Implications: The results confirm that the Dynes formula is not just a phenomenological tool for tunneling but a necessary framework for describing the optical electrodynamics of disordered superconductors. The temperature independence of $\Gamma$ in THz measurements suggests that the microscopic origin of the pair-breaking (likely magnetic impurities or interface effects in ALD films) is distinct from inelastic electron scattering mechanisms.
• Material Science: The study validates Atomic Layer Deposition (ALD) as a viable technique for producing high-quality, uniform superconducting NbN films with controllable properties.
• Applications: Understanding sub-gap absorption and pair-breaking mechanisms is critical for the development of superconducting quantum circuits and cryogenic detectors (e.g., superconducting nanowire single-photon detectors), where in-gap states can lead to unwanted dissipation and decoherence. The findings suggest that for disordered transition-metal nitrides, the Dynes formalism is the appropriate strategy for quantifying sub-gap losses.