$\textit{Ab initio}$ Identification of Hydrogen Tunneling as Two-Level Systems in Nb$_2$O$_5$ and Ta$_2$O$_5$

This study identifies hydrogen tunneling as the microscopic origin of two-level systems in amorphous Nb$_2$O$_5$ and Ta$_2$O$_5$, explaining their detrimental impact on superconducting qubit coherence and the experimentally observed higher loss in niobium oxide compared to tantalum oxide.

The Gist

Imagine you are trying to build the world's most sensitive radio receiver. You want it to hear a whisper from a galaxy away, but every time you turn it on, there's a static hiss that drowns out the signal. In the world of quantum computers and ultra-sensitive sensors, this "static" is a major problem. It comes from tiny, invisible defects in the metal coatings (made of Niobium or Tantalum) that cover these devices.

Scientists call these defects Two-Level Systems (TLS). Think of a TLS like a tiny, invisible switch that is stuck halfway between "on" and "off." It's not fully one or the other; it's flickering back and forth. This flickering creates electrical noise that ruins the performance of the machine.

For years, scientists knew these switches existed, but they didn't know what was flipping them. Was it a missing atom? A stray electron? A molecule spinning around? It was a mystery.

This paper solves the mystery by pointing the finger at a very small suspect: Hydrogen.

The Detective Work: How They Found the Culprit

The researchers, Cristóbal Méndez and Tomás Arias, used a clever combination of high-speed computer simulations (like a super-fast video game engine) and precise quantum physics calculations to investigate the inside of these metal oxides.

Here is how they cracked the case, using some everyday analogies:

1. The "Hill and Valley" Problem
Imagine the atoms in the metal oxide are like a landscape of hills and valleys. A hydrogen atom (which is tiny and light) can sit in a valley. Sometimes, there are two valleys right next to each other, separated by a small hill.

• The Tunnel: Because hydrogen is so light, it doesn't need to climb over the hill. Thanks to quantum mechanics, it can "tunnel" straight through the hill, jumping from one valley to the other.
• The Flicker: When it jumps back and forth, it acts like that flickering switch (the TLS), creating the static noise.

2. Why Hydrogen and Not Oxygen or Nitrogen?
The team checked other atoms like Oxygen and Nitrogen.

• The Heavy Hiker Analogy: Imagine trying to jump through a wall. A feather (Hydrogen) can easily flutter through a small gap. A boulder (Oxygen or Nitrogen) is too heavy; it would need a massive gap or a huge amount of energy to get through.
• The math showed that only Hydrogen is light enough and close enough to the right spots to jump back and forth at the specific speed (frequency) that causes the noise in these machines. The heavier atoms are just too sluggish.

3. The Niobium vs. Tantalum Riddle
Scientists have long noticed that devices coated in Niobium (Nb) oxide are "noisier" (have more static) than those coated in Tantalum (Ta) oxide. Why?

• The Sponge Analogy: The researchers found that Niobium oxide is like a wet sponge that loves to soak up hydrogen. Tantalum oxide is more like a dry sponge; it doesn't hold onto hydrogen as well.
• Because Niobium holds more hydrogen, it has more of these "flickering switches" running around, creating more noise. Tantalum holds less hydrogen, so it's quieter.

The Big Picture: What This Means

The paper concludes that hydrogen atoms tunneling through the metal oxide are the primary source of this noise.

This is a huge breakthrough because:

1. It explains the difference: It finally explains why Niobium is noisier than Tantalum (Niobium just holds more hydrogen).
2. It offers a solution: Now that we know the enemy is hydrogen, engineers can try to bake the hydrogen out of the materials or design coatings that keep hydrogen away.
3. It's a new tool: The method they used (a mix of AI and physics) is like a new metal detector. It can be used to scan other materials to find similar "flickering switches" before they ruin our future quantum computers.

In short: The "ghost in the machine" causing static in our most advanced quantum devices is just a tiny, invisible hydrogen atom doing a quantum backflip. Once we know that, we can finally stop the noise.

Technical Summary

1. Problem Statement

Superconducting qubits and Superconducting Radio-Frequency (SRF) cavities operate in the microwave frequency range (0.1–10 GHz), where performance is critically limited by dielectric loss and noise. A primary source of this dissipation is attributed to ensembles of Two-Level Systems (TLS) residing in the native oxide layers of superconducting metals, specifically Niobium (Nb) and Tantalum (Ta).

Despite the critical nature of this issue, the microscopic origin of these TLS remains unclear. While several mechanisms have been proposed (electronic defects, charge trapping, molecular reorientations), no single origin has been definitively identified. Experimental trends provide constraints but not unique solutions:

• TLS loss increases with amorphous disorder.
• Loss is higher in the pentoxide phase ($M_2O_5$) than in sub-oxides.
• Nb oxide exhibits significantly higher loss (approx. 30% higher loss tangent) than Ta oxide.
• Loss correlates with oxygen vacancy content.

The challenge is to identify a specific atomic-scale mechanism that explains these statistical trends, particularly the difference between Nb and Ta oxides, within the relevant microwave frequency window.

2. Methodology

The authors employed a hybrid computational approach combining Machine-Learned Interatomic Potentials (MLIP) for high-throughput sampling and Density Functional Theory (DFT) for targeted validation.

• Structural Sampling:

  • Used the FairChem MLIP to relax hydrogen (H) configurations in amorphous $Nb_2O_{5-\delta}$ and $Ta_2O_{5-\delta}$ structures (derived from Pritchard et al., spanning various oxygen vacancy concentrations $\delta$).
  • Generated trial interstitial positions on a uniform 3D grid (1 Å spacing) within unit cells.
  • Clustered relaxed coordinates to identify unique interstitial sites.
  • Validated a subset of results using DFT (JDFTx code with PBE functional) to ensure accuracy.

• Diffusion and Tunneling Analysis:

  • Connected neighboring unique sites to generate diffusion pathways.
  • Calculated Minimum-Energy Paths (MEP) and activation barriers ($V_0$) using the Nudged Elastic Band (NEB) method.
  • Extracted site-to-site separation distances ($r$) and barrier heights.

• Theoretical Modeling (WKB Approximation):

  • Since direct calculation of tunnel splitting ($\Delta$) from total energy differences is numerically impossible for the $\mu$eV scale, the authors used a semiclassical Wentzel–Kramers–Brillouin (WKB) approach.
  • Modeled the potential as a symmetric quartic double well.
  • Calculated tunnel splitting $\Delta$ using the barrier height $V_0$, mass $m$, and separation $r$:
    $$\Delta \propto \exp\left(-\frac{S_0}{\hbar}\right)$$
  • Identified defects falling within the 0.1–10 GHz frequency band as candidate TLS.

• Statistical Ensemble Modeling:

  • Analyzed the statistical distribution of all distinct H-minima pairs with energy asymmetry $\epsilon \le 10$ meV and separation $r \le 1.65$ Å.
  • Fitted distributions of $\Delta$ and $\epsilon$ to estimate the effective TLS density ($\rho_{eff}$).
  • Calculated the intrinsic dielectric loss tangent ($\tan \delta_0$) using the standard TLS loss formula, incorporating dipole moments derived from Bader charge analysis.

3. Key Contributions

• Identification of Hydrogen as the Primary Candidate: The study systematically ruled out heavier interstitials (O, N) as primary TLS sources in bulk Nb, demonstrating that only Hydrogen possesses the necessary barrier–distance combinations ($V_0 \approx 100\text{--}200$ meV, $r \approx 1.17$ Å) to produce tunnel splittings in the microwave regime.
• Quantitative Explanation of Nb vs. Ta Loss: The work provides the first atomistic explanation for why Nb oxide is more lossy than Ta oxide. It attributes this to a higher thermodynamic tendency for H incorporation in Nb pentoxide and a higher probability of forming microwave-active H-tunneling pairs.
• Transferable Framework: The authors developed a workflow (Grid Seeding $\to$ MLIP Relaxation $\to$ NEB $\to$ WKB Estimation) that can be applied to screen other disordered dielectrics and interfaces for TLS-active motifs.

4. Key Results

• Bulk Nb Benchmark: In crystalline bcc Nb, H hops between tetrahedral sites yielded barrier/distance combinations near the 0.1–10 GHz window. Heavier species (O, N) had barriers too high or separations too large to tunnel at microwave frequencies.
• Amorphous Oxide Statistics:
  • Formation Energy: H formation energies are systematically lower in $Nb_2O_{5-\delta}$ than in $Ta_2O_{5-\delta}$, indicating H is more readily incorporated into Nb oxide.
  • Tunnel Splitting Distributions: The statistical sampling revealed a higher density of H-pairs in Nb oxide that satisfy the criteria for microwave-frequency tunneling ($\Delta \in [0.1, 10]$ GHz) compared to Ta oxide.
• Loss Tangent Predictions:
  • For Nb oxide at $\sim 2$ at.% H concentration, the model predicts $\tan \delta_0 \approx 8.6 \times 10^{-4}$. This aligns with experimental ranges ($8.1 \times 10^{-4}$ to $1.8 \times 10^{-3}$).
  • For Ta oxide at $\sim 1$ at.% H concentration (reflecting lower H solubility), the model predicts $\tan \delta_0 \approx 6.7 \times 10^{-4}$.
  • Result: The model successfully reproduces the experimental observation that Ta oxide is roughly 30% less lossy than Nb oxide.

5. Significance

This paper resolves a long-standing bottleneck in the materials science of superconducting devices by pinpointing hydrogen tunneling as the microscopic origin of TLS in native Nb and Ta oxides.

• Material Design: The findings suggest that reducing hydrogen incorporation (e.g., via processing improvements or surface treatments) could directly improve qubit coherence times and SRF cavity quality factors.
• Fundamental Physics: It shifts the paradigm from searching for isolated "candidate defects" to analyzing defect ensembles and their statistical distributions, providing a rigorous link between atomistic defect physics and macroscopic microwave loss.
• General Applicability: The MLIP-accelerated workflow offers a scalable method for screening other amorphous dielectrics and interfaces, potentially accelerating the development of low-loss materials for quantum computing and particle accelerator technologies.