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

Abstract

Two-level systems (TLS) in native Nb and Ta oxides limit superconducting-qubit coherence and SRF-cavity quality factors in the microwave frequency range, yet their microscopic origin remains unclear. We combine MLIP-accelerated sampling of hydrogen configurations and diffusion pathways in amorphous Nb and Ta pentoxides with targeted $\textit{ab initio}$ validation. Hydrogen is the only light interstitial with barrier-distance combinations near the $\sim0.1-10$ GHz tunneling regime, and its ensemble statistics in amorphous oxides produce effective TLS densities and loss estimates consistent with the experimentally observed higher loss in Nb oxide than in Ta oxide. Our results point to H tunneling as a plausible microscopic TLS source in these materials.

Figures (5)

• Figure 1: Workflow for identifying hydrogen-related TLS in Nb and Ta pentoxide. (1) Place H on a uniform three-dimensional grid (1 Å spacing) and relax each seed to the nearest minimum; cluster the results to obtain unique interstitial sites. (2) Connect neighboring sites and compute minimum-energy paths and diffusion barriers using the NEB method. (3) For each hop, extract the site-to-site separation $r$, barrier height $V_0$, and estimate the tunnel splitting $\Delta$ (and frequency $f$) via the quartic WKB expression (Eq. \ref{['eq:delta_quartic']}).
• Figure 2: Minimum-to-minimum distance $r$ versus barrier height $V_{0}$ for hops in bcc Nb, obtained from MLIP-based NEB, for (a) H, (b) O, and (c) N. Colors denote the local environment: pristine Nb (blue), Nb with an interstitial O at an octahedral site (green), and Nb with an Nb vacancy (yellow). A DFT cross-check for pristine Nb is indicated by a star. The shaded band (bounded by dashed lines) shows the quartic--WKB isofrequency region corresponding to $f=0.1$--$10$ GHz, computed using the tunneling mass of the diffusing species ($m_{\mathrm H}$, $m_{\mathrm O}$, $m_{\mathrm N}$). Points within the band yield microwave-scale tunnel splittings, whereas points well above it correspond to exponentially suppressed splittings.
• Figure 3: Distribution of H interstitial formation energies over the set of distinct relaxed interstitial sites identified in each amorphous cell (median and interquartile range shown in black) as a function of the oxygen vacancy composition $\delta$.
• Figure 4: Statistical distributions of hydrogen TLS parameters in amorphous Nb$_2$O$_{5-\delta}$ and Ta$_2$O$_{5-\delta}$. (a,b) Probability density of tunnel splittings $\Delta$ for all distinct pairs of relaxed H minima with endpoint energy difference $\le 10$ meV and separation $\le 1.65$ Å. Solid curves show Gaussian-mixture fits to $\log(\Delta)$. (c,d) Probability density of asymmetries $\varepsilon$ for the same pair ensemble. Because endpoint energy differences below 0.05 meV are not reliably resolved in the MLIP total energies, the solid curves show grouped two-component half-normal-mixture fits used to extrapolate the low-$\varepsilon$ tail relevant to TLS statistics.
• Figure 5: Effective TLS density $\rho_{\mathrm{eff}}$ in amorphous Nb$_2$O$_{5-\delta}$ and Ta$_2$O$_{5-\delta}$ as a function of hydrogen concentration, obtained from the statistical pair model described in the text. The larger $\rho_{\mathrm{eff}}$ predicted for Nb reflects the higher probability of microwave-active H tunneling defects extracted from the atomistically derived pair statistics. The shaded region marks the physically relevant H-concentration range considered in this work.