Bulk and Surface Tunneling Hydrogen Defects in Alumina

We perform ab initio calculations of hydrogen-based tunneling defects in alumina to identify deleterious two-level systems (TLS) in superconducting qubits. The defects analyzed include bulk hydrogenated Al vacancies, bulk hydrogen interstitial defects, and a surface OH rotor. The formation energies of the defects are first computed for an Al- and O-rich environment to give the likelihood of defect occurrence during growth. The potential energy surfaces are then computed and the corresponding dipole moments are evaluated to determine the coupling of the defects to an electric field. Finally, the tunneling energy is computed for the hydrogen defect and the analogous deuterium defect, providing an estimate of the TLS energy and the corresponding frequency for photon absorption. We predict that hydrogenated cation vacancy defects will form a significant density of GHz frequency TLSs in alumina.

Figure 1

Defect formation energies ($E_f$) under O-rich (left) and Al-rich (right) growth conditions for interstitial H (purple), surface OH (orange), Al vacancy (red), and hydrogenated Al vacancy (blue) defects in $\alpha \mathrm{\text{−}}{\mathrm{Al}}_2{\mathrm{O}}_3$ as a function of the Fermi level. Only the lowest energy charge state is shown for each defect type within the band gap for $\alpha \mathrm{\text{−}}{\mathrm{Al}}_2{\mathrm{O}}_3$. Interstitial ${\mathrm{H}}_2$ (green) provided as a reference. Calculations were conducted at the $\Gamma$ point using 120 and 84 atom bulk and surface supercells with 500 eV plane wave cutoff energies.

Figure 2

(a) Tunneling bulk and surface H (blue) rotor defects identified in α-${\mathrm{Al}}_2{\mathrm{O}}_3$ viewed partially off axis in the [0001] direction. Dashed circles indicate rotor MEP. Blue encircled O atoms (red) indicate the local rotor minima. (b) View along the rotational axis of the H tunneling defects and the O nearest neighbors that create the rotor local minima. (c) View perpendicular to the H defect rotational axis and O nearest neighbors illustrating rotational plane parallel to the (0001) plane in $\alpha \mathrm{\text{−}}{\mathrm{Al}}_2{\mathrm{O}}_3$. (left) OH rotor bound to a surface Al (gray) resulting in a threefold degenerate rotor. (center) Hydrogenated Al bulk vacancy defect resulting in the formation of a threefold degenerate ${\mathrm{H}}^{+}$ rotor. (right) Interstitial ${\mathrm{H}}^{+}$ with six O nearest neighbors that form a sixfold degenerate rotor. (d) Three-fold degenerate relaxed PES of hydrogenated charged Al vacancies in ${\mathrm{Al}}_2{\mathrm{O}}_3$. PES calculated along the MEP for transitions of H between minima and fit to an analytic function to solve for the TLS properties (Table ).