Hydrogen as a source of flux noise in SQUIDs

Superconducting qubits are hampered by flux noise produced by surface spins from a variety of microscopic sources. Recent experiments indicated that hydrogen (H) atoms may be one of those sources. Using density functional theory calculations, we report that H atoms either embedded in, or adsorbed on, an $\alpha \text{−}\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3\left(0001\right)$ surface have sizable spin moments ranging from 0.81 to $0.87{\mu }_B$ with energy barriers for spin reorientation as low as $\sim 10\mathrm{mK}$. Furthermore, H adatoms on the surface attract gas molecules such as ${\mathrm{O}}_2$, producing new spin sources. We propose coating the surface with graphene to eliminate H-induced surface spins and to protect the surface from other adsorbates.

Figure 1

(a) Left panel shows the spin density of an H atom embedded in $\alpha \text{−}\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3\left(0001\right)$. The gray and orange balls represent Al and O atoms, respectively. The spin density of the embedded H atom is represented by red isosurfaces $\left(0.05e/{\AA }^3\right)$. Black crosses show the positions along the diffusion path (blue arrows) for the embedded H atom heading toward the surface, with A, B, C, D, and E denoting the interstitial sites in different layers. (b) Left axis shows the relative total energy of an H atom diffusing from interior sites to the surface. Energies at ${\mathrm{TS}}_{\mathrm{A}\text{−}\mathrm{B}}$, ${\mathrm{TS}}_{\mathrm{B}\text{−}\mathrm{C}}$, ${\mathrm{TS}}_{\mathrm{C}\text{−}\mathrm{D}}$, and ${\mathrm{TS}}_{\mathrm{D}\text{−}\mathrm{E}}$ indicate the diffusion barriers between two adjacent interstitial sites. Right axis shows the calculated ESR values corresponding to each interstitial site. The horizontal blue dashed line represents the experimental ESR value [8].

Figure 2

(a) Schematic geometries of an H atom adsorbed on the O site of an $\alpha \text{−}\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3\left(0001\right)$ surface. Only atoms near the adsorption site are shown. The gray, orange, and green balls depict Al, O, and H atoms. Charge depletion and accumulation are represented by blue and red, respectively. The lime-green isosurface depicts the distribution of spin density. (b) The same as (a) but with an H atom adsorbed on the Al site of an $\alpha \text{−}\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3\left(0001\right)$ surface. (c) Reaction pathway of the H adatom hopping from the ${\mathrm{H}}_{\mathrm{atop}\text{−}\mathrm{Al}}$ site to the ${\mathrm{H}}_{\mathrm{atop}\text{−}\mathrm{O}}$ site. The horizontal dashed lines indicate the energy barrier. Insets show the top and side views of atomic arrangements corresponding to different states. (d). Relative total energy versus the polar angle $\theta$ of the spin direction with respect to the surface normal for the ${\mathrm{H}}_{\mathrm{atop}\text{−}\mathrm{O}}$ (red line) and ${\mathrm{H}}_{\mathrm{atop}\text{−}\mathrm{Al}}$ geometries (blue line) on an $\alpha \text{−}\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3\left(0001\right)$ surface. The left and right insets show the isoenergy surfaces of the MAE versus the polar and azimuthal angles that are sketched in the central inset.

Figure 3

(a) The atomic geometry and charge redistribution of an ${{\mathrm{O}}_2}^{-}$ molecule adsorbed on ${\mathrm{H}}_{\mathrm{atop}\text{−}\mathrm{O}}/\alpha \text{−}\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3\left(0001\right)$. Al, O, and H atoms are colored as in Fig. 1. $E_{\mathrm{tot}}(\theta ,\phi )$ is given in the right figure, with an arrow indicating the easy axis. Charge depletion and accumulation is represented by blue and red colors, respectively. (b) The partial density of states of ${{\mathrm{O}}_2}^{-}$ molecules adsorbed on ${\mathrm{H}}_{\mathrm{atop}\text{−}\mathrm{O}}/\alpha \text{−}\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3\left(0001\right)$. The inset gives the isosurface of the spin density. (c) Calculated XAS and XMCD spectra of the oxygen $K$ edge for ${O_2}^{-}$ molecules associated with ${\mathrm{H}}_{\mathrm{atop}\text{−}\mathrm{O}}/\alpha \text{−}\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3\left(0001\right)$.

Figure 4

(a) Atomic geometry and charge redistribution of $\mathrm{graphene}/{\mathrm{H}}_{\mathrm{atop}\text{−}\mathrm{O}}/\alpha \text{−}\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3\left(0001\right)$. Charge depletion and accumulation are represented by blue and red, respectively. (b) Electronic band structure of $\mathrm{graphene}/{\mathrm{H}}_{\mathrm{atop}\text{−}\mathrm{O}}/\alpha \text{−}\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3\left(0001\right)$ in a folded two-dimensional Brillouin zone for the $2\times 2$ supercell. The color bar indicates their relative weights in graphene and ${\mathrm{H}}_{\mathrm{atop}\text{−}\mathrm{O}}/\alpha \text{−}\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3\left(0001\right)$. Horizontal black line represents the Fermi level. (c) The relative total energy as the H atom diffuses across graphene over $\alpha \text{−}\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3\left(0001\right)$, accompanied by the corresponding results for H diffusing through a freestanding graphene (gray line). Insets are the top and side views of atomic configurations for H being below and above graphene. Bars at the bottom show the calculated magnetic moments of $\mathrm{graphene}/{\mathrm{H}}_{\mathrm{atop}\text{−}\mathrm{O}}/\alpha \text{−}\mathrm{A}{\mathrm{l}}_2{\mathrm{O}}_3\left(0001\right)$ in different configurations.