Electric-field noise from carbon-adatom diffusion on a Au(110) surface: First-principles calculations and experiments

The decoherence of trapped-ion quantum gates due to heating of their motional modes is a fundamental science and engineering problem. This heating is attributed to electric-field noise arising from the trap-electrode surfaces. In this work, we investigate the source of this noise by focusing on the diffusion of carbon-containing adsorbates on the surface of Au(110). We show by density functional theory, based on detailed scanning probe microscopy, how the carbon adatom diffusion on the gold surface changes the energy landscape and how the adatom dipole moment varies with the diffusive motion. A simple model for the diffusion noise, which varies quadratically with the variation of the dipole moment, predicts a noise spectrum, in accordance with the measured values.

Figure 1

Electric-field noise spectral density at $\omega /2\pi$ = 4.7 MHz vs the cumulative dose for incremental treatments. $S_E$ is determined from Eq. (1) and the motional heating rate of a ${}^{25}\mathrm{Mg}^{+}$ ion trapped 63 $\mu \mathrm{m}$ above the nearest electrode. The ion-bombardment treatments to the electrodes employ 500 eV argon ions with typical current densities of $\sim 0.2–0.5\mu \mathrm{A}/{\mathrm{cm}}^2$. As the treatments proceed, the coverage of the contamination decreases. The line connecting the points is intended only to indicate sequential data, not to indicate a trend in the behavior. This nonmonotonic behavior in the noise with successive treatments has been observed in each of the six ion traps we have investigated with ion-bombardment treatments. As the treatments proceed, the estimated coverage of the contamination decreases from approximately 3 ML to less than 0.05 ML.

Figure 2

(a) Derivative STM image ($17\times 10{\mathrm{nm}}^2$) of an untreated ion-trap surface covered by $\sim 2.5\mathrm{ML}$ of adventitious carbon. (b) Derivative STM image ($17\times 10{\mathrm{nm}}^2$) of the same trap after treatment by ${\mathrm{Ar}}^{+}$ bombardment, revealing a clean $\mathrm{Au}\left(110\right)$ surface. (c) Top view of the $\mathrm{Au}\left(110\right) (2\times 1)$ reconstructed superstructure relaxed with DFT (top-row Au surface atoms are shown in light yellow) and (d) its simulated STM image. In (b), (c), and (d), the $(2\times 2)$ simulation supercell along the $\left[1\overline{1}0\right]$ and $\left[001\right]$ directions is represented by a red rectangle, while the $〈1\overline{1}0〉$ rows are indicated by red dashed lines. The reconstructed $(2\times 1)$ structure features alternately missing $〈1\overline{1}0〉$ rows. Adjacent $〈1\overline{1}0〉$ top rows are separated by $\simeq 8.2\AA$ (i.e., twice the lattice constant of bulk Au) along the $\left[001\right]$ direction.

Figure 3

Carbon-atom adsorption ($\theta =0.25\mathrm{ML}$ coverage) on Au(110) ($1\times 1$) at (a) the atop site, (b) bridge site, and (c) fcc site and on $\mathrm{Au}\left(110\right) (2\times 1)$ at (d) the atop site, (e) bridge site, and (f) fcc site. The top view and side view are illustrated in the top and bottom panels, respectively. Color legend: C, dark brown; Au, gold and light yellow. For Au(110) ($1\times 1$), total-energy calculations indicate that a single C atom adsorbs preferentially at the bridge site (b) slightly more energetically favorable than at the fcc site (c) by 0.14 eV and at the atop site (a) by 1.686 eV. Fourfold C coordination at the bridge site exhibits C-Au bond distances of 2.06 and 2.11 Å, facilitating metal-to-C charge transfer. Similar to the ($1\times 1$) structure, the most stable adsorption site for ($2\times 1$) is the bridge site (e), with fourfold coordinated C showing bond lengths of 1.99 and 2.10 Å. The atop site (d) is the least favorable by 2.06 eV, with a similar C-Au bond of 1.84 Å, while adsorption at the fcc site (f) is slightly more favorable by 0.377 eV.

Figure 4

Calculated carbon-adatom diffusion path on (a) $\mathrm{Au}\left(110\right) (1\times 1)$ and (b) $\mathrm{Au}\left(110\right) (2\times 1)$ along the $\left[1\overline{1}0\right]$ direction ($\theta =0.25\mathrm{ML}$ coverage). Color legend: C, dark brown; Au, gold and light yellow. Starting from the most energetically favorable bridge sites for C adsorption on Au(110) $(1\times 1)$ and $(2\times 1)$, the energetics of the C diffusion pathways along the $\left[1\overline{1}0\right]$ direction was computed at $\theta$ = 0.25 ML coverage. The predicted energy barrier heights $E_a$ for C diffusion from the bridge site are 0.20 and 0.57 eV for $(1\times 1)$ and $(2\times 1)$, respectively. The fcc site corresponds to a local energy minimum located 0.07 and 0.37 eV above the bridge site for $(1\times 1)$ and $(2\times 1)$, respectively (i.e., local C diffusion barriers of $E_a=0.13$ and 0.20 eV surround the fcc site for those structures).

Figure 5

(a) Calculated adsorption energy landscape and (b) dipole map for a carbon adatom diffusing on the $\mathrm{Au}\left(110\right) (1\times 1)$ surface and (c) adsorption energy landscape and (d) dipole map on the $\mathrm{Au}\left(110\right) (2\times 1)$ surface, along the $\left[001\right]$ and $\left[1\overline{1}0\right]$ directions at $\theta =0.25\mathrm{ML}$ carbon coverage. The diffusion path in the energetically favorable trench along the $\left[1\overline{1}0\right]$ direction, plotted in Fig. 4, can be seen. At the bottom of each panel is a contour plot of the energy and dipole 3D maps; the contour line values are given on the right vertical axis in each panel and are color coded.

Figure 6

Measured frequency-normalized electric-field noise spectral densities for the lowest-noise room-temperature surface-electrode ion traps plotted as a function of the ion-electrode distance. These data are overlaid on top of our estimated range from Eq. (4), shown as a light gray band, and reasonable values for the diffusion constant and patch size. Data from Fig. 1 are also plotted here in red for comparison.