Stability, Energetics, and Magnetic States of Cobalt Adatoms on Graphene

We investigate the stability and electronic properties of single Co atoms on graphene with near-exact many-body calculations. A frozen-orbital embedding scheme was combined with auxiliary-field quantum Monte Carlo calculations to increase the reach in system sizes. Several energy minima are found as a function of the distance $h$ between Co and graphene. Energetics only permit the Co atom to occupy the top site at $h=2.2\AA$ in a high-spin $3d^{8}4s^1$ state, and the van der Waals region at $h=3.3\AA$ in a high-spin $3d^{7}4s^2$ state. The findings provide an explanation for recent experimental results with Co on free-standing graphene.

Figure 1

Binding energy of Co on ${\mathrm{C}}_6{\mathrm{H}}_6$ as a function of Co adsorption height $h$ at the sixfold site for GGA, $\mathrm{GGA}+U$, and exact FP-AFQMC calculations. For FP-AFQMC, left, middle, and right curves correspond to nominal $3d^{9}4s^0$, $3d^{8}4s^1$, and $3d^{7}4s^2$ Co configurations, respectively. For GGA and $\mathrm{GGA}+U$, the left and right curves correspond to $3d^{9}4s^0$ and $3d^{8}4s^1$ Co configurations, respectively. The shaded area on the AFQMC Morse fits reflects 1 standard deviation which includes both statistical and systematic errors. The inset shows $\mathrm{GGA}+U$ errors as a function of $U$. The difference in the calculated binding energy between $\mathrm{GGA}+U$ and the exact results is plotted for $\mathrm{Co}/{\mathrm{C}}_6{\mathrm{H}}_6$, $\mathrm{Co}/{\mathrm{C}}_2{\mathrm{H}}_4$, and $\mathrm{Co}/{\mathrm{C}}_4{\mathrm{H}}_8$, at their respective minimum position $h$.

Figure 2

The error in the calculated binding energy from DFT, for $\mathrm{Co}/{\mathrm{C}}_6{\mathrm{H}}_6$, $\mathrm{Co}/{\mathrm{C}}_2{\mathrm{H}}_4$, and $\mathrm{Co}/{\mathrm{C}}_4{\mathrm{H}}_8$, respectively. The low-spin ($S=1/2$) and high-spin ($S=3/2$) states correspond to $3d^{9}4s^0$ and $3d^{8}4s^1$ Co configurations, respectively. The results from $\mathrm{GGA}+U$ and $\mathrm{LDA}+U$ for $U=4\mathrm{eV}$ are also shown for comparison.

Figure 3

Binding energy of Co on graphene as a function of $h$. Squares, diamonds, circles, and triangles correspond to hollow ($H$), bridge ($B$), and top ($T$) sites, and the vdW region, respectively. The dashed line indicates the low-spin $H$ site (open squares). Extrapolation to the CBS limit has been included. Shaded areas are one-$\sigma$ estimates of uncertainties, including the statistical errors in AFQMC.