First-principles calculation of the spin-orbit splitting in graphene

Recent success in making macroscopic graphene samples has stimulated interest in possible unusual electron physics near the Brillouin zone (BZ) vertex $K$, notably the prediction of a spin quantum Hall effect. Observability depends critically on the size of the spin-orbit gap ${\ensuremath{\Delta}}_{\mathit{SO}}$ at $K$. Prior approximate calculations give results from $1.2\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ $(\ensuremath{\approx}0.1\phantom{\rule{0.3em}{0ex}}\mathrm{meV})$ down to $10\phantom{\rule{0.3em}{0ex}}\mathrm{mK}$ $(\ensuremath{\approx}0.00086\phantom{\rule{0.3em}{0ex}}\mathrm{meV})$. We report fully first-principles all-electron calculations of this splitting using large Gaussian basis sets and the Douglas-Kroll-Hess methodology in the density functional theory fitting function code GTOFF. Our result ${\ensuremath{\Delta}}_{\mathit{SO}}\ensuremath{\approx}0.6\phantom{\rule{0.3em}{0ex}}\mathrm{K}$ or $0.05\phantom{\rule{0.3em}{0ex}}\mathrm{meV}$ is robust against the choice of the approximate exchange-correlation functional and against variations of the lattice constant, density of the BZ scan, basis set enrichment, and key numerical convergence parameters.