Dynamical Jahn-Teller Effect at a Vacancy Center in Graphene

We study the substitutional vacancy center in graphene from density-functional LAPW calculations and show that it is magnetic and at the same time forms a dynamical Jahn-Teller center. A net magnetic moment of $2 \mu_B$ is found, which is explained in terms of the occupation of the $sp^2\sigma$ dangling bond state and the zero-mode state derived from the $\pi$ bands. The adiabatic potential surface resulting from the $ E \otimes e$ vibronic coupling was computed and subsequently the Schr\"odinger equation was solved for the nuclear motion of the carbon atoms. Our calculations show the tunneling splitting $3 \Gamma$ to be about 80 cm$^{-1}$, which is substantially larger than the typical strain fields, leading to a dynamical Jahn-Teller effect (JTE). This explains the puzzling behavior of why in the STM measurements a symmetric carbon triangle is observed around the vacancy, while at the same time we predict the splitting of the vacancy-induced electron states by the static JTE {\it in spite of} the triangular symmetry.