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Ferrimagnetic Slater Insulator Phase of the Sn/Ge(111) Surface

We have performed the semilocal and hybrid density-functional theory (DFT) studies of the Sn/Ge(111) surface to identify the origin of the observed insulating ${\sqrt{3}}{\times}{\sqrt{3}}$ phase below ${\sim}$30 K. Contrasting with the semilocal DFT calculation predicting a metallic 3${\times}$3 ground state, the hybrid DFT calculation including van der Waals interactions shows that the insulating ferrimagnetic structure with ${\sqrt{3}}{\times}{\sqrt{3}}$ structural symmetry is energetically favored over the metallic 3${\times}$3 structure. It is revealed that the correction of self-interaction error with a hybrid exchange-correlation functional gives rise to a band-gap opening induced by a ferrimagnetic order. The results manifest that the observed insulating phase is attributed to the Slater mechanism via itinerant magnetic order rather than the hitherto accepted Mott-Hubbard mechanism via electron correlations.