Valley-Polarized Metals and Quantum Anomalous Hall Effect in Silicene

Silicene is a monolayer of silicon atoms forming a two-dimensional honeycomb lattice, which shares almost every remarkable property with graphene. The low energy structure of silicene is described by Dirac electrons with relatively large spin-orbit interactions due to its buckled structure. The key observation is that the band structure is controllable by applying the electric field to a silicene sheet. We explore the phase diagram of silicene together with exchange field $M$ and by applying electric field $E_{z}$. There appear quantum anomalous Hall (QAH) insulator, valley polarized metal (VPM), marginal valley polarized metal (M-VPM), quantum spin Hall (QSH) insulator and band insulator (BI). They are characterized by the Chern numbers and/or by the edge modes of a nanoribbon. It is intriguing that electrons have been moved from a conduction band at the K point to a valence band at the K' point for $E_{z}>0$ in the VPM. We find in the QAH phase that flat gapless edge modes emerge and that spins form a momentum-space skyrmion to yield the Chern number. It is remarkable that a topological quantum phase transition can be induced simply by changing electric field in a single silicene sheet.