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 electric field to silicene. We explore the phase diagram of silicene together with exchange field M and by applying electric field Ez. 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 Ez > 0 in the VPM. We find in the QAH phase that almost 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.Silicene, a monolayer of silicon atoms forming a twodimensional honeycomb lattice, has been synthesized [1][2][3] and attracts much attention [4][5][6][7][8] recently. Almost every striking property of graphene could be transferred to this innovative material. It has additionally a salient feature, that is a buckled structure [4,5] owing to a large ionic radius of silicon. Silicene has a relatively large spin-orbit (SO) gap of 1.55meV, which provides a mass to Dirac electrons. Furthermore, we may control experimentally the mass [7] by applying the electric field E z . Silicene undergoes a topological phase transition from a quantum spin Hall (QSH) state to a band insulator (BI) as |E z | increases [7]. A QSH state is characterized by a full insulating gap in the bulk and helical gapless edges [9][10][11][12].There exits another state of matter in graphene [13][14][15], that is a quantum anomalous Hall (QAH) state [18,19], characterized by a full insulating gap in the bulk and chiral gapless edges. Unlike the quantum Hall effect, which arises from Landau-level quantization in a strong magnetic field, the QAH effect is induced by internal magnetization and SO coupling.In this paper we analyze the band structure of silicene together with exchange field M and by applying electric field E z to silicene. We explore the phase diagram in the E z -M plane. Silicene has a rich varieties of phases because the electric field E z and the exchange field M have different effects on the conduction and valence bands characterized by the spin and valley indices. There are insulator phases, which are the QSH, QAH and BI phases. There emerges a new type of metal phase, the valley-polarized metal (VPM) phase, where electrons have been moved from a conduction band at the K point to a valence band at the K' point for E z > 0. Such a phase is utterly unknown in liter...

Silicene is a monolayer of silicon atoms forming a two-dimensional (2D) honeycomb lattice and shares almost all the remarkable properties of graphene. The low-energy structure of silicene is described by Dirac electrons with relatively large spin-orbit interactions owing to its buckled structure. A key observation is that the band structure can be controlled by applying an electric field to a silicene sheet. In particular, the gap closes at a certain critical electric field. Examining the band structure of a silicene nanoribbon, we show that a topological phase transition occurs from a topological insulator to a band insulator with an increase of electric field. We also show that it is possible to generate helical zero modes anywhere in a silicene sheet by adjusting the electric field locally to this critical value. The region may act as a quantum wire or a quantum dot surrounded by topological and/or band insulators. We explicitly construct the wave functions for some simple geometries based on the lowenergy effective Dirac theory. These results are also applicable to germanene, which is a 2D honeycomb structure of germanium.

Recently phosphorene, monolayer honeycomb structure of black phosphorus, was experimentally manufactured and attracts rapidly growing interests. Here we investigate stability and electronic properties of honeycomb structure of arsenic system based on first principle calculations. Two types of honeycomb structures, buckled and puckered, are found to be stable. We call them arsenene as in the case of phosphorene. We find that both the buckled and puckered arsenene possess indirect gaps. We show that the band gap of the puckered and buckled arsenene can be tuned by applying strain. The gap closing occurs at 6% strain for puckered arsenene, where the bond angles between the nearest neighbour become nearly equal. An indirect-to-direct gap transition occurs by applying strain. Especially, 1% strain is enough to transform the puckered arsenene into a directgap semiconductor. Our results will pave a way for applications to light-emitting diodes and solar cells.Graphene, a planar honeycomb monolayer of carbon atoms, is one of the most fascinating materials 1, 2 . It has high mobility, heat conductance and mechanical strength. However, it lacks an intrinsic band gap, which makes electronic applications of graphene difficult. The finding of graphene excites material search of other monolayer honeycomb systems with intrinsic gaps. Recently, honeycomb structures of the carbon group attract much attention, which are silicene, germanene and stanene 3 . The geometric structures of these systems are buckled due to the hybridization of sp 2 and sp 3 orbitals. Accordingly we can control the band gap by applying perpendicular electric field [4][5][6] . These are topological insulators owing to spin-orbit interactions 7 . Although silicene and germanene have already been manufactured on substrates [8][9][10] , their free-standing samples are not yet available, which makes experiments difficult to reveal their exciting properties. Phosphorene, a monolayer of black phosphorus, was recently manufactured by exfoliating black phosphorus [11][12][13][14][15][16] . It has already been shown that it acts as a field-effect transistor 11 . The experimental success evokes recent flourish of studies of phosphorene [17][18][19][20][21][22][23] . The structure is puckered, which is different from the planar graphene and the buckled silicene. Furthermore, the buckled phosphorene named "blue phosphorene" is shown to be stable by first principle calculations 24 .In this paper, motivated by recent studies on phosphorene, we have investigated stability and electronic properties of arsenene, which is a honeycomb monolayer of arsenic, by employing density functional theory (DFT) based electronic structure calculations. First we show two types of honeycomb structures, namely buckled and puckered, are stable by investigating phonon spectrum and cohesive energy. Our calculations show that the buckled arsenene is slightly more stable than the puckered arsenene. Though both these two systems possess indirect band gaps, it is possible to make a transition fro...

A second-order topological insulator in d dimensions is an insulator which has no d-1 dimensional topological boundary states but has d-2 dimensional topological boundary states. It is an extended notion of the conventional topological insulator. Higher-order topological insulators have been investigated in square and cubic lattices. In this Letter, we generalize them to breathing kagome and pyrochlore lattices. First, we construct a second-order topological insulator on the breathing Kagome lattice. Three topological boundary states emerge at the corner of the triangle, realizing a 1/3 fractional charge at each corner. Second, we construct a third-order topological insulator on the breathing pyrochlore lattice. Four topological boundary states emerge at the corners of the tetrahedron with a 1/4 fractional charge at each corner. These higher-order topological insulators are characterized by the quantized polarization, which constitutes the bulk topological index. Finally, we study a second-order topological semimetal by stacking the breathing kagome lattice.

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