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 electric field to silicene. We explore the phase diagram of silicene together with exchange field and by applying electric field . 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 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 two-dimensional 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 meV, which provides a mass to Dirac electrons. Furthermore, we may control experimentally the mass(7) by applying the electric field . Silicene undergoes a topological phase transition from a quantum spin Hall (QSH) state to a band insulator (BI) as increases(7). A QSH state is characterized by a full insulating gap in the bulk and helical gapless edges(11); (12); (9); (10).
There exits another state of matter in graphene(13); (14); (15), that is a quantum anomalous Hall (QAH) state(19); (18), 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 and by applying electric field to silicene. We explore the phase diagram in the - plane. Silicene has a rich varieties of phases because the electric field and the exchange field 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 . Such a phase is utterly unknown in literature as far as we are aware of. There are also metallic states on phase boundaries, which are metal (M), marginal-VPM (M-VPM) and spin VPM (SVPM) states. All these phases and states are characterized by the Chern numbers and/or by the edge modes of a nanoribbon. It is possible to materialize any one of them by controlling at an appropriate value of . Furthermore, as we have pointed out elsewhere(7), by applying an inhomogeneous field , it is possible to materialize some of these topological phases together with states on the phase boundaries simultaneously in a single silicene sheet.
Silicene consists of a honeycomb lattice of silicon atoms with two sublattices made of A sites and B sites. The states near the Fermi energy are orbitals residing near the K and K’ points at opposite corners of the hexagonal Brillouin zone. We refer to the K or K’ point also as the K point with the valley index . We take a silicene sheet on the -plane, and apply the electric field perpendicular to the plane. Due to the buckled structure the two sublattice planes are separated by a distance, which we denote by with Å. It generates a staggered sublattice potential between silicon atoms at A sites and B sites.
The silicene system is described by the four-band second-nearest-neighbor tight binding model,
where creates an electron with spin polarization at site , and run over all the nearest/next-nearest neighbor hopping sites. We explain each term. (i) The first term represents the usual nearest-neighbor hopping with the transfer energy eV. (ii) The second term represents the effective SO coupling with meV, where is the Pauli matrix of spin, with if the next-nearest-neighboring hopping is anticlockwise and if it is clockwise with respect to the positive axis. (iii) The third term represents the first Rashba SO coupling associated with the nearest neighbor hopping, which is induced by external electric field(20); (14). It satisfies and becomes of the order of eV at the critical electric field meVÅ. (iv) The forth term represents the second Rashba SO coupling with meV associated with the next-nearest neighbor hopping term, where for the A (B) site, and with the vector connecting two sites and in the same sublattice. (v) The fifth term is the staggered sublattice potential term. (vi) The sixth term represents the exchange magnetization: Exchange field may arise due to proximity coupling to a ferromagnet such as depositing Fe atoms to the silicene surface or depositing silicene to a ferromagnetic insulating substrate, as has been argued for graphene(13); (14); (15). The Hamiltonian (1) can also be used to describe germanene, which is a honeycomb structure of germanium(5); (6), where various parameters are eV, meV, meV and Å.
In this paper we derive the topological phase diagram in the - plane and make its physical interpretation. The topological quantum numbers are the Chern number and the index. If the spin is a good quantum number, the index is identical to the spin-Chern number . They are defined when the state is gapped and when the Fermi level is taken within the gap, and given by and , where is the summation of the Berry curvature in momentum space over all occupied states of electrons with . They are well defined even if the spin is not a good quantum number(21); (15). In the present model the spin is not a good quantum number because of spin mixing due to the Rashba couplings and , and the resulting angular momentum eigenstates are indexed by the spin chirality . We can calculate these numbers at each point in the - plane by using the standard formulas(13); (14); (15).
We present our result on the phase diagram in Fig.1. We show later how to derive the phase boundaries based on the low-energy Dirac theory. We have also calculated the band structure of a silicene nanoribbon with zigzag edges, which we give in Fig.2 for typical points in the phase diagram. The topological numbers are in the BI phase, in the QSH phase, in the QAH phase with and in the QAH phase with . In all these states the band gap is open, where the Fermi level is present, and they are insulators.
We first discuss the system at and compare our results with those previously obtained in graphene(13); (14); (15). The main difference is the appearence of almost flat edge modes in our system (Fig.3). This occurs because the Rashba interactions are different between these two systems. We have for and at the K and K’ points in silicene, but and in graphene. Nevertheless, the difference is only quantitative. As far as the topological properties are concerned, there exists no difference. Indeed, in these two systems, the Chern number is identical in each corresponding phase together with quantized Hall conductivity, and the edge states support the edge current. However, the group velocity of the edge modes is extremely small due to the almost flat gapless modes in silicene.
Our most important result is the VPM phase, which appears in such regions that and occupies a major part of the phase diagram. A part of the conduction (valence) band is above (below) the Fermi level at the K (K’) point for , as is observed in Fig.2(VPM). Hence, electrons are moved from the K valley to the K’ valley, as implies the valley polarization. The phase is characterized by the property that it is a metallic state though gaps are open both at the K and K’ points. We note that the Chern and spin-Chern numbers are ill-defined in the VPM phase, since the Fermi level does not lie inside the band gaps at the K and K’ points simultaneously.
There exist M-VPM states on phase boundaries indicated by heavy lines in the phase diagram, where the conduction and valence bands touch the Fermi surface at the K and K’ points, respectively, for . On the other hand, in SVPM states the conduction and valence bands touch the Fermi surface both at the K and K’ points. We expect topological quantum critical phenomena in these states.
In order to explore the physics underlying the phase diagram, we analyze the low-energy effective Hamiltonian derived from the tight binding model (1). It is described by the Dirac theory around the point as
with , where is the Pauli matrix of the sublattice pseudospin, is the Fermi velocity, and Å is the lattice constant.
The Hamiltonian explicitly reads
in the basis , where , and the diagonal elements are
with the spin and the sublattice pseudospin . They are not good quantum numbers in general. However, since and are very small with respect to the other parameters, it is a good approximation to set in most cases. Thus the spin is almost a good quantum number in general. An exceptional case occurs when two Dirac cones collapse and cross each other, forming a QAH state after taking into account the effect of , as we soon discuss.
We diagonalize the Hamiltonian (3) and obtain four energy levels. When two energy levels coincide, the band gap becomes zero, as found in Fig.2(M,VMP3,VMP2,SVPM). This occurs at the K and K’ points, where . Let us temporarily neglect because it is very small. Then, the band closes when with (4). They yield four lines described by for , for , and two lines by outside the square. They are illustrated by dotted lines in Fig.1. These lines are modified by the nonzero effect of , but the modification is too small to be recognized in Fig.1. See also (5) for the typical order of correction.
The Hamiltonian can be diagonalized analytically in some cases. First, along the -axis in the phase diagram [Fig.1], we have already demonstrated(7) that a topological phase transition occurs along the -axis from the QSH insulator [Fig.2(QSH)] to the band insulator [Fig.2(BI)]. The critical point is given by
where we have set with Å. Note that the effect of is negligible, . The SVPM realizes at the critical point, where helical currents flow in the bulk.
Second, along the -axis, the first Rashba interaction vanishes (), and the energy spectrum reads
We study a topological phase transition along the -axis based on this formula (6). When , there are two spin-degenerate Dirac cones for conduction and valence bands with a gap between them [Fig.2(QSH1)]. As increases, the spin-up (spin-down) Dirac cones are pushed upward (downward) [Fig.2(QSH2)]. When , the band gap is given as at , and it closes at : This is a topological phase transition point [Fig.2(M)]. Let us temporally assume . Then, as increases further, the two Dirac cones cross each other making a circle around each K point. Actually, the Rashba interaction () mixes up and down spins, turning the crossing points into the anticrossing points, and opens a gap to form the QAH insulating state [Fig.2(QAH) and Fig.3].
When , the gap is given by
We present the energy spectrum (6) and the Berry curvature calculated by using the corresponding wave function at in Fig.4. As explained there, spins rotates across the anticrossing point, generating a skyrmion spin texture in the momentum space. This is consistent with the previous study for graphene(14). The radius of the anticrossing circles is given by (8). We comment that the gap (7) is of the order of eV when is of the order of meV.
We now examine a point in the phase diagram such that . In all regions where the effects of and are negligible, the energy spectrum is derived as
The effect of is to change the mass of the Dirac electron. Let us increase from at a fixed value of . The mass decreases (increases) for the Dirac cone characterized by () until , but the behavior becomes opposite after . As a result the tip of each Dirac cone is pushed either downward or upward as indicated in Fig.2. Consequently the valley symmetry is broken. Note that the energy difference at each momentum between the conduction and valence bands with the same spin is given by
for , and this is independent of . Thus, the difference is smaller for the up-spin Dirac cones at the K point, but this is opposite at the K’ point.
We finally determine the phase boundary. It is determined as a boundary between insulating and metallic states. The Chern and spin-Chern numbers are quantized in insulating states, while they are ill-defined in metallic states. As we have seen, each Dirac cone moves upward or downward oppositely at the K and K’ points. Because of this phenomenon the system can become metallic though the gap is open both at the K and K’ points. This is the VPM state. It occurs when one valence band crosses the Fermi level. The condition yields four heavy lines () in the phase diagram (Fig.1). On the other hand, the gap formula (7) determines the boundary between the QAH phase and the VPM phase, which are the parabolic curves in the phase diagram (Fig.1). In passing we comment that the VPM phase is metallic in nature and does not have mobility gap. Thus the transition from insulator to VPM might accompany a mobility gap closing.
I am very much grateful to N. Nagaosa for many fruitful discussions on the subject. This work was supported in part by Grants-in-Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture No. 22740196.
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