Abstract
As a topological insulator, the quantum Hall (QH) effect is indexed by the Chern and spinChern numbers and . We have only in conventional QH systems. We investigate QH effects in generic monolayer honeycomb systems. We search for spinresolved characteristic patterns by exploring Hofstadter's butterfly diagrams in the lattice theory and fan diagrams in the lowenergy Dirac theory. It is shown that the spinChern number can takes an arbitrary high value for certain QH systems. This is a new type of topological insulators, which we may call high spinChern insulators. Samples may be provided by graphene on the SiC substrate with ferromagnetic order, transitionmetal dichalcogenides with ferromagnetic order, transitionmetal oxide with antiferromagnetic order and silicene with ferromagnetic order. Actually high spinChern insulators are ubiquitous in any systems with magnetic order. Nevertheless, the honeycomb system would provide us with unique materials for practical materialization.
Introduction
The quantum Hall (QH) effect is one of the most fascinating phenomena in physics^{1,2,3}. It is characterized by the topological index^{4}. The integer QH state at the filling factor ν = n has the Chern number n. The concept of topological insulator stems from QH systems^{5,6}. The topological insulator is characterized by the Chern number and the index in the presence of the timereversal symmetry, while by the Chern number and the spinChern number when the spin s_{z} is a good quantum number^{7,8,9}. We investigate the spinChern number in QH systems. In the conventional monolayer QH system it takes 1/2 and 0 alternately as upspin and downspin electrons fill Landau levels successively. The phenomenon occurs because the cyclotron energy is always larger than the Zeeman energy Δ_{Z} in general. Indeed, we have and in GaAa samples, where B is the magnetic field in Tesla. In the bilayer QH system it can be at most 1 due to the layer degree of freedom^{3}. There are no conventional QH states with higher spinChern numbers.
In this work we investigate an intriguing possibility to materialize QH states carrying higher spinChern numbers. For this purpose we make use of a state which contains either upspin or downspin electrons near the Fermi level in the absence of external magnetic field. Such a state is available in a system when it has strong exchange interactions produced by magnetic order. The magnetic order may be intrinsically present or introduced externally in the system. When the energy spectrum is split into Landau levels in magnetic field they are expected to carry high spinChern numbers, yielding a new type of topological insulators. We call them high spinChern insulators.
As realistic systems we analyze Dirac electrons in monolayer honeycomb systems. Dirac electrons are massless in graphene while they are massive in other materials. We have previously presented a generic Hamiltonian for honeycomb systems^{10}, which contains eight interaction terms mutually commutative in the Dirac limit. Among them four contribute to the Dirac mass. The other four contribute to the shift of the energy spectrum. We are able to make a full control of the Dirac mass and the energy shift independently at each spin and valley by varying these parameters, and materialize various topological phases^{11,12}. Without external magnetic field, the spinvalley dependent Chern number takes ±1/2. Thus we have only 16 types of topological insulators^{10}. In this classification, the Chern number can only be −2, −1, 0, 1, 2, while the spinChern number can be −1, −1/2, 0, 1/2, 1. However, once we switch on magnetic field, the Chern number can take almost all integer values.
The magnetic order can be ferromagnetic or antiferromagnetic. Examples are given by graphene on the SiC substrate with ferromagnetic order, transitionmetal dichalcogenides (MX_{2})^{13,14,15} with ferromagnetic order, perovskite transitionmetal oxide grown on [111] direction (TMO)^{16,17} with antiferromagnetic order, and silicene^{18,19} with ferromagnetic order. It is interesting that the magnetic order is present intrinsically in MX_{2} and TMO. In the absence of magnetic field, graphene, MX_{2} and TMO are trivial insulators while silicene is a quantum spinHall (QSH) insulator.
We explore Hofstadter's butterfly diagrams^{20,21,22,23,24,25} to see a global pattern of spin resolution. We also explore fan diagrams to see a detailed pattern of spin resolution in the lowmagnetic field regime. We then calculate the Chern and spinChern numbers based on the bulkedge correspondence^{21} and on the Kubo formula^{26,27}. They show a perfect agreement in this regime. Our main result is the prediction of high spinChern insulators that may have arbitrarily high spinChern numbers.
High spinChern insulators are actually ubiquitous. The condition is that the band structure has a finite region in the vicinity of the Fermi level where spins are perfectly polarized. Consequently, we would be able to generate them also in conventional QH systems, where electrons are described by the Schrödinger equation. A typical example is given by a squarelattice system, where we can show that high spinChern insulators arise precisely as in honeycomb systems. However, there exists no realistic monolayer squarelattice materials. We may also think of ordinary QH systems such as GaAs samples. It might be possible to introduce magnetic order into the system by doping Mn atoms since Mn atoms form a ferromagnetic order. However the mobility becomes drastically lower by the inhomogeneous doping of Mn, and QH effects would hardly be observed. There exists a recent realization of ordinary QH effects on the interface of the oxides MgZnO/ZnO^{28}. Even in this material it is impossible to introduce the magnetic order since the electron layer is formed on the interface. On the contrary, there are realistic materials for monolayer honeycomb systems. They have very high mobility because samples have no impurities. They are excellent candidates for practical materialization of high spinChern insulators.
Results
The honeycomb lattice consists of two sublattices made of A and B sites. We consider a buckled system with the layer separation 2ℓ between these two sublattices. The states near the Fermi energy are π orbitals residing near the K and K′ points at opposite corners of the hexagonal Brillouin zone. The lowenergy dynamics in the K and K′ valleys is described by the Dirac theory. In what follows we use notations s_{z} = ↑↓, t_{z} = A, B, η = K, K′ in indices while for α = ↑↓, for i = A, B, and η_{i} = ±1 for i = K, K′ in equations. We also use the Pauli matrices σ_{a} and τ_{a} for the spin and the sublattice pseudospin, respectively.
We investigate the honeycomb system in perpendicular magnetic field B by introducing the Peirls phase, , with A the magnetic potential. Any hopping term from site i to site j picks up the phase factor . The magnetic field is given by in unit of e/h, where a is the lattice constant and Φ is the magnetic flux penetrating one hexagonal area. Note that Φ = 1 implies B = 1.6 × 10^{5} Tesla in the case of graphene.
The relevant tightbinding model is given by^{10,29,30}, where creates an electron with spin polarization α at site i in a honeycomb lattice, and run over all the nearest/nextnearestneighbor hopping sites. The first term represents the nearestneighbor hopping with the transfer energy t. The second term represents the chemical potential μ. The third term represents the SO coupling^{29} with λ_{SO}, where ν_{ij} = +1 (−1) if the nextnearestneighboring hopping is anticlockwise (clockwise) with respect to the positive z axis. The fourth term represents the staggered sublattice potential term^{29} with λ_{V}. It may be present intrinsically^{13,14,15} or generated^{31} externally by applying external electric field E_{z}, where λ_{V} = ℓE_{z}. The fifth term is the mean exchange term^{11} with λ_{X}. The sixth term represents the staggered exchange term^{10} with the difference λ_{SX} between the A and B sites. The last term is the staggered SO term^{10} with λ_{SSO}.
The lowenergy Dirac Hamiltonian at the K_{η} point is^{10} where is the Fermi velocity, and is the covariant momentum. We divide the potential terms into two groups, one proportional to τ_{z} and the other not. When the spin s_{z} is diagonalized, they are given by , with and Here, is the Dirac mass and shifts the energy spectrum.
Electrons make cyclotron motion under perpendicular magnetic field and fill the energy levels. We evaluate the energy spectrum numerically based on the tightbinding Hamiltonian (1) and analytically based on the lowenergy Dirac theory (2). To see a global pattern of spin resolution we explore Hofstadter's butterfly diagrams^{21,22,23,24,25} in the lattice theory. To see a detailed pattern of spin resolution in the lowmagnetic field regime we explore fan diagrams in the Dirac theory. We then calculate the Chern and spinChern numbers based on the bulkedge correspondence^{21} in the lattice theory and on the Kubo formula^{26,27} in the Dirac theory. They show a perfect agreement in the lowmagnetic field regime. We have explicitly applied these methods to QH states and reveal high spinChern insulators in various honeycomb systems.
The Hall and spinHall conductivities are given by using the TKNN formula^{4} as with^{5,6} where is the summation of the Berry curvature in the momentum space over all occupied states of electrons with s_{z} = ↑↓. We summarize the results on high spinChern insulators in Fig. 1 for systems we have studied.
Graphene with magnetic order
The first example is given by graphene. The staggered potential is introduced by attaching boronnitride^{32} or siliconcarbide on to graphene. Experimentally observed gap opening is 0.26 eV on the SiC substrate^{33}. On the other hand, the firstprinciple calculation shows that we can introduce ferromagnetic exchange interaction in the order of 5 meV^{9,34,35}. We thus study the Hamiltonian (1) together with t = 2.7 eV, λ_{V} = 0.26 eV, λ_{X} = 5 meV, λ_{SO} = λ_{SX} = λ_{SSO} = 0. Due to the ferromagnetic order (λ_{X} ≠ 0), the energy levels of upspin and downspin electrons are shifted in opposite directions, as illustrated in the band structure of a zigzag nanoribbon [Fig. 2(b)]. Thus there appear only upspin electrons and downspin holes near the Fermi level both for the K and K′ points at Φ = 0. The band structure also shows that the system is a trivial insulator.
The Hofstadter diagram is displayed in Fig. 2(a). Perfectly polarized upspin electrons and downspin holes are found all over Φ. The experimentally accessible regime is given by Φ < 10^{−3} or B < 16 Tesla. We have calculated the Chern and spinChern numbers (green) and (violet) as functions of μ at Φ = 10^{−3} in Fig. 2(d). QH plateaux emerge at ν = 0, ±1, ±3, ±5, ±7, …, for which the spinChern numbers are High spinChern insulators thus emerge. The Hall and spinHall conductivities are illustrated in Fig. 1(a) for ν ≤ 10.
Transition metal dichalcogenides with ferromagnetic order
The second example is given by monolayer transition metal dichalcogenide (MX_{2}). Recently there are experimental report on the ferromagnetism in MX_{2}^{36,37}. It is a trivial insulator due to a large staggered potential even though there exists a SO coupling, as is demonstrated by the band structure of nanoribbon [Fig. 3(b)]. It is described by massive Dirac fermions. Transition metal dichalcogenides have many varieties depending on the combination of transition metal and chalcogen. The examples are MoS_{2}, MoSe_{2}, WS_{2}, WSe_{2}. Among them molybdenite (MoS_{2}) is a most typical material. We take the sample parameters^{13} t = 1.1 eV, λ_{SO} = λ_{SSO} = 75 meV, λ_{V} = 830 meV, λ_{X} = λ_{SX} = 0 in the Hamiltonian (1). We note that the magnitudes of the SO term and staggered SO term are exactly identical since the intrinsic SO interaction exists only at one sublattice.
The Hofstadter diagram is displayed in Fig. 3(a). We have calculated the Chern and spinChern numbers (green) and (violet) as functions of μ at Φ = 10^{−3} in Fig. 3(d). The QH plateaux reads ν = 0, ±1, ±2, ±3, ±4, …, for which the spinChern numbers are High spinChern insulators thus emerge. The Hall and spinHall conductivities are illustrated in Fig. 1(b) for ν ≤ 10.
Perovskite transitionmetal oxides
The third example is given by TMO, where t ≈ 0.2 eV, λ_{SO} = 7.3 meV, λ_{V} = ℓE_{z}, λ_{SX} = 141 meV, λ_{X} = λ_{SSO} = 0 for LaCrAgO^{17}. A salient property is that the material contains an intrinsic staggered exchange effect ∝ λ_{SX}. It has antiferromagnetic order yielding Dirac mass. We can control the band structure by applying electric field due to the buckled structure. When the electric field is off (λ_{V} = 0), upspin and downspin electrons are degenerate. The degeneracy is resolved as λ_{V} increases, and there appear only downspin electrons and holes near the Fermi level both for the K and K′ points at Φ = 0, as found in the nanoribbon band structure [Fig. 4(b)].
The Hofstadter diagram is displayed in Fig. 4(a). We have calculated the Chern and spinChern numbers (green) and (violet) as functions of μ at Φ = 10^{−3} in Fig. 4(d). The QH plateaux reads ν = 0, ±2, ±4, ±6, ±8, …, for which the spinChern number reads High spinChern insulators thus emerge. The Hall and spinHall conductivities are illustrated in Fig. 1(c) for ν ≤ 10.
Silicene with ferromagnetic order
The final example is given by silicene, provided that we could introduce a ferromagnetic order. This could be done by a proximity coupling to a ferromagnet such as depositing Fe atoms to the silicene surface or depositing silicene to a ferromagnetic insulating substrate. Recently the effects of absorption of transition metal atoms are studied by the firstprinciple calculation, where the up and down band shifted oppositely^{38}. We thus have t = 1.6 eV, λ_{SO} = 3.9 meV, λ_{X} ≠ 0, λ_{V} = λ_{SX} = λ_{SSO} = 0 in the Hamiltonian (1).
The nanoribbon band structure is given at Φ = 0 in Fig. 5(b). The spinresolved Hofstadter diagram is displayed in Fig. 5(a). We have calculated the Chern and spinChern numbers (green) and (violet) as functions of μ at Φ = 10^{−3} in Fig. 5(d). The QH plateaux reads ν = 0, ±2, ±4, ±6, ±8, …, for which the spinChern number reads High spinChern insulators thus emerge. The Hall and spinHall conductivities are illustrated in Fig. 1(d) for ν ≤ 10.
It is intriguing that at ν = 0. This is a QSH insulator without the timereversal symmetry. The emergence of this state is natural since silicene is a QSH insulator, to which the present system is reduced adiabatically in the limit Φ → 0. The QSH insulator under broken timereversal symmetry has already been discussed theoretically^{7,8,9,39} and recently found experimentally^{40}.
Discussion
High spinChern insulators may arise in QH states due to strong exchange interactions generated by magnetic order. The magnetic order may be intrinsically present or introduced externally. They give a new type of spinChern insulators. Honeycomb systems are most realistic for practical materialization. We have presented four examples, graphene on the SiC substrate with ferromagnetic order, MX_{2} with ferromagnet order, TMO in electric field and silicene with ferromagnet order. Similarly these QH states may occur in other honeycomb systems such as boronnitride and silicon carbide provided that magnetic order is introduced.
The condition for such QH states to appear is given essentially by the band structure in the absence of magnetic field. We arrange the band structure to contain either upspin or downspin electrons near the Fermi level by implementing appropriate magnetic order. For small magnetic field, there are many Landau levels. When there are N spinpolarized energy levels, the maximum energy must be smaller than the energy gap ΔE between the two Dirac cones with the opposite spins, where or with the Dirac mass (3). The maximum value of the spinChern number is given by , when there exists no degeneracy in the spectrum. It increases as B decreases.
Methods
We have employed the following methods to make the analysis of QH systems in various honeycomb systems and derive their Chern and spinChern numbers.
Hofstadter butterfly
We compute the bulk band structure numerically by applying periodic boundary conditions to the honeycomb system. This requires that the magnetic flux Φ to be a rational number, Φ = p/q (p and q are mutually prime integers). Then, the system is periodic in both spatial directions. We use the Bloch theorem to reduce the Schrödinger equation to a 2q × 2q matrix equation for each s_{z} = ↑↓, where the factor 2 is due to the sublattice (A, B) degrees of freedom. In so doing we choose a generalized gauge of the one used in graphene^{21} so as to include the link connecting the nextnearest neighbor hopping sites. It is given in such a way that the magnetic flux becomes 1/6 for each isosceles triangle whose two edges are given by the neighbor hopping: See Fig. 6. The resulting band structure is the Hofstadters butterfly diagram.
We have given the Hofstadter diagram for graphene in Fig. 2(a), for dichalcogenide in Fig. 3(a), for TMO in Fig. 4(a), and for silicene in Fig. 5(a). We also present a closer look of the Hofstadter butterfly (Φ < 0.05) in Fig. 7 for graphene (a1), for dichalcogenide (b1), for TMO (c1), and for silicene (d1).
Fan diagram
We introduce a pair of ladder operators, satisfying , where is the magnetic length. The Hamiltonian H_{η} is block diagonal and given by with the diagonal elements being in the basis {ψ_{A}, ψ_{B}}^{t}. Here, is the cyclotron frequency.
It is straightforward to solve the eigen equation of . The eigenvalues are for , which depend on Φ. We also have corresponding to N = 0, which is independent of Φ. The eigenstate describes electrons when and holes when . Note that, in the energy spectrum (15a), ± corresponds to electrons or holes provided is zero or sufficiently small.
We refer to each energy spectrum as a fan. There are four fans indexed by valley K_{η} and spin s_{z}. Each fan consists of two parts, one for electrons and the other for holes. These two parts are connected at one pivot when , and otherwise one fan has two pivots. The separation between these two pivots is given by , while the average distance of the two pivots from the Fermi level is given by . Let us call the energy level (15b) the lowest Landau level. In this convention there exists one lowest Landau level in each fan. Thus there are four lowest Landau levels in one fan diagram.
We illustrate the fan diagram in Fig. 7 for graphene (a1), for dichalcogenide (b1), for TMO (c1), and for silicene (d1). We also present a closer look of the Hofstadter butterfly (Φ < 5/100) on the same figure. They agree with one another very well for Φ < 1/100. We can see the degeneracy of each Landau level from the fan diagram, by noting which pivots the lowest Landau levels are attached to. Thus the lowest Landau levels are nondegenerate in graphene (a1), for dichalcogenide (b1) and for TMO (c1), but 2fold degenerate for silicene (d1).
Buldedge correspondence
The most convenient way to determine the topological charge in the lattice formulation is to employ the bulkedge correspondence^{21}. The edgestate analysis can be performed for a system with boundaries such as a cylinder. When solving the Harper equation on a cylinder, the spectrum consists of bulk bands and topological edge states. See Fig. 7(b). We typically find a few edge states within the bulk gaps, some of which cross the gap from one bulk band to another. Each edge state contributes one unit to the quantum number for each s_{z} = ↑↓ at the filling ν = N. More precisely, in order to evaluate , we count the edge states, taking into account their location (right or left edges) and direction (up or down) of propagation^{21}. The location of each state is derived by computing the wave function, while the direction of propagation can be obtained from the sign of its momentum derivative dE/dk, with k the momentum parallel to the edge. We focus on one edge. Edge states with opposite directions contribute with opposite signs. The resultant formula reads where and denote the number of up and downmoving states with spin s_{z}, respectively, at the right edge.
Our main task is to determine the Chern and spinChern numbers for each Landau level. We presents the edgestate analysis at Φ = 1/100 in Fig. 7 for graphene (a2), for dichalcogenide (b2), for TMO (c2), and for silicene (d2). According to the formula (16) we count the number of edge modes, from which we derive the topological numbers (magenta) and (cyan) based on (6).
Kubo formula
We use the Kubo formulation in the Dirac theory to derive the Hall conductivity for each spin s_{z} in each valley K_{η}. Such a formula has been derived for graphene^{26,27}. We may generalize it to be applicable to the Dirac system (2), for each spin s_{z}. It is straightforward to calculate as a function of the chemical potential μ with the use of formulas (3), (4) and (15a).
We show curves and at Φ = 1/100 in Fig. 7 for graphene (a3), for dichalcogenide (b3), for TMO (c3), and for silicene (d3). We can explicitly check that the Kubo formula (17) presents the correct values that are obtained based on the bulkedge correspondence. We have confirmed the validity of the Kubo formula for various systems.
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Acknowledgements
I am very much grateful to N. Nagaosa and H. Aoki for many fruitful discussions on the subject. This work was supported in part by GrantsinAid for Scientific Research from the Ministry of Education, Science, Sports and Culture No. 22740196.
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Department of Applied Physics, University of Tokyo, Hongo 731, 1138656, Japan
 Motohiko Ezawa
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M.E. performed all calculations and made all contribution to the preparation of this manuscript.
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