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High Spin-Chern Insulators with Magnetic Order

Scientific Reports volume 3, Article number: 3435 (2013) | Download Citation

Abstract

As a topological insulator, the quantum Hall (QH) effect is indexed by the Chern and spin-Chern numbers and . We have only in conventional QH systems. We investigate QH effects in generic monolayer honeycomb systems. We search for spin-resolved characteristic patterns by exploring Hofstadter's butterfly diagrams in the lattice theory and fan diagrams in the low-energy Dirac theory. It is shown that the spin-Chern number can takes an arbitrary high value for certain QH systems. This is a new type of topological insulators, which we may call high spin-Chern insulators. Samples may be provided by graphene on the SiC substrate with ferromagnetic order, transition-metal dichalcogenides with ferromagnetic order, transition-metal oxide with antiferromagnetic order and silicene with ferromagnetic order. Actually high spin-Chern 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 physics1,2,3. It is characterized by the topological index4. The integer QH state at the filling factor ν = n has the Chern number n. The concept of topological insulator stems from QH systems5,6. The topological insulator is characterized by the Chern number and the index in the presence of the time-reversal symmetry, while by the Chern number and the spin-Chern number when the spin sz is a good quantum number7,8,9. We investigate the spin-Chern number in QH systems. In the conventional monolayer QH system it takes 1/2 and 0 alternately as up-spin and down-spin electrons fill Landau levels successively. The phenomenon occurs because the cyclotron energy is always larger than the Zeeman energy ΔZ 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 freedom3. There are no conventional QH states with higher spin-Chern numbers.

In this work we investigate an intriguing possibility to materialize QH states carrying higher spin-Chern numbers. For this purpose we make use of a state which contains either up-spin or down-spin 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 spin-Chern numbers, yielding a new type of topological insulators. We call them high spin-Chern 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 systems10, 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 phases11,12. Without external magnetic field, the spin-valley dependent Chern number takes ±1/2. Thus we have only 16 types of topological insulators10. In this classification, the Chern number can only be −2, −1, 0, 1, 2, while the spin-Chern 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, transition-metal dichalcogenides (MX2)13,14,15 with ferromagnetic order, perovskite transition-metal oxide grown on [111] direction (TMO)16,17 with antiferromagnetic order, and silicene18,19 with ferromagnetic order. It is interesting that the magnetic order is present intrinsically in MX2 and TMO. In the absence of magnetic field, graphene, MX2 and TMO are trivial insulators while silicene is a quantum spin-Hall (QSH) insulator.

We explore Hofstadter's butterfly diagrams20,21,22,23,24,25 to see a global pattern of spin resolution. We also explore fan diagrams to see a detailed pattern of spin resolution in the low-magnetic field regime. We then calculate the Chern and spin-Chern numbers based on the bulk-edge correspondence21 and on the Kubo formula26,27. They show a perfect agreement in this regime. Our main result is the prediction of high spin-Chern insulators that may have arbitrarily high spin-Chern numbers.

High spin-Chern 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 square-lattice system, where we can show that high spin-Chern insulators arise precisely as in honeycomb systems. However, there exists no realistic monolayer square-lattice 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/ZnO28. 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 spin-Chern insulators.

Results

The honeycomb lattice consists of two sublattices made of A and B sites. We consider a buckled system with the layer separation 2 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 low-energy dynamics in the K and K′ valleys is described by the Dirac theory. In what follows we use notations sz = ↑↓, tz = A, B, η = K, K′ in indices while for α = ↑↓, for i = A, B, and ηi = ±1 for i = K, K′ in equations. We also use the Pauli matrices σa and τa 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 × 105 Tesla in the case of graphene.

The relevant tight-binding model is given by10,29,30, where creates an electron with spin polarization α at site i in a honeycomb lattice, and run over all the nearest/next-nearest-neighbor hopping sites. The first term represents the nearest-neighbor hopping with the transfer energy t. The second term represents the chemical potential μ. The third term represents the SO coupling29 with λSO, where νij = +1 (−1) if the next-nearest-neighboring hopping is anticlockwise (clockwise) with respect to the positive z axis. The fourth term represents the staggered sublattice potential term29 with λV. It may be present intrinsically13,14,15 or generated31 externally by applying external electric field Ez, where λV = ℓEz. The fifth term is the mean exchange term11 with λX. The sixth term represents the staggered exchange term10 with the difference λSX between the A and B sites. The last term is the staggered SO term10 with λSSO.

The low-energy Dirac Hamiltonian at the Kη point is10 where is the Fermi velocity, and is the covariant momentum. We divide the potential terms into two groups, one proportional to τz and the other not. When the spin sz 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 tight-binding Hamiltonian (1) and analytically based on the low-energy Dirac theory (2). To see a global pattern of spin resolution we explore Hofstadter's butterfly diagrams21,22,23,24,25 in the lattice theory. To see a detailed pattern of spin resolution in the low-magnetic field regime we explore fan diagrams in the Dirac theory. We then calculate the Chern and spin-Chern numbers based on the bulk-edge correspondence21 in the lattice theory and on the Kubo formula26,27 in the Dirac theory. They show a perfect agreement in the low-magnetic field regime. We have explicitly applied these methods to QH states and reveal high spin-Chern insulators in various honeycomb systems.

The Hall and spin-Hall conductivities are given by using the TKNN formula4 as with5,6 where is the summation of the Berry curvature in the momentum space over all occupied states of electrons with sz = ↑↓. We summarize the results on high spin-Chern insulators in Fig. 1 for systems we have studied.

Figure 1: Plateaux for the Hall conductivity and the spin-Hall conductivity for |B| < 16 Tesla.
Figure 1

The vertical axis is the filling factor ν, which is equal to σxy (green) in unit of . We have also shown (violet) in unit of . When they agree, the curves are in grey. The horizontal axis represents the chemical potential μ. Graphene, dichalcogenide, TMO are trivial insulators at ν = 0, while silicene is a QSH insulator at ν = 0.

Graphene with magnetic order

The first example is given by graphene. The staggered potential is introduced by attaching boron-nitride32 or silicon-carbide on to graphene. Experimentally observed gap opening is 0.26 eV on the SiC substrate33. On the other hand, the first-principle calculation shows that we can introduce ferromagnetic exchange interaction in the order of 5 meV9,34,35. We thus study the Hamiltonian (1) together with t = 2.7 eV, λV = 0.26 eV, λX = 5 meV, λSO = λSX = λSSO = 0. Due to the ferromagnetic order (λX ≠ 0), the energy levels of up-spin and down-spin electrons are shifted in opposite directions, as illustrated in the band structure of a zigzag nanoribbon [Fig. 2(b)]. Thus there appear only up-spin electrons and down-spin 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.

Figure 2: Graphene with ferromagnetic order.
Figure 2

We have set λV = 0.2t and λX = 0.1t for illustration. The vertical axis is the energy in unit of t in all figures. (a) Spin-resolved Hofstadter's diagram. The horizontal axis is the magnetic flux Φ. We have taken Φ = p/q with q ≤ 100. The contribution from up(down)-spin electrons is shown in magenta (cyan). (b) Band structure of a zigzag nanoribbon at Φ = 0. Spins are polarized near the Fermi level. The horizontal axis is the momentum k. (c) Topological number calculated based on the Dirac formula (17) as a function of μ at Φ = 1/1000. It is found that begins to increase monotonously when the bulk band having spin sz begins to be filled in (b). (d) Chern and spin-Chern numbers (green) and (violet) as a function of μ derived based on (6) at Φ = 1/1000. We have taken μ as the vertical axis and Chern numbers as the horizontal axis.

The Hofstadter diagram is displayed in Fig. 2(a). Perfectly polarized up-spin electrons and down-spin 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 spin-Chern 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 spin-Chern numbers are High spin-Chern insulators thus emerge. The Hall and spin-Hall 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 (MX2). Recently there are experimental report on the ferromagnetism in MX236,37. 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 MoS2, MoSe2, WS2, WSe2. Among them molybdenite (MoS2) is a most typical material. We take the sample parameters13 t = 1.1 eV, λSO = λSSO = 75 meV, λV = 830 meV, λX = λSX = 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.

Figure 3: Dichalcogenide.
Figure 3

We have set λSO = λSSO = 0.1t and λV = 0.4t for illustration. See also the caption of Fig. 2.

The Hofstadter diagram is displayed in Fig. 3(a). We have calculated the Chern and spin-Chern 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 spin-Chern numbers are High spin-Chern insulators thus emerge. The Hall and spin-Hall conductivities are illustrated in Fig. 1(b) for |ν| ≤ 10.

Perovskite transition-metal oxides

The third example is given by TMO, where t ≈ 0.2 eV, λSO = 7.3 meV, λV = ℓEz, λSX = 141 meV, λX = λSSO = 0 for LaCrAgO17. A salient property is that the material contains an intrinsic staggered exchange effect λSX. 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 (λV = 0), up-spin and down-spin electrons are degenerate. The degeneracy is resolved as λV increases, and there appear only down-spin 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)].

Figure 4: TMO in electric field.
Figure 4

We have set λSO = 0.2t, λV = 0.1t and λSX = 0.4t for illustration. See also the caption of Fig. 2.

The Hofstadter diagram is displayed in Fig. 4(a). We have calculated the Chern and spin-Chern 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 spin-Chern number reads High spin-Chern insulators thus emerge. The Hall and spin-Hall 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 first-principle calculation, where the up and down band shifted oppositely38. We thus have t = 1.6 eV, λSO = 3.9 meV, λX ≠ 0, λV = λSX = λSSO = 0 in the Hamiltonian (1).

The nanoribbon band structure is given at Φ = 0 in Fig. 5(b). The spin-resolved Hofstadter diagram is displayed in Fig. 5(a). We have calculated the Chern and spin-Chern 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 spin-Chern number reads High spin-Chern insulators thus emerge. The Hall and spin-Hall conductivities are illustrated in Fig. 1(d) for |ν| ≤ 10.

Figure 5: Silicene with ferromagnetic order.
Figure 5

We have set λSO = 0.2t and λX = 0.1t for illustration. See also the caption of Fig. 2.

It is intriguing that at ν = 0. This is a QSH insulator without the time-reversal 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 time-reversal symmetry has already been discussed theoretically7,8,9,39 and recently found experimentally40.

Discussion

High spin-Chern 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 spin-Chern insulators. Honeycomb systems are most realistic for practical materialization. We have presented four examples, graphene on the SiC substrate with ferromagnetic order, MX2 with ferromagnet order, TMO in electric field and silicene with ferromagnet order. Similarly these QH states may occur in other honeycomb systems such as boron-nitride 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 up-spin or down-spin electrons near the Fermi level by implementing appropriate magnetic order. For small magnetic field, there are many Landau levels. When there are N spin-polarized 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 spin-Chern 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 spin-Chern 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 sz = ↑↓, 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 graphene21 so as to include the link connecting the next-nearest 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.

Figure 6: Illustration of Peirls phase in honeycomb lattice.
Figure 6

It is convenient to choose the gauge in which the Peirls phase is given by Φij = 2πΦlij between the i and j sites with lij the number assigned to the link; , and so on. The horizontal links are indexed as , with n = 1 at the lowest corner of one magnetic unit cell. This gauge is constructed by generalizing the one used in Ref. 22 to include the link between the next-nearest neighbor sites.

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).

Figure 7: Fan diagram, bulk-edge spectrum and Chern numbers.
Figure 7

(a1 ~ d1) Closer look of the Hofstadter's butterfly for Φ = p/400 with , and the fan diagram for the Landau levels. The horizontal axis is the magnetic flux Φ. (a2 ~ d2) Energy spectrum with zigzag edges derived based on the lattice theory at Φ = 1/100. The horizontal axis is the momentum k. We can determine by counting the number of edge modes. (a3 ~ d3) Chern numbers calculated based on the Kubo formula (17) as a function of μ at Φ = 1/100. The horizontal axis is the topological number . The same results on are obtained from the methods (a2 ~ d2) and (a3 ~ d3). See also the caption of Fig. 2.

Fan diagram

We introduce a pair of ladder operators, satisfying , where is the magnetic length. The Hamiltonian Hη is block diagonal and given by with the diagonal elements being in the basis {ψA, ψB}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 Kη and spin sz. 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 2-fold degenerate for silicene (d1).

Buld-edge correspondence

The most convenient way to determine the topological charge in the lattice formulation is to employ the bulk-edge correspondence21. The edge-state 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 sz = ↑↓ 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 propagation21. 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 down-moving states with spin sz, respectively, at the right edge.

Our main task is to determine the Chern and spin-Chern numbers for each Landau level. We presents the edge-state 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 sz in each valley Kη. Such a formula has been derived for graphene26,27. We may generalize it to be applicable to the Dirac system (2), for each spin sz. 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 bulk-edge 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 Grants-in-Aid for Scientific Research from the Ministry of Education, Science, Sports and Culture No. 22740196.

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  1. Department of Applied Physics, University of Tokyo, Hongo 7-3-1, 113-8656, 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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https://doi.org/10.1038/srep03435

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