Chern Insulator and Chern Half-Metal States in the Two-Dimensional

Letter pubs.acs.org/JPCL

Chern Insulator and Chern Half-Metal States in the Two-Dimensional Spin-Gapless Semiconductor Mn2C6S12 Aizhu Wang,†,‡ Xiaoming Zhang,† Yuanping Feng,*,§ and Mingwen Zhao*,† †

School of Physics and State Key Laboratory of Crystal Materials, Shandong University, Jinan 250100, Shandong, China Department of Electrical and Computer Engineering and Department of Physics, National University of Singapore, Singapore 117579, Singapore § Department of Physics & Centre for Advanced Two-Dimensional Materials, National University of Singapore, Singapore 117542, Singapore ‡

S Supporting Information *

ABSTRACT: Two-dimensional metal−organic frameworks (2DMOFs) with exotic electronic structures are drawing increasing attention. Here, using first-principles calculations, we demonstrate a spin-gapless MOF, namely, Mn2C6S12, with the coexistence of a spin-polarized Dirac cone and parabolic degenerate points. The Curie temperature evaluated from Monte Carlo simulations implies Mn2C6S12 possessing stable ferromagnetism at room temperature. Taking the spin−orbit coupling into account, the Dirac cone is gapped and the degenerate points are lifted, giving rise to multiple topologically nontrivial states with nonzero Chern number, which imply the possibility of Mn2C6S12 to be a Chern insulator and a Chern half-metal. Our results offer versatile platforms for achieving spin filtering or a quantum anomalous Hall effect with promising application in spintronics devices.

T

and the conduction bands is shown in Figure 1a. As the spin− orbit coupling (SOC) is taken into account, the Dirac cone will

he synthesis of graphene, a natural topological insulator (TI), has opened a word of Dirac cones and the physics that derives from them.1,2 Since then, Dirac materials have attracted huge interest in fundamental physics and practical applications. Indeed, graphene holds promise to revolutionize nanotechnology due to it being ultrathin, light, transparent, and resilient to bending.2−6 In addition, Dirac Fermions in graphene have been of great importance to the field of topological quantum matter and are promising for achieving fractional7−9/half-integer6,10 quantum Hall effects. Actually, the search for realistic two-dimensional (2D) Dirac materials, including graphynes, silicene, germanene, organic frameworks, metal−organic frameworks (MOFs), and so on, has benefited from the fruitful interplay between experiments and theories,11−21 which provides a versatile platform for hosting nontrivial topological states usable for the applications of electronic devices. More detail can be found in the Supporting Information. However, the space between 2D Dirac materials dreams and immediate reality was packed with applications. Manipulation of the spins of graphene through the inherent defects limited the applications. On the basis of the novel designs of band structure, the concept of spin-gapless semiconductors (SGSs) with linear or parabolic energy dispersions was proposed theoretically,22,23 inspiring considerable efforts in search for realistic materials with spin-gapless states.24−28 A typical spingapless state with the same spin direction for both the valence © XXXX American Chemical Society

Figure 1. Band structures of (a) a Dirac SGS with linear dispersion, (b) a spin semiconductor with parabolic dispersion, and (c) the coexistence state. The red and blue arrows indicate spin up and spin down, respectively.

be gapped, leading to the quantum anomalous Hall effect (QAHE).23,29 Subsequently, the QAHE has been predicted in SGSs, such as Mn2CoAl, Mn(C6H5)3, C14N12, and C10N9,24−26 which opened up a brand new avenue for applications in nextgeneration electronic and spintronics devices. The magnetic TIs, also referred to as Chern insulators, are characterized by nonzero integer Chern numbers.30,31 The Received: May 13, 2017 Accepted: July 30, 2017 Published: July 30, 2017 3770

DOI: 10.1021/acs.jpclett.7b01187 J. Phys. Chem. Lett. 2017, 8, 3770−3775

Letter

The Journal of Physical Chemistry Letters

each Mn atom bonds to six S atoms, while S atoms are connected via a C2 dimer, forming a 2D hexagonal pattern on the base plane. The optimized 2D lattice, with a chemical formula of Mn2C6S12 in each unit cell, is buckled with C atoms moving alternately up and down out of the base plane. The distance between two C atoms is about 1.497 Å, exhibiting the C−C single-bond feature. The length of the Mn−S bond is about 2.440 Å. The optimized lattice constant is 11.178 Å. The dynamic stability is confirmed by the phonon spectrum calculated along the highly symmetric directions in the Brillouin zone (BZ), as shown in Figure 2b. There are no modes with imaginary frequencies in the spectrum, and the monolayer is therefore expected to be dynamically stable. From the experimental point of views, the bimetallic oxalate networks formed by oxalate ligands and magnetic ions37−39 exhibit similar structural features. Benefiting from the well-developed bimetallic oxalate network technology and recent progress in nanotechnology, realization of the Mn2C6S12 lattice is plausible in the near future. We then turned to the electron spin polarization of the Mn2C6S12 monolayer. Self-consistent calculations gave a spinpolarized state with 8 μB magnetic moments in one primitive cell. The spin-resolved band structures are shown in Figure 3a. Clearly, the degeneracy of the two spin channels is lifted with remarkable spin-splitting in the bands close to the Fermi level. The valence and conduction bands of the spin-up channel meet at a single point at the Fermi level, whereas the spin-down channel has a band gap of about 1.03 eV, exhibiting clear features of SGSs.22,23 Interestingly, the lowest conduction band of the spin-down channel passes through the touching point (Dirac point) of the spin-up channel, as shown in Figure 3a. This unique band structure distinguishes the Mn2C 6S12 monolayer from the conventional reported SGSs.24−26 We also calculated the electronic band structure of the Mn2C6S12 monolayer using a hybrid functional (HSE06) and found that the SGS features remain intact. We evaluated the Fermi velocity (νf) of the Mn2C6S12 lattice by fitting the Dirac bands at k=K + q to the expression of νf = E(q)/h|q|. The Fermi velocities of electrons and holes are identical in our calculations, which are 0.03 × 106 m/s. The value is approximately 3.5% of that of graphene, 0.86 × 106 m/s from the present calculations. The low Fermi velocity of the Mn2C6S12 lattice compared with that of graphene is attributed to the longer lattice constant and larger voids in the porous framework. However, this value is almost of the same order as the electron saturation velocity in silicon crystals (∼105 m/s),

topologically nontrivial gaps due to SOC of Chern insulators reported so far can be classified into two typical groups, according to the strength of intrinsic magnetization. The type-I TI gap was produced by interspin SOC under moderate magnetization, while the type-II TI gap was induced by intraspin SOC under large magnetization.32 Most Chern insulators reported in the previous literature process the typeI TI gap.33−35 The type-II TI gap has also been predicted in 2D organic materials.25,26,36 However, the coexistence of these two types of gaps in 2D materials has never been reported. In this contribution, we propose a new 2D MOF, namely, Mn2C6S12, and demonstrate the intriguing electronic structures of this material. Using first-principles calculations, we show that the 2D Mn2C6S12 lattice has intrinsic electron spin polarization and stable ferromagnetism. The Curie temperature estimated by Monte Carlo (MC) simulations within the Ising model is considerably higher than room temperature. In the absence of SOC, the Mn2C6S12 lattice exhibits the features of a SGS with linear dispersion in one spin channel and parabolic dispersion in the other spin channel. The semiconducting spin channel has a band gap of 1.03 eV, while the Fermi velocity of the conducting spin channel, 0.3 × 105 m/s, is almost of the same order as the electron saturation velocity in bulk silicon. Taking SOC into account, type-I and type-II band gaps with nonzero Chern numbers are opened up, leading to topologically nontrivial states existing in the Chern insulator and Chern half-metal. Realization of the Mn2C6S12 lattice will greatly broaden the scientific and technological impact of spintronics. The relaxed geometric structure of the 2D Mn2C6S12 monolayer is shown in Figure 2a. In the 2D MOF lattice,

Figure 2. Honeycomb structure of the Mn2C6S12 lattice (a) and corresponding phonon spectrum along highly symmetric points in the Brillouin zone (b).

Figure 3. (a) Spin-resolved band structures and the electron density of states (PDOS) projected onto C, S, and Mn atoms of the Mn2C6S12 lattice. (b) Isosurfaces of the Kohn−Sham wave functions corresponding to the states (indicated by the black dotted region) nearest to the Fermi level. The energy at the Fermi level was set to zero. 3771


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Figure 4. Orbital-resolved electron density of states projected onto different atomic orbitals of C (a), S (b), and Mn (c) atoms. Spin-up and spindown channels are plotted in the top and bottom panels, respectively. (d) Average magnetic moment per Mn atom obtained from MC simulations as a function of temperature. The corresponding spin-resolved total electron density of states and the temperature-dependent heat capacity are presented in the insets of this figure. The energy at the Fermi level was set to zero.

single bond in the C2 unit. Therefore, the valences of Mn, C, and S in the framework are +3, +3, and −2, respectively. Interestingly, the exchange interaction between the local magnetic moments of Mn ions is hinted to by the Kohn− Sham electron wave function (Figure 3b), where the conduction electrons of the C atoms act as the intermediate of the magnetic interactions. The relative stability of the two magnetic coupling states can be evaluated from the energy difference (ΔE) between the FM and AFM states, ΔE = EFM − EAFM. The value of ΔE obtained from the present calculations is about −0.72 eV. The negative value implies that the FM ordering is energetically more favorable for the 2D MOF lattice. The stability of the FM state was further confirmed by using MC simulations within the Ising model. Without an external field, the Hamiltonian of the Ising model is written as the following

which implies high electron mobility in the conducting spin channel. High electron mobility in one spin channel and relatively large band gap in the opposite spin channel meet the demand of filtering the current into a single spin channel and thus are quite promising for spintronic device applications. To visualize the origins of the Dirac bands, we plotted the electron density of states (PDOS) projected onto different atoms (Figure 3a) and atomic orbitals (Figure 4a−c). We can see clearly that the Dirac bands come mainly from the pz orbitals of S atoms and the dxy/dyz/dxz/dx2−y2 orbitals of Mn atoms, while the contribution from C atoms is neglectable. Those characters are also consistent with the isosurfaces of the Kohn−Sham wave functions, as plotted Figure 3b. To study the possible magnetic coupling between local magnetic moments, we adopted two different initial spin configurations and got two stable magnetic orderings: ferromagnetic (FM) and antiferromagnetic (AFM) states. To visualize the spatial distribution of spins in the Mn2C6S12 lattice, we plotted the spin-polarized electron densities Δρ, which were calculated from the difference between the electron densities of two spin channels, Δρ = ρ↑ − ρ↓, as shown in the inset of Figure 4d and Figure S3. It is evident that the Mn ions carry most of the magnetic moments, with a small contribution of S atoms, which is consistent with the 3d orbital shapes of the Δρ isosurfaces, as well as the spin-polarized electron density of states. Each Mn atom provides three electrons to bond with S, leaving four unpaired d electrons with the same spin orientation, which leads to the local magnetic moments of 4 μB. Three electrons of a C atom transfer to the nearestneighbor S atoms, while the last one contributes to the C−C

Ĥ = −J0 ∑ m̂ i ·m̂ j i,j

where m̂ i and m̂ j are the local magnetic moments at nearestneighbor sites i and j, respectively. The nearest-neighbor exchange parameter, J0, of local magnetic moments can be determined from ΔE using the formula J0 = |ΔE|/6m2. We employed a 70 × 70 supercell containing 9800 local magnetic moments. The simulations lasted for 1 × 109 loops. In each loop, each spin was changed, and the possible values of m were 4, 2, 0, −2, and −4 because the calculated local magnetic moment assigned to each Mn atom was about 4.0 μB. The temperature-dependent magnetic moment per Mn atom 3772


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It is interesting to see that, besides the type-I gaps, SOC also leads to two type-II gaps (Δ3 and Δ4) arising from the intraspin interaction. This is the first report on the coexistence of type-I and type-II gaps in 2D materials. When the Fermi level is moved into these two gaps, the Mn2C6S12 lattice will be tuned to a Chern insulator and a Chern half-metal, respectively. The above topologically nontrivial states in the band structure of the Mn2C6S12 lattice are verified by calculating the Chern numbers (C) from the k-space integral of Berry curvature (Ω(k)) of the occupied bands using the Kubo formula31,42

obtained from MC simulations is shown in Figure 4d. It can be seen that the magnetic moment decreases to 3 μB at a temperature of ∼1240 K and becomes 0 μB when the temperature is higher than 1310 K. To make the FM− paramagnetic transition more clear, we calculated the heat capacity (Cv) of the Mn2C6S12 lattice using the expression ΔE T ΔT where ΔET is the change of the total energy of the system as the temperature is increased from T to T + ΔT. The calculated Cv as a function of temperature is shown in the inset of Figure 4. It is obvious that Curie temperature (TC) is about 1280 K and the FM−paramagnetic transition is a second-order phase transition. Although MC simulations may overestimate the TC value, such a high TC implies the stable ferromagnetism of the Mn2C6S12 lattice at room temperature. The stability of the ferromagnetism and the spin-gapless state is quite crucial for applications in spintronic devices. In the presence of SOC, the spin-gapless states would be gapped at the meeting points, leading to QAHE states.23,25,40,41 We therefore involved the SOC effects in the electronic structure calculations of the Mn2C6S12. The bands nearest to the Fermi level arise mainly from 3d orbitals of Mn and p orbitals of S, and the SOC effects are expected to be more pronounced than other MOFs.25 It is found that the SOC lifts the degeneracy of the two spin channels at the Dirac point, resulting in two gaps (Δ1 = 7.79 meV and Δ2 = 15.10 meV) as shown in Figure 5a. The two gaps belong to the type-I band Cv = lim

ΔT → 0

C=

1 2π

∫BZ Ω(k ⃗) d2k

Ω(k ⃗) =

∑ fn Ωn(k ⃗) n

Ωn(k ⃗) = − ∑ 2Im n ′≠ n

⟨ψnk|νx|ψn ′ k⟩⟨ψn ′ k|νy|ψnk⟩ (εn ′ k − εnk)2

where n is the band index, εnk and ψnk are the eigenvalue and eigenstate of band n, respectively, νx/y is the velocity operator, and f n is Fermi distribution function. The calculated anomalous Hall conductivity (AHC) of the Mn2C6S12 lattice as a function of the electron filling is presented in Figure 5b. We found that the Chern numbers of SOC gaps Δ1 and Δ3 acquire an integer value of −1, implying that the gaps are topologically nontrivial. However, as the Fermi level is moved to gaps Δ2 and Δ4, the Chern numbers deviate slightly from the integer values of ±1 due to the contribution from the conducting spin channel with nonzero Berry curvature. The change of the AHC induced by electrons filling the band between Δ1 and Δ2 is related to the unique Berry curvature distribution of the band, as shown in Figure S4d. The Berry curvature has positive values near the corners of the BZ, which will be added to the total Berry curvature of the occupied state according to Kubo formula as the Fermi level moving up from Δ1 to Δ2. Consequently, the AHC calculated from the integral of the Berry curvature will undergo a sharp positive turn of about 2. The existence of topological edge states is an important signature of 2D TIs. We calculated the edge states of a Mn2C6S12 nanoribbon by employing the WANNIER90 package43 to establish the maximally localized Wannier functions (MLWFs). The band structure reproduced by Wannier interpolation fit well with the DFT results, as shown in Figure S2. On the basis of a recursive strategy,44 the edge Green’s function of the semi-infinite lattice was constructed from the MLWFs. The local density of states of the edges calculated from the Green’s function is plotted in Figure 5c. From this figure, we can see that the bulk states are connected by the topologically nontrivial edge states, which is a feature of Chern insulators. More importantly, the bulk SOC band gaps (Δ1 and Δ3) are well separated from other bands and thus implementable for achieving the QAHE at room temperature. As the Fermi level locates in gap Δ2 (or Δ4), the Mn2C6S12 nanoribbon will carry quantized spin-up (or -down) current along the edges characterized by AHC, together with current in the opposite spin direction in the interior region arising from the conducting band. These unique electron transport properties offer new opportunities for the design of spin filters and spintronics devices. Briefly, we demonstrate in this work a 2D spin-gapless MOF, namely, Mn2C6S12, with the coexistence of Chern insulator and

Figure 5. (a) Band structures of the Mn2C6S12 lattice with SOC. (b) Calculated AHC and (c) semi-infinite edge states based on MLWFs. The energy at the Fermi level was set to zero.

gap because they are related to the interaction between two spin channels. The Fermi level is pinned at ∼22 meV below gap Δ1 due to the existence of a tiny electron pocket in the spin-up channel. Under a small gate voltage, the Fermi level can be tuned into gap Δ1 or gap Δ2. As the Fermi level locates in gap Δ1, the Mn2C6S12 lattice will behave as a Chern TI that is available for achieving QAHE. More interestingly, when the Fermi level is moved upward to gap Δ2, the Mn2C6S12 lattice will become a Chern half-metal, that is, one spin channel is insulting while the opposite spin channel is conducting.32 The coexistence of Chern insulator and Chern half-metal states offers versatile platforms for novel phenomena, which are crucial for device applications. 3773



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Chern half-metal states on the basis of first-principles calculations. The Curie temperature evaluated by the MC simulations within the Ising model is considerably higher than room temperature. Both the type-I and type-II band gaps in the range of 7.79−15.10 meV are formed due to the interspin and intraspin SOC. The topologically nontriviality of these band gaps is confirmed by the nonzero Chern numbers and the edge states of nanoribbons. These results offer versatile platforms for achieving SGSs and topological phases in 2D MOFs, which are crucial for applications in nanoscale electronic devices.

COMPUTATIONAL METHODS The first-principles calculations within density functional theory (DFT) were implemented by the Vienna Ab initio Simulation Package (VASP).45 A generalized gradient approximation (GGA) in the form of Perdew−Burke−Ernzerhof (PBE)46 was adopted to describe the electron−electron interactions. In our work, we adopted a GGA+U strategy to describe the strong Coulomb interaction between the partially filled 3d shells of Mn. The values of the Coulomb parameter U and exchange parameter J were set to 4.0 and 0.0 eV, as implemented in many previous works.47−50 The energy cutoff employed for planewave expansion of electron wave functions was set to 520 eV, and the electron−ion interactions were treated using projectoraugmented wave (PAW) potentials.51 The system was modeled by unit cells repeated periodically on the x−y plane, while a vacuum region of about 25 Å was applied along the z direction to avoid mirror interaction between neighboring images. Structural optimization was carried out using a conjugate gradient (CG) method until the remaining force on each atom was less than 0.05 eV/ Å. ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.7b01187. Common 2D materials with Dirac cones, comparison of band structures based on DFT and Wannier90, spinpolarized electron density at FM and AFM ordering, and mutation of the Chern number and the Berry curvature (PDF)

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REFERENCES

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (Y.F.). *E-mail: [email protected] (M.Z.). ORCID

Aizhu Wang: 0000-0003-2297-5426 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is supported by the National Natural Science Foundation of China (No. 21433006), the 111 project (No. B13029), the National Key Research and Development Program of China grant 2016YFA0301200, and the National Super Computing Centre in Jinan. 3774


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