Two Dimensional Antiferromagnetic Chern Insulator: NiRuCl6 - Nano

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Two Dimensional Antiferromagnetic Chern Insulator: NiRuCl6 Pan Zhou, Chang Qing Sun, and Lizhong Sun Nano Lett., Just Accepted Manuscript • Publication Date (Web): 20 Sep 2016 Downloaded from http://pubs.acs.org on September 20, 2016

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Two Dimensional Antiferromagnetic Chern Insulator: NiRuCl6 P. Zhou,† C. Q. Sun,‡ and L. Z. Sun∗,‡ Key Laboratory of Low-dimensional Materials and Application Technology, School of Material Sciences and Engineering, Xiangtan University, Xiangtan 411105, China, and Hunan Provincial Key laboratory of Thin Film Materials and Devices, School of Material Sciences and Engineering, Xiangtan University, Xiangtan 411105, China E-mail: [email protected]

Abstract Density functional theory (DFT) and Berry curvature calculations show that quantum anomalous Hall effect (QAHE) can be realized in two-dimensional(2D) anti-ferromagnetic (AFM) NiRuCl6 . The results indicate that NiRuCl6 behaves as an AFM Chern insulator and its spinpolarized electronic structure and strong spin-orbit coupling (SOC) are responsible for the QAHE. By tuning SOC, we found that the topological property of NiRuCl6 arises from its energy band inversion. Considering the compatibility between the AFM and insulators, AFM Chern insulator provides a new way to archive high temperature QAHE in experiments due to its different magnetic coupling mechanism from that of ferromagnetic (FM) Chern insulator. Keywords:Quantum anomalous Hall effect, antiferromagnetic, Chern Insulator, spin-polarization

∗ To

whom correspondence should be addressed Laboratory of Low-dimensional Materials and Application Technology, School of Material Sciences and Engineering, Xiangtan University, Xiangtan 411105, China ‡ Hunan Provincial Key laboratory of Thin Film Materials and Devices, School of Material Sciences and Engineering, Xiangtan University, Xiangtan 411105, China † Key

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The quantum anomalous Hall effect (QAHE), arising from spin-orbit coupling (SOC) and time reversal symmetry (TRS) broken, shows voltage transverse to the electric current even in the absence of an external magnetic field. Dissipationless boundary states exist in the two edges of materials with QAHE. 1–3 The zero-field dissipationless chiral edge transport channels of QAHE behaving as an information express way for next generation electronics can be used in high speed, low power consumption electronics. 1,4 Although the fundamental principle of QAHE has been proposed in a honeycomb lattice model 5 long ago, only latest experiment proves that the QAHE can be realized in Cr-doped (Bi,Sb)2 Te3 . 4 In the system, its ferromagnetic ordering and SOC are sufficiently strong to give rise to a topological nontrivial phase with a finite Chern number. However, some extreme experimental conditions including ultra-low temperature hinder its practical application. Nowadays, searching for materials with QAHE properties working under high temperature and with higher quantum plateau are important issue in the scope. The direct approach to realize QAHE is to introduce FM order in quantum spin Hall insulators (QSHI) to break its time-reversal symmetry and turn its helical edge states to chiral ones. 6 Such approach requires Chern insulator, or ferromagnetic (FM) insulator with a non-zero Chern number. Unfortunately, finding FM Chern insulator is very challenging because there are few ferromagnetic insulators in nature. In conventional diluted magnetic semiconductors, long-range FM order is determined by the Ruderman-Kittel-Kasuya-Yosida(RKKY) interaction that damages QAHE through producing foreign conduction channels. The success observation of QAHE in Cr or V doped (Bi,Sb)2 Te3 thanks to the van Vleck mechanism. 4 According to present experimental situation, besides the nonuniform TM doping, the limitation for improving the Curie temperature is the van Vleck mechanism that is not strong enough to overcome thermal fluctuation due to its second order perturbation nature. As mentioned above, to archive QAHE, researchers pay more attentions to break the time reversal symmetry of topological insulator by importing ferromagnetic order. This reality easily allows people produce illusion that QAHE must connect with ferromagnetism. In the present letter, by taking two dimensional transition metal halides (TMHs) as prototype model, we prove


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Figure 1: (color online) Crystal structure for single layer NiRuCl6 from the top (a) and side view (b). that QAHE can be realized in AFM material as long as the material shows spin-polarization and its time reversal symmetry is broken. The 2D TMHs mainly composed of transition metal atoms and halogen atom with the general formula Mn Ym , where M is a transition metal and X is halogen element (Cl, Br and I). A recent work 7 reveal that Dirac electronic state can happen in VCl3 with this crystal structure. To produce spin-polarization, we construct a new type 2D TMHs of M1 M2 Y6 , where the M1 and M2 represent 3d and 4d TM atoms, respectively, and Y is Cl. Our results indicate that under 3%-10% compress strain in plane the single layer NiRuCl6 is 2D AFM QAHE insulator, or AFM Chern insulator. Moreover, the pristine single layer NiRuCl6 is a half-metal AFM (HMAFM) 8,9 with totally compensating magnetic moment. Manipulation of spin degrees of freedom in electronic devices to realize information storage, logical calculation, and other functional devices is the premise and main task of spintronics. However, the materials with spin-polarization such as half-metal (HM) and half-semiconductor (HS) often show ferromagnetic order which will restrict the generation of spin-polarized current due to the magnetic domains and stray field accompanied FM order. 9 Moreover, the Curie temperatures of the HM and HS are usually below the room temperature due to the generally weak FM exchange interaction. Using spin-polarized AFM materials, such as HMAFM,


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could be a feasible solution. The magnetic ground state of NiRuCl6 is AFM, whereas its valence band maximum (VBM) and conduction band minimum (CBM) are spin polarized, which is an excellent candidate for spin-polarized AFM materials.

Figure 2: (color online) Energy band (a) and density of states (b) for pristine NiRuCl6 . (c) and (d) are the energy band under 5% compress stress without and with SOC, respectively. The electronic structures of NiRuCl6 was studied with projector augmented wave (PAW) 10 formalism implemented in the Vienna ab initio simulation package. 11,12 The Perdew-Burke-Ernzerh version of general gradient approximation was used to describe the exchange and correlation functional. The plane-wave cutoff energy was set to be 600 eV and a vacuum space of larger than 15 Å was set to avoid the interaction between two adjacent layers. The convergence criterion for the total energy was 10−7 eV. The crystal lattice and atoms were all relaxed without any restrictions until the Hellmann-Feynman force on each atom was smaller than 0.01 eV/Å. 4 ACS Paragon Plus Environment

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Figure 3: (color online) Schematic representation of the quasi-superexchange interaction between Ni2+ - Cl-3pσ - Ru4+ in the 90◦ case. The structure of NiRuCl6 as shown in Fig. 1 was derived from single layer RuCl3 13–18 by replacing the one Ru atoms in the primitive cell with Ni atoms. The Ni and Ru atoms are surrounded by a distorted octahedron of Cl atoms. We calculated phonon spectrum 19 of the structure and there are no imaginary frequency, which confirms its dynamic stability. The total energy results indicate that under AFM state (the spin directions of Ni and Ru are reversal) the NiRuCl6 is 54 meV more stable than it is under the FM state (the spin directions of Ni and Ru are identical). Therefore, the magnetic ground state of the NiRuCl6 is AFM and its exchange-coupling constant in the Heisenberg model is negative. We plot energy band and density of states (DOS) of NiRuCl6 under AFM state in Fig.2(a) and (b). The results show that the 2D NiRuCl6 is a kind of typical zero magnetic moment spinpolarized HM, namely the total magnetic moment in the unit cell is zero and the electronic states around Fermi level is 100% spin-polarized. Through closely inspecting the projected DOS (PDOS) of the system, we found that the three energy bands around Fermi level mainly come from the 4d orbitals of Ru. Under octahedron surrounding the d orbital of TM would be split into eg and t2g . ′

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If a further triangle crystal field is present, t2g is decomposed into ag , eg1 , and eg2 ; and eg breaks into eg1 and eg2 . The PDOS of the system indicates that the three energy bands around Fermi level 5 ACS Paragon Plus Environment

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Figure 4: (color online) (a) Anomalous Hall conductivity when we shift the Fermi level around its original Fermi level; (b) The distribution of the Berry curvature in momentum space for NiRuCl6 ; (c) Calculated edge state for semi-infinite boundary of NiRuCl6 . All results are obtained under 5% compress strain. ′

′

are ag , eg1 , and eg2 of Ru in spin-down channel as shown in Fig.2, and the Fermi level cross the ′

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energy bands of eg1 and eg2 producing the HM nature in the systems. In normal TMCl2 or TMCl3 (TM is 3d transition metal atom) materials, their magnetic order is determined by the superexchange interaction, 20–22 by which TM atom interaction follows the GKA rule 23,24 through the intermediate Cl atom. However, the superexchange builds on highly localized 3d TM materials. For NiRuCl6 , the magnetic distribution is deviated from normal superexchange due to the existence of more decentralized 4d orbits of Ru. The magnetic coupling mechanism in NiRuCl6 can be named as quasi-superexchange as illustrated in Fig.3. The bondangle of Ni-Cl-Ru as shown in the figure is close to 90◦ (96.38◦ ). Along with the bonding, firstly, 4s electrons of Ni transfer to Cl and form Cl− ion. For the interaction between Cl− and Ni, the pσ of Cl− is only non-orthogonal with the dx2 −y2 of Ni forming a partial covalent bond between them. Then the electron virtual hopping occurs from the pσ orbital of Cl− to the eg of Ni2+ in the spin-up channel when the system is excited. The remaining spin-down electron of Cl− with the form of pπ interacts with the t2g orbital of Ru4+ . According to the superexchange mechanism and GKA rule, the magnetic moment of Cl ion should be negative (spin-down), ferromagnetic coupling with Ni2+ 6 ACS Paragon Plus Environment

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and anti-ferromagnetic coupling with Ru4+ . However, our first-principle calculations show that the Cl ion shows tiny positive (spin-up) magnetic moment. The distinct phenomenon can be attributed to the relative delocalization of the 4d orbital of Ru in comparison with that of 3d of Ni. The conclusion can be verified by the Bader charge 25 (the Bader charge of Cl, Ni, and Ru is 0.365, -0.939, and -1.225e, respectively) that Ru loses more electrons than Ni in the bonding process. According to the quasi-superexchange mechanism, the actual spin orientation of Cl atom is decided by the competition between the interaction strength of Cl-Ni and Cl-Ru due to the charge transfer derived from a virtual hopping. In NiRuCl6 , the interaction strength between Cl and Ru is larger than that between Cl and Ni that produces t2g orbits of Ru4+ transfer more spin-up electrons to Cl atom and finally leads to a weak spin-up magnetic moment for Cl. The quasi-superexchange mechanism in NiRuCl6 produces the anti-ferromagnetic coupling between Ru and Ni in NiRuCl6 . The difference of exchange splitting between 3d of Ni and 4d of Ru produces the 100% spin-polarization around the Fermi level. Although the pristine 2D NiRuCl6 is a typical HMAFM with zero magnetic moment, a global band gap can be opened by applied 3% to 10% compress strain in the plane. The compress strain puts the system transforming from zero magnetic moment HMAFM to zero magnetic moment half-semiconductor (HS) AFM. In the present work, we chose 5% compress strain (which can be easily realized in experiments through proper substrate as discussed below) as example to analyze its topological properties. Although we mainly analyze the results of 5% compressed NiRuCl6 , for pristine 2D NiRuCl6 , according to the Wilson loop method, 1,26 it is a kind of topological Chern metal. The band structures of the system under 5% compress strain in plane are shown in Fig.2 (c) and (d). The results in Fig.2(c) indicate that under 5% strain without SOC the system is still HMAFM. When the SOC effect is considered and the system is under 5% strain, it show about 30 meV global band gap, as shown in Fig.2(d). With the help of our QAHE code 27 and wannier90 code, 28 we calculate the anomalous Hall conductivity with the formula:

σxy =

e2 C h¯


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C = −∑ n

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∫

dκ f (κ )Ωn,z (κ ), 3 n BZ (2π )

where C is the first Chern number, it will be finite integer value if QAHE can be realized in the material. Ωn,z (κ ) is the Berry curvature in reciprocal space, we can get the Chern number from the integral of Berry curvature in total first Brillouin zone. Our results show that the Chern number of the system is 2 when the Fermi level locates in the band gap. Meanwhile the system remains HSAFM under the 5% compress strain in the plane; namely, the system behaves as AFM Chern insulator. Its anomalous Hall conductivity (AHC) as shown in Fig.4(a) indicates that the QAHE gap can reach 20 meV when 5% compress strain is applied. The non-quantized AHE part in the low energy range in Fig.4 (a) derives from the crossover between Fermi level and the energy bands ′

around K and K point, the details can be found in Figure S1 in the supplementary materials. The Berry curvature distribution for the QAHE state is displayed in Fig.4 (b). The distribution shows ′

that the main contribution of Berry curvature comes from the high symmetric line Γ-K and Γ-K ′

close to the two valley K and K in the reciprocal space. Due to the space inversion symmetry ′

is broken, K and K are independent of each other and their local maxima of Berry curvature is ′

different. The integration of the Berry curvature around the K or K were performed separately as indicated in the Figrure S2 in the supplementary materials. The results show that CK ≈ 2 and CK ′ ≈ 0 (0.13/2π ) indicating the Chern number mainly derives from the contribution of the Berry curvature around K. The reason will be discussed below. It is known that nonzero integer Chern number for a 2D material guarantees the existence of quantized edge states when the material is cut into one-dimensional ribbon. To check whether similar quantized edge states exist for NiRuCl6 , we calculated the edge states by constructing semi-infinite boundary NiRuCl6 with Maximally Localized Wannier functions Hamilton. The results are shown in Fig.4(c). There are two topologically protected chiral edge states between the valence band and the conduction band. The results clearly indicate the Chern number of the system is C = 2. The above results indicate that NiRuCl6 behaves as AFM Chern insulator. To understand the ′

special topological phenomenon, taking K and K as a reference, we calculate the local orbital pro-


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′

′

jection of wavefunction to label the contribution of ag and eg1 (or eg2 ) to the three bands around the Fermi level as showed in Fig. 5 (The calculation detail can be found in the supplementary materials). From low to high energy, the three bands around the Fermi level in the figure are labeled as v1, c1, and c2, respectively. Firstly, we consider the electronic state along Γ-K. As analysis above, ′

′

the three 4d states of Ru (ag , eg1 , and eg2 ) located around the Fermi level for pristine NiRuCl6 determine its topological properties. The results indicate that around K the c2 band mainly comes ′

′

from ag state, whereas c1 and v1 bands are nearly equal proportional hybrid by eg1 and eg2 as ′

′

shown in Fig.5 (a1) and (b1). Because the overlap integration between ag and eg1 (or eg2 ) under momentum operator must be zero (the details are provided in supplementary materials), the topo′

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logical properties of NiRuCl6 should be originated from the inversion between eg1 and eg2 due to strong SOC effect. When the SOC is not turned on, large part of the three bands are equal linear ′

′

combination of eg1 and eg2 which is depicted as magenta in Fig.5 (a1) and (b1). When the SOC ′

′

is fully considered, eg1 (blue) and eg2 (red) bands are clearly separated each other as shown in ′

′

Fig.5 (a2) and (b2). The separation between eg1 and eg2 around K plays a role of energy inversion and leads to sharp peaks in Berry curvature space. These peaks finally result in the AFM Chern ′

′

insulator nature of the system. The eg1 and eg2 own same symmetry and different wavefunction phase, 29 that the band gap must close if they exchange each other. To find out the close of the band gap along Γ-K, we calculate the energy band of NiRuCl6 under different SOC strengths and different compress strains in plane as shown in Figure S3 in the supplementary materials. The close of ′

′

the band gap between eg1 and eg2 around K point is clearly shown in Figure S3(a) when the SOC ′

′

strength increases. Moreover, when SOC is considered, a global energy gap between eg1 and eg2 will be induced by external compress strain. The results in Figure S3(b) indicate that the inversion ′

′

between eg1 and eg2 and the global energy gap are remained when the compress strain is within [4% to 10%]. When the compress strain is larger than 10% the above band inversion disappears. ′

For the case of K , although the dispersion character of energy band around Fermi level are very similar to that around K as shown in Fig.5, according to the projected percentage listed in Table S1, the electronic state order of c1 and c2 reverse to that around K. Moreover, as the increase in


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′

the SOC, the bands around the K do not show close-reopen feature that it does not exhibit topo′

logical inversion along Γ-K . The conclusion agrees with the results of the integration of the first Chern number. To quantitatively describe the inversion strength, following the work of Liu et al., 30 we calculate the spin-orbit spillage of NiRuCl6 and the results are shown in Fig.5 (a3) and (b3). The details of this method can be found in the supplementary materials. The spillage γ (k) ≥1 in some k-point contributes the topological phase transformation between normal insulator and Chern insulator. The results indicate that the topological phase transformation mainly comes from the inversion around K and some point between Γ and K in reciprocal space. The results also indicate ′

that although sharing similar energy band dispersion between Γ-K and Γ-K , their intrinsic electronic properties are absolutely different. This is coincide with the Berry curvature contribution ′

to quantized Chern number around K and K discussed above. To verify the spin polarization of the bands around the Fermi level when the SOC is turned on, we calculated the real spin-projected energy band along high symmetry lines around Berry curvature extreme value in Figure S4. We only chose the three bands around the Fermi level because the non-zero Chern number absolutely comes from the sublattice of Ru, and there are no pseudospin exist in NiRuCl6 . Moreover, the localized spin-projected density of states (LDOS) of edge obtained from the semi-infinite nanosheets with the basis of maximal localized Wannier function is also calculated. The results are shown in the Figure S5 in the supplementary materials. The results indicate that, after SOC is turned on, the bands around Fermi level as well as the edge states are almost totally spin-polarized. According to the work of Weng et al. 1 and Liang et al., 31 the 2D NiRuCl6 is a spin-polarized QAHE system. Finally, let us compare the QAHE in graphene and NiRuCl6 . For graphene, when exchange ′

field is weak, the QAHE come from the two separate skyrmions around K and K points in reciprocal space. 32 When the limit of M/λR ≫ 1 (M and λR represent the size of exchange field and Rashba spin-orbit coupling, respectively) is achieved, both real-spin skyrmions and A/B sublattice pseudospin merons together contribute to the formation of QAHE in graphene. 33 These two situations can not be applied to 2D NiRuCl6 due to its single spin texture and lack pseudospin freedom. The low energy states around Fermi level of NiRuCl6 mainly come from 4d orbits of Ru, whereas 3d


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′

orbitals of Ni locate in the deep energy level, namely the states around K/K completely come from the Ru sublattice. This is absolutely different from the situation of graphene, where A/B sublat′

tices are equivalent and respectively contribute to the valley K and K . The situation of NiRuCl6 is similar with that of spin-polarized QAHE, 1,31 the energy band of spin-up and spin-down are totally separated, and only one spin energy exist around Fermi level due to large exchange splitting. For device application of 2D NiRuCl6 , besides non-dissipative electronic transport in future electronics, its spin-polarized QAHE can be devised a kind of electric double-layer transistors (EDLTs) with the help of liquid ion gate. 34 The Electric double-layer transistor based on NiRuCl6 we proposed is shown schematically in figure S6. The liquid ion gate can effectively tune the Fermi level of NiRuCl6 nanoribbon and then act as on-off function. If we use a kind of piezoelectric substrate 35 as illustrated in Figrue S6, a horizontal electric field can induce homogeneous biaxial strain to the substrate, and then this strain can be controllably exert on the superficial 2D materials (NiRuCl6 ). Moreover, when the Fermi level is in the bulk gap, the electronic transport is dissipationless. The fundamental application of spin-polarized QAHE can also refer to previous works. 31,36 The Néel temperature of NiRuCl6 will limit its future applications in spintronics (HMAFM

Figure 5: (color online) The energy band structure along high symmetry line Γ-K without (a1) ′ and with SOC (a2) and Γ-K without(b1) and with SOC(b2). The chromaticity of the three bands ′ derive from the compound through the three primary colours representd ag (green), eg1 (blue), and ′ eg2 (red).(a3)(b3) spin-orbit spillage along high symmetry line. and HSAFM) and QAHE. To this end, we used Monte Carlo (MC) and Ising model to calculate its 11 ACS Paragon Plus Environment

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ˆ i ·m ˆ j , where m ˆi Néel temperature. In the calculations Hamiltonian can be written as Hˆ = − ∑i j J m ˆ j are the magnetic moments at sites i and j, respectively. The term J is the exchange parameand m ter which is determined by the exchange energy Eex (54meV) with the formula of J = (1/3)Eex /2m2 . Where m =| m | and the factor of 1/3 is included because there are 3 magnetic coupling interactions in one unit cell. Therefore, J for NiRuCl6 is 2.25 meV. A 50 × 50 supercells and 105 loops are carried out during the MC simulations. The Specific Heat Capacity(SHC) as the function of temperature T were calculated and the results are showed in Fig.6. The SHC reaches the highest value when the temperature is close to 120 K suggesting a thermal-induced antiferromagnetic to paramagnetic phase transition. The temperature is several order higher than current experimental temperature. Considering the strong coupling mechanism of AFM, it is expected show QAHE in experiments under high temperature. Wehling et al. 37 previously pointed out that the electronic structure and magnetic properties

Figure 6: (color online) The Specific Heat Capacity of NiRuCl6 as the function of temperature T for 50 × 50 lattices. of TM doped systems are very sensitive to local Coulomb interaction U of TM atom. In present letter, we tested the result of U=3.0 and 5.0eV for Ni which does not affect the main conclusion in present work because the energy bands around the Fermi level mainly come from 4d orbitals of Ru. We also test the U of Ru up to 2.5 eV, which is close to recent constrained random phase


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approximation’s value of SrRu2 O6 . 38 As the U increases, the maximum of energy band v1 shifts to Γ and concrete analysis can be found in supplementary materials. Although the different U of Ru leads to different details of electronic structure of NiRuCl6 as shown in Figure S7 in the supplementary materials, the system is remains AFM Chern insulator. Moreover, the recent proposed modified Becke-Johnson(mBJ) potential 39 is also used to validate the conclusion in our present work, which can be found in the Figure S8 in the supplementary materials. When the system is under 7% compress strain, reversal topological order are exist for K on both sides of Fermi level, the system behaves as AFM Chern insulator. The results derived from mBJ further confirm the conclusion obtain from PBE. To prove the experimental synthesis of two dimensional monolayer NiRuCl6 , we calculate the Gibbs free energy of bulk-form multilayer NiRuCl6 and α -RuCl3 , whose structures are similar to graphite. α -RuCl3 is the parent material of bulk-form NiRuCl6 and synthetized in the early 1930s. 13 The results show that the Gibbs free energy of multilayer NiRuCl6 is lower than its counterpart of RuCl3 . Moreover, the Helmholtz free energy indicates that the stability of single-layer NiRuCl6 is comparable with its parent single-layer RuCl3 . The dependence of the stability of NiRuCl6 on the mole fraction of Ru and disorder arrangement in the cation sites is investigated. The results are shown in Table S2, Figure S9, Figure S10, and Figure S11 in the supplementary materials. The results indicate that the three dimensional layered NiRuCl6 can be synthesized with the similar method of producing RuCl3 : mixing the metal powder of Ru and Ni and react with chlorine under appropriate temperature and carbon monoxide. When the multilayer NiRuCl6 synthesized in experiments, the monolayer NiRuCl6 can be obtained through exfoliation. In summary, 2D NiRuCl6 is an AFM Chern insulator. The quantum anomalous hall effect (QAHE), zero magnetic moment AFMHM, and zero magnetic moment AFMHS can be archived in the system. Its topological properties under compress strain derive from the reversal of 4d states of Ru around the Fermi level. When external compress strain produces global energy gap, QAHE can be archived in the system. Considering its magnetic coupling is stronger than that of RKKY and van Vleck mechanism, AFM Chern insulator is hoping to realize high temperature QAHE, AFMHM,


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and AFMHS in experiments. Our results broaden the outlook of Chern insulator and QAHE as well as pave a new way to realize high temperature QAHE.

ASSOCIATED CONTENT Supporting Information Supporting Information Available: The details of the crossover between Fermi level and energy bands around the Fermi level; The first Brillouin zone for the integration of first Chern number; ′

′

′

′

The method of projection for ag , eg1 , and eg2 ; overlap integration between ag and eg1 (or eg2 ) under momentum operator; Energy band with different SOC strength under 5% compress strain and different compress stress with SOC; The details of spin-orbit spillage calculations; The spin-projected energy band and LDOS for semi-infinite edge when SOC is turn on; Electric double-layer transistor based on NiRuCl6 ; Energy band structure and Berry curvature distribution for different U of Ru and different strain;The energy band structure of mBJ potential with different strain; The lowest energy changes with different concentration of Ru in 2×2 supercell; All structures and energy with different distribution of Ni and Ru in 2×2 supercell; And the Helmholtz free energy of monolayer RuCl3 and NiRuCl6 as a function of temperature. This material is available free of charge via the Internet at http://pubs.acs.org.

Notes The authors declare no competing financial interests.


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Acknowledgement P. Zhou thank useful discussions with Jianpeng Liu and Wei Ku. This work is supported by the National Natural Science Foundation of China (Grant No. 11574260) and scientific research innovation project of Hunan Province(CX2015B219).

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Two Dimensional Antiferromagnetic Chern Insulator: NiRuCl6 - Nano

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