Quantum anomalous Hall effect in stable 1T-YN$_2$ monolayer with a large nontrivial band gap and high Chern number
The quantum anomalous Hall (QAH) effect is a topologically nontrivial phase, characterized by a non-zero Chern number defined in the bulk and chiral edge states in the boundary. Using first-principles calculations, we demonstrate the presence of the QAH effect in 1T-YN monolayer, which was recently predicted to be a Dirac half metal without spin-orbit coupling (SOC). We show that the inclusion of SOC opens up a large nontrivial band gap of nearly eV in the electronic band structure. This results in the nontrivial bulk topology which is confirmed by the calculation of Berry curvature, anomalous Hall conductance and the presence of chiral edge states. Remarkably, a high Chern number is found, and there are three corresponding gapless chiral edge states emerging inside the bulk gap. Our results open a new avenue in searching for QAH insulators with high temperature and high Chern numbers, which can have nontrivial practical applications.
The discovery of quantum Hall effect brought about a new fundamental concept, the topological phase of matter, to condensed matter physics Klitzing et al. (1980). The quantum Hall effect is obtained in a two-dimensional (2D) electron gas in the presence of a strong perpendicular external magnetic field, which drives the electrons to fill in the discrete Landau levels, resulting in the quantized Hall conductance. Nevertheless, Landau levels are not the necessary ingredient for realization of the quantum Hall effect, where the integer Hall plateaus are actually interpreted by Chern numbers, a type of topological invariants defined in the 2D momentum space Thouless et al. (1982). The topological interpretation implies that the realization of quantum Hall effect may be achieved without external magnetic field. The first toy model for the quantum Hall effect without Landau level, i.e. the quantum anomalous Hall (QAH) effect was proposed by Haldane in 1988 in a 2D honeycomb lattice with next-nearest-neighboring hopping modulated by staggered flux Haldane (1988). While there had been theoretical studies of QAH effect in solid state materials Onoda and Nagaosa (2003); Qi et al. (2006), little progress was made in experiment until the discovery of time-reversal (TR) invariant topological insulators Hasan and Kane (2010); Qi and Zhang (2011). Realization of QAH effect combines several basic ingredients that the system should be in 2D regime, insulating, TR symmetry breaking with a finite magnetic ordering, and has a non-zero Chern number in the valence bands Weng et al. (2015); Zhang et al. (2016). Numerous theoretical schemes for realizing QAH insulators have been proposed, including the magnetically doped quantum well-based 2D topological insulators Liu et al. (2008), graphene with transition metal (TM) adatoms (3, 4 and 5) Qiao et al. (2010); Zhang et al. (2012a); Acosta et al. (2014); Hu et al. (2015), buckled honeycomb-lattice systems of group IV or V elements with TM adatoms Ezawa (2012); Zhang et al. (2012b, 2013), half-functionalized honeycomb-lattice systems Huang et al. (2014); Wu et al. (2014a), heterostructure quantum wells Garrity and Vanderbilt (2013), organic metal frameworks Wang et al. (2013), and also ultracold atom systems Wu (2008); Liu et al. (2010). In particular, following the proposal in Ref. Yu et al. (2010), the QAH effect was firstly realized in experiment using Cr-doped magnetic thin-film topological insulator, with the quantized Hall conductance being observed Chang et al. (2013). The QAH states have also been reported in experiment for ultracold atoms, with the Haldane model Ess () and a minimized spin-orbit coupled model Wu et al. (2016a) being realized, respectively. Realization of QAH phases is of great interests in both fundamental theory and potential applications. For example, the chiral edge states of a spin-orbit coupled QAH insulator may exhibit novel topological spin texture in real space, which can have applications in designing spin devices Wu et al. (2014b). More interestingly, the heterostructure formed by a QAH insulator and conventional -wave superconductivity can realize chiral topological superconductors, which hosts Majorana zero modes binding to vortices and chiral Majorana edge modes in the boundary Qi et al. (2010); Liu et al. (2014a). Remarkably, a recent experiment observed such chiral Majorana edge modes based on the QAH insulator/-wave superconductor heterostructure, where a -plateau of tunneling conductance was observed He et al. (2017).
Albeit the important experiment progress made for realization, there are limitations for the current study of real QAH materials. First, so far the QAH phases in solid state materials are observed with quite low temperature, typically in the order of , due to small topological bulk gap Chang et al. (2013). To enable broad studies of fundamental physics and potential future applications, it is important to have QAH insulators with large bulk gap and observable with high temperature. Furthermore, the current experimental studies are focused on the QAH states with low Chern number . The QAH insulators with high Chern numbers are different topological phases which are expected to bring about new fundamental physics and interesting applications, but are yet to be available in experiment. This motivates us to search for new QAH insulators based on the transition metal compounds.
Transition metal is defined as an element whose atom has a partially filled sub-shell or which can give rise to cations with an incomplete sub-shell McNaught and McNaught (1997). Due to the partially filled shell, TMs can have many different oxidation states when forming compounds and show many appealing electronic, magnetic, and catalytic properties. Strong electronegative elements (groups V, VI, and VII) are easily combined with TMs to form stable compounds. One of the most attracting compounds are the transition metal dichalcogenides (TMDs) with chemical composition TMX, where TM stands for the TM and X is a chalcogen element such as S, Se, or Te Chhowalla et al. (2015); Liu et al. (2015a). TMD monolayers have many new physical properties as compared to their bulk counterparts due to their reduced dimensions. As confirmed by experiment, TMDs can exhibit three different structures, called 1H, 1T, and 1Tﬂ Qian et al. (2014). Using first-principles calculations, TMD monolayers have been predicted to show both semiconducting (1H-MoS) Mak et al. (2010) and conducting (1T-PtSe) Wang et al. (2015a) properties, spin polarization effect (1H-VS) Ma et al. (2012) and QSH effect (1Tﬂ-WTe) Qian et al. (2014), where some of them have been realized experimentally Chhowalla et al. (2015); Liu et al. (2015a). However, the QAH effect has not yet been predicted in TMD structures. A recent expansion of the TMD-like compounds has been realized through MoN, a nitrogen-rich TM nitrides (TMN) that has been synthesized through a solid-state ion-exchange reaction under high pressure Wang et al. (2015b). First-principles calculations predict that 1H-MoN monolayer is a high temperature 2D FM material with Curie temperature of nearly 420 K Wu et al. (2015). This suggests that combining TM and N atoms can lead to stable 2D monolayers exhibiting novel band structures. A natural question arises: can we find the QAH effect in the TMNs?
Previous calculations focused on TMN monolayers with the chemical composition TMN (TM = Y, Zr, Nb and Tc) which demonstrated that the most energetically and dynamically stable phase is 1T Wu et al. (2016b); Wang and Ding (2016); Liu et al. (2017). Interestingly, the strong nonlocal orbitals of the N atoms in 1T-YN result in a Curie temperature of 332 K. The three unpaired electrons in the two N atoms give 1T-YN a FM ground state with a total magnetic moment of 3 per unit cell. Remarkably, the electronic band structures as obtained from DFT calculations show that 1T-YN is a -state Dirac half metal (DHM) in the absence of spin-orbit coupling (SOC). A DHM is defined as a metal in which a Dirac cone exists at the Fermi level in one spin channel and a band gap opens in the other channel Wang (2008, 2017). The 100% spin-polarization and massless Dirac fermions in DHMs attracteda lot of attention due to potential applications in high-speed spintronic devices Liu et al. (2017). In this paper, we investigate the stable 1T-YN monolayer using first-principles calculations with the inclusion of SOC. We obtain a relatively large nontrivial band gap ( 0.1 eV) in the Dirac cone. The nontrivial properties of the 1T-YN monolayers are further confirmed by the calculation of the Berry curvature, the anomalous Hall conductance (AHC), the Chern number, and the corresponding edge states. The large nontrivial band gap and high Chern number are very interesting for practical applications in future nanodevices.
Ii Computational Methods
The first-principles calculations were done with the Vienna ab initio simulation package (VASP) using the projector augmented wave (PAW) method in the framework of Density Functional Theory (DFT) Kresse and Furthmüller (1996); Kresse and Hafner (1993); Kresse and Joubert (1999). The electron exchange-correlation functional was described by the generalized gradient approximation (GGA) in the form proposed by Perdew, Burke, and Ernzerhof (PBE) Perdew et al. (1996). The structure relaxation considering both the atomic positions and lattice vectors was performed by the conjugate gradient (CG) scheme until the maximum force on each atom was less than 0.01 eV/Å, and the total energy was converged to eV. To avoid unnecessary interactions between the YN monolayer and its periodic images, the vacuum layer is set to at least 17 Å. The energy cutoff of the plane waves was chosen as 500 eV. The Brillouin zone (BZ) integration was sampled by using a 31 31 1 -centered Monkhorst-Pack grid. To obtain a more reliable calculation for the electronic band structure, especially the band gap, the screened Heyd-Scuseria-Ernzerhof Hybrid functional method (HSE06) Heyd et al. (2003, 2006) was also used with a 15 15 1 -centered Monkhorst-Pack grid for BZ integration. SOC is included by a second variational procedure on a fully self-consistent basis. An effective tight-binding Hamiltonian constructed from the maximally localized Wannier functions (MLWF) was used to investigate the surface states Mostofi et al. (2014); Kong et al. (2017). The iterative Greenâs function method Sancho et al. (1985) was used with the package Wannier_tools Wu et al. (2017).
SOC is a relativistic effect which describes the interaction of the spin of an electron with its motion Dresselhaus et al. (2007); Winkler (2003). Though SOC is a small perturbation in a crystalline solid and has little effect on structure and energy, it plays a more important role in the band structure near the Fermi level in the case of the heavy elements. SOC will cause a spin splitting of the energy bands in inversion-asymmetric systems, and more importantly, it can result in a band opening and a band inversion that gives rise to fascinating phenomena. Previous studies did not consider SOC in 1T-YN, although the SOC effect of the Y and N atoms are not negligible. In the following, we will take SOC into account and study the electronic band structure of stable 1T-YN monolayer by first-principles calculations.
The space group of YN is (No.164, D), and its Wyckoff Positions are Y (0, 0, 0) and N (1/3, 2/3, ) with . As shown in Fig. 1(a), the Y atom is the inversion center in the octahedral structure unit and the three N atoms in the upper layer will turn into the three N atoms in the lower layer under inversion. The optimised lattice parameter and Y-N bond distance are 3.776 and 2.350 Å, respectively, which is in good agreement with previous calculations Liu et al. (2017). Without SOC, we observe a distorted Dirac cone along the M-K high-symmetry line both at the PBE and HSE06 level. Similar band structures can also be observed in other 2D non-magnetic systems, such as the TaCX Zhou et al. (2016) and the distorted hexagonal frameworks GaBi-X (X = I, Br, Cl) Li et al. (2017). However, distorted Dirac cones in 2D magnetic systems are rare. The calculated band structures with SOC are shown in Fig. 1(b). The blue lines indicate the band structure at the PBE level and a band gap (29.7 meV) can be observed at the Fermi level along the M-K high-symmetry line. The semilocal approximations to the exchange-correlation energy at the PBE level underestimate the band gap with respect to experiment and overestimate electron delocalization effects for many -element compounds. Therefore, calculations at the HSE06 level are necessary to give more reliable band gaps and band structures. We found that the band gap at the HSE06 level is larger than that at the PBE level. To see the difference more clearly, the enlarged band structure are shown in Fig. 1(c). The calculated band gap at the PBE level is E = 29.7 meV, while that at the HSE06 level E increases to 97.5 meV. By projecting the wavefunctions onto the spherical harmonics, we analysed the occupation of the different atomic orbitals both with and without SOC near the Fermi level. Though there are minor contributions of orbitals from the Y atom near the Fermi level, the major contribution of the atomic orbitals comes from the orbitals of the N atoms. This is consistent with previous study Liu et al. (2017). Next, we constructed a tight-binding Halmitonian with 12 Wannier functions by projecting the , , and orbitals of the two N atoms in the unit cell of 1T-YN to further examine the electronic band structure. Despite the difference in the band gap, the main character close to the Fermi level at the PBE and HSE06 level with SOC are qualitatively the same. In the following analysis, we will consider the results at the PBE level with SOC.
To investigate the topological properties of 1T-YN, we first calculated the gauge-invariant Berry curvature in momentum space. The Berry curvature in 2D can be obtained by analyzing the Bloch wave functions from the self-consistent potentials:
where, is the Fermi-Dirac distribution function, is the velocity operator, is the Bloch wave function, is the eigenvalue and the summation is over all occupied bands below the Fermi level ( indicates the unoccupied bands above the Fermi level). In Fig. 2 the reproduced band structure (blue curves) and Berry curvature (red curves) along the high symmetry direction K-M-Kﬂ, calculated by Wannier interpolation Mostofi et al. (2014), are shown. As can be observed, the nonzero Berry curvature is mainly distributed around the avoided band crossings at the Fermi level. The peaks in the Berry curvature at the two sides of the M point have the same sign. Furthermore, a plot of the Berry curvature peaks over the whole Brillioun zone (see inset of Fig. 2) indicates that all 6 peaks have the same sign because of inversion symmetry.
The Chern number is obtained by integrating the Berry curvature over the BZ,
The Chern number is an integer and gives rise to the quantized charge Hall conductance: . is also known as the AHC, and the calculated chemical-potential-resolved AHC is shown in Fig. 3(a). A nontrivial gap of about 30 meV and a Chern number can be deduced from the plateau near the Fermi level. The non-zero Chern number can also be confirmed by evaluating the electronic polarization at discrete points in one primitive reciprocal lattice vector Soluyanov and Vanderbilt (2011); Yu et al. (2011); Gresch et al. (2017). In another primitive reciprocal lattice vector , the hybrid Wannier charge centers (WCCs) can be defined as
where is the periodic part of the Bloch function . The sum of the hybrid WCCs will give the electronic polarization ( stands for the electronic charge) which is gauge invariant modulo a lattice vector. This brings about a well-defined physical observable under a continuous deformation of the system. Thus the Chern number is given by
As can be seen from Fig. 3(b), the electronic polarization shifts upwards with the winding number 3, so the Chern number .
According to the bulk-edge correspondence Hatsugai (1993), the non-zero Chern number is closely related to the number of nontrivial chiral edge states that emerge inside the bulk gap of a semi-infinite system. With an effective concept of principle layers, an iterative procedure to calculate the Green’s function for a semi-infinite system is employed. The momentum and energy dependence of the local density of states at the edge can be obtained from the imaginary part of the surface Green’s function:
and the results are shown in Fig. 4. It is clear that there are three gapless chiral edge states that emerge inside the bulk gap connecting the valance and conductance bands and corresponding to a Chern number .
Iv Discussions and Conclusions
By the calculations of Berry Curvature, the AHC, the Chern number, and the corresponding edge states, we confirm the topological nontrivial properties of the 1T-YN monolayer. The most significant points of our research can be summarized as follows:
The 1T-YN monolayer has a nontrivial band gap of nearly 0.1 eV which is sufficient to the realization of QAH effect at room temperature in experiment. The found magnitude of the nontrivial band gap compares favorably to previous proposals Ren et al. (2016); Zhang et al. (2016). The largest band gap among the large number of QSH insulators is 1.08 eV Song et al. (2014); Liu et al. (2014b), but a band gap in QAH insulators of this order of magnitude has not been discovered. In recent reviews Ren et al. (2016); Zhang et al. (2016), some systems have been proposed exhibiting large nontrivial band gaps ( 0.1 eV), but most of them are realized by functionalization methods, such as half functionalization of stanene with I atoms ( 340 meV) Wu et al. (2014a) and half functionalization of Bi (111) bilayer with H atoms ( 200 meV) Niu et al. (2015); Liu et al. (2015b). However, their stability still needs to be confirmed, and realizing such functionalization seems to be difficult to control experimentally. In contrast, note that the stability of 1T-YN has been demonstrated by energetic and dynamical analysis Liu et al. (2017). Since MoN has been realized in experiment Wang et al. (2015b) and its monolayer stability has been predicted Wu et al. (2015), it seems likely that the realization of 1T-YN monolayers is also possible.
Unlike the quantum Hall effect in which the Chern number can be tuned by changing the magnetic field or varying the Fermi level, the Chern number for intrinsic QAH effects in realistic materials are mostly limited to 1 and 2 Ren et al. (2016); Zhang et al. (2016); Sheng and Nikolić (2017). QAH insulators with high Chern number and corresponding nontrivial chiral edge states are rarely reported. These nontrivial edge states will provide strong currents and signals which are significant in experiments. For example, more evidences should be observed in the QAH insulator/-wave superconductor heterostructure for the chiral Majorana edge modes, which is predicted to realize the robust topological quantum computing He et al. (2017). Thus, the stable 1T-YN monolayer with a high Chern number is expected to lead to important applications in experiments.
To our knowledge, this is the first time that the QAH effect is predicted in a 2D TMN material. At present, the topological QSH effect was predicted in 1Tﬂ TMDs Qian et al. (2014). On the other hand, some TMD monolayers with square lattice Ma et al. (2015) and hexagonal lattice Ma et al. (2016) have also been predicted to be 2D topological insulators, but such structures have yet not been observed experimentally. Here, we report that the QAH effect can be realized in the 1T structure of TMN, which is a common structure for TMDs in exeperiment.
In conclusion, we investigated the electronic band structure of 1T-YN monolayer by first-principles calculations in the case of SOC and observed an intrinsic QAH effect. A large nontrivial band gap ( 0.1 eV) and high Chern number () were found. We calculated the Berry curvature and AHC to demonstrate the nontrivial topological properties. Three nontrivial gapless chiral edge states were found which provide strong evidence for the realization of the QAH effect in experiments. The prediction of the QAH effect in the 1T-YN monolayer provides a different type of structure and material for the investigation of QAH insulators in the TM compounds.
Acknowledgements.This work is supported by Ministry of Science and Technology of China (MOST) (Grant No. 2016YFA0301604), National Natural Science Foundation of China (NSFC) (No. 11574008), the Thousand-Young-Talent Program of China, and Fonds Wetenschappelijk Onderzoek (FWO-Vl). The computational resources and services used in this work were provided by the VSC (Flemish Supercomputer Center), funded by the Research Foundation - Flanders (FWO) and the Flemish Government - department EWI, and the National Supercomputing Center in Tianjin, funded by the Collaborative Innovation Center of Quantum Matter. W. Wang acknowledge financial support from the National Natural Science Foundation of China (Grant No. 11404214) and the China Scholarship Council (CSC).
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