Half-Dirac semimetals and the quantum anomalous Hall effect in Kagome Cd₂N₃ lattices

Abstract

Half-Dirac semimetals (HDSs), which possess 100% spin-polarizations for Dirac materials, are highly desirable for exploring various topological phases of matter as low-dimensionality opens unprecedented opportunities for manipulating the quantum state of low-cost electronic nanodevices. The search for high-temperature HDSs is still a current hotspot and yet challenging experimentally. Herein based on first-principles calculations, we propose the realization of Half Dirac semimetals (HDS) in two-dimensional (2D) Kagome transition-metal nitride Cd₂N₃, which is robust against strain engineering. Monte Carlo simulations reveal that Cd₂N₃ possesses a Curie temperature reaching up to T_C = 225 K, which is much higher than that of the reported monolayers CrI₃ (T_C = 45 K) and Cr₂Ge₂Te₆ (T_C = 20 K). The band crossings in Cd₂N₃ are gapped out by the spin–orbit coupling, which brings about the quantum anomalous Hall (QAH) effect with a sizeable band gap of E_g = 4.9 meV, characterized by the nonzero Chern number (C = 1) and chiral edge states. A tight-binding model is further used to clarify the origin of HDSs and nontrivial electronic properties. The results suggest monolayer transition-metal nitrides as a promising platform to explore fascinating physical phenomena associated with novel 2D emergent HDSs and QAH insulators toward realistic spintronics devices, thus stimulating experimental interest.

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1. Introduction

Spintronics, using an electron's spins instead of its charge to carry information, has attracted extensive attention because of faster transport and dissipation-less properties in magnetic materials.^1,2 To build spintronic devices, the selection of ferromagnetic (FM) materials and control of ferromagnetism, such as the generation and injection of spin-polarized carriers, the manipulation and detection of the spin direction, and the long-distance spin-polarized transport, are crucial and yet challenging. Many concepts of spintronic materials have been put forward to solve these problems, including the use of half-metals (HMs),^3–5 topological insulators (TIs),^6–9 magnetic semiconductors,^10–12 topological nodal line semimetals (TNLSs),^13–15 and so on. Interestingly, two-dimensional (2D) FM ordering has been experimentally discovered in van der Waals crystals CrI₃ (T_C = 45 K)¹⁶ and Cr₂Ge₂Te₆ (T_C = 20 K),¹⁷ though their T_C is lower than room temperature, opening the door for the application of 2D spintronics at the nanoscale. Spin-gapless semiconductors (SGS), as shown in Fig. 1(a), possessing a band gap in one of the spin channels while the others keeping a sizeable zero gap in the Fermi level, are more interesting and have been proposed by Wang in 2008.¹⁸ Owing to the special zero band-gap character of SGSs, these types of materials exhibit many novel properties, including (i) only a tiny amount of energy is required to excite electrons from the valence band (VB) to the conduction band (CB); (ii) the excited charge carriers, both electrons and holes, can be 100% spin-polarized simultaneously; (iii) by using the Hall effect, 100% spin-polarized electrons and holes can be easily separated. Many 2D and 3D SGSs,^19–26 including TMF₃ (ref. ¹⁵) and MX (M = Li, Na, K, Rb, Cs; X = S, Se, Te)²⁸ exhibit various topological phases. Also, some examples have been synthesized such as Mn₂CoAl,¹⁹ and CoFeMnSi.²⁰ All these provide a new playground and opportunities for spintronic, electronic and optic devices.

Fig. 1 Schematic rule for designing HDSs. (a) SGSs, (b) Dirac semimetals, and (c) HDSs. The Dirac point of HDSs is exactly located at the Fermi level, unlike that of SGSs. Owing to the intersection of the Dirac-like node and the Fermi level, HDSs represent the real gapless properties rather than half-metals in one spin channel.

Dirac semimetal is a new type of quantum state with non-trivial topological properties. Its body energy band generally has a three-dimensional Dirac point near the Fermi surface. The representative example is monolayer graphene composed of a purely 2D honeycomb network of carbon atoms.^27,29 Low-energy excitation in graphene is governed by π electrons in 2p orbitals, whose energy dispersion has linear band crossings with the Dirac-like node at the Fermi level, called the Dirac cones. It exhibits unique electronic properties such as ballistic charge transport,³⁰ high carrier mobility,³¹ and the quantum Hall effect.³² Subsequently, many 2D lattices such as germanene,³³ carbon allotropes,^34–37 boron allotropes,³⁸ SiC₃,²⁷ and C₄N,³⁹ have been identified as Dirac materials. However, the Dirac cone in these materials is intrinsically spin-degenerated, which actually limits its utilization in massless spintronics. Thus, the investigation of the controllability of the band crossings and topological phases by using the p-orbital degrees of freedom is desirable, but such an interesting possibility has rarely been investigated.

The ultimate goals for future spintronic devices are to require ultra-fast transport and ultra-lower energy consumption, that is, eliminating the effective mass of electrons or holes and enabling massless charges to be completely spin-polarized. In comparison with parabolic-like SGSs (Fig. 1(a)) and four-degenerated band Dirac cones (Fig. 1(b)), Dirac-like SGSs, termed HDSs, as illustrated in Fig. 1(c), are the better class of magnetic materials for use in spintronic devices as HDSs possess more exotic spin and charge states. Thus, it is highly desirable to find stable HDSs with an intrinsic 100% spin-polarized Dirac state. Normally, HDSs can be divided into two types: type-I, the d-state HDSs, in which the Dirac state is contributed by the d-orbital of transition-metal atoms, and type-II, the p-state HDSs, whose states are from the p-orbital of transition-metal atoms. To date, the majority of reported HDS members belong to type-I HDSs,^39–47 while there has been barely any reports on 2D type-II HDS materials. It is highly desirable to design type-II HDSs with robust spin ordering and a high Curie temperature simultaneously, which may make experimental synthesis largely accessible.

Meanwhile, among the 2D materials family, 2D Kagome lattices have been attracting great interest due to their rich properties. Most members of the family are carbides and oxides (e.g., Be₃C₂, V₂O₃, Nb₂O₃, and LiFeSe),^40–45 consisting of one flat layer with D_6h symmetry. Considering that nitrogen has one electron more than the oxygen atom, it is natural to ask whether transition-metal nitrides can also be stabilized in a 2D crystal Kagome lattice. If so, this may open the door to a new family of 2D materials, i.e., 2D Kagome transition-metal nitrides, which will also grow into a big family with fascinating quantum properties.

In this work, we try to address the above challenge and question by proposing a new family of 2D transition-metal nitrides with the Kagome structure. Based on first-principles calculations, we show that Cd₂N₃ can be stabilized in this 2D lattice as shown in Fig. 2(a), which is dynamically and thermally stable. Monte Carlo simulations on the basis of the 2D Heisenberg Hamiltonian model show that the Curie temperature T_C = 225 K, having potential for high temperature applications. The coexistence of fully spin-polarization and linear dispersion Dirac states can be realized in Cd₂N₃, which features ultra-clean energy dispersion and ultra-high Fermi velocity. Distinguished from most proposed nonmagnetic Dirac materials, the four-fold degeneracy neck crossing-point traces out two-fold degeneracy lines emerged in the single-spin channel and the profile of HDSs is revealed by a tight-binding (TB) model. To the best of our knowledge, the Cd₂N₃ monolayer represents the first example that hosts the 2D HDS in the equilibrium state. When turning on the spin–orbit coupling (SOC), Cd₂N₃ becomes a quantum anomalous Hall (QAH) insulator with its energy gap of E_g = 5.0 meV, larger than recently predicted E_g = 2.3 meV in Mn-DCA,³⁸ which is characterized by the nonzero Chern number (C = 1) and chiral edge states. In comparison to Cr- or V- doped (Bi, Sb)₂Te₃ film,s such a lattice without any magnetic doping is easier to synthesize and has much higher homogeneity. Thus, our discovery not only predicts a family of new 2D HDS materials, which vastly enriches the SGS family, but also provides an experimentally feasible platform to explore the new emergent QAH effect and their fascinating fundamental physical effects.

Fig. 2 (a) Top and side views with lattice vectors a₁ and a₂ in the x–y plane. Rhombus shows the unit cell. (b) Calculated phonon spectrum of Cd₂N₃. (c) MD simulations at temperature 300 K.

2. Computational details and method

All the spin-polarized calculations are performed by using density-functional theory (DFT), as implemented in the Vienna ab initio simulation package (VASP).⁴⁶ The projector-augmented-wave (PAW) potential,^47,48 Perdew–Burke–Ernzerhof (PBE) exchange-correlation functional⁴⁹ and plane-wave basis with a kinetic energy cutoff of 500 eV are employed. An all-electron description, the projector augmented wave method, is used to describe the electron-ion interaction. The Brillouin zone is sampled by using a 9 × 9 × 1 Gamma-centered Monkhorst–Pack grid, and the SOC is included by a second variational procedure on a fully self-consistent basis. During structural optimization, all atomic positions and lattice parameters are fully relaxed, and the maximum force allowed on each atom is less than 0.02 eV Å⁻¹. Spin polarization is included through all the calculations. Since standard DFT may fail to describe the magnetism, the Heyd–Scuseria–Ernzerhof (HSE06) functional⁵⁴ has been carried out to examine the magnetism and electronic structures. Phonon dispersion is calculated using the finite displacement method, as implemented in the PHONOPY code.⁵⁵

3. Results and discussion

Because most elements and compounds are not favorable to directly form a 2D Kagome lattice, we propose a new transition-metal nitride, Cd₂N₃, to substitute the Kagome lattice, as illustrated in the top and side views of Fig. 2(a). The Cd₂N₃ monolayer forms a honeycomb lattice with a point symmetry of D_6h by sharing one nitrogen (N) bridge with each neighbor, reminiscent of the crystal structure in the 2D organic Mn-DCA lattice.^{50–53,57,58} The optimized lattice parameters for Cd₂N₃ are a₁ = a₂ = 7.29 Å, and three N atoms locate around each Cd atom and the bond angle N–Cd–N is 120°, with a Cd–Cd distance of d = 4.21 Å. To characterize the thermodynamic stability of Cd₂N₃, we calculate the cohesive energy ΔE_f defined as

E_f = E (Cd₂N₃) − 2μ (Cd) −3/2μ (N₂) (1)

where E (Cd₂N₃) is the total energy, and μ (Cd) and μ(N₂) are the chemical potentials taken from bulk Cd and gas phase N₂. The calculated result for Cd₂N₃ is E_f = −0.67 eV per atom, indicating an exothermic chemical reaction. These values are comparable to those of monolayers CrI₃ (−0.903 eV per atom)⁵⁹ and Cr₂Ge₂Te₆ (−0.552 eV per atom).⁶⁰ The moderate formation energy also indicates that Cd₂N₃ can be realized with a strong bonding network.

It is important to examine whether the experimentally fabricated Cd₂N₃ is stable and whether the magnetic state survives at room temperature. To test the kinetic stability of Cd₂N₃, we perform phonon spectrum calculations for Cd₂N₃, as shown in Fig. 1(b). Apparently, there is no imaginary mode in the whole Brillouin zone, indicating that the Cd₂N₃ lattice is dynamically stable. Additionally, the thermal stability of the Cd₂N₃ lattice is assessed by performing ab initio molecular dynamics (MD) simulations. We use a 4 × 4 supercell to carry out the individual simulations at a temperature of 300 K. As seen from Fig. 2(c), the crystal structure of Cd₂N₃ does not collapse throughout 8 ps MD simulation. In fact, Wang et al⁵⁶ reported that the Y₂O₃ lattice forms a complete monolayer on platinum-supported graphene. The Y₂O₃ monolayer interacts weakly with graphene, but is stable at high temperature. Scanning tunnelling microscopy reveals that Y₂O₃ exhibits a 2D hexagonal lattice rotated by 30° relative to the hexagonal graphene lattice. The above results reveal that the Cd₂N₃ monolayer has very good thermal stability and could possibly be prepared in the experiment and survive at room temperature.

We now focus on the magnetic structures of Cd₂N₃. Spin-polarized results reveal a spin moment of 5.0 μ_B per cell as a result of large Hund's rule, which can be further illustrated by spin-polarized electron density, Δρ = ρ_↑ − ρ_↓, the difference between the electron densities of two spin channels, shown in the inset in Fig. 3(c). A large amount of spins is localized at the N site (∼1.21 μ_B) with a slight contribution from the Cd ion (∼0.08 μ_B), indicating that the N ions mainly contribute to magnetism, also verified by Bader charge analysis.⁶¹ As is known, the Mermin–Wagner theorem⁶² reveals that the long-range magnetic ordering is absent by thermal fluctuations in 2D lattices. However, it doesn't hold true for all 2D structures, especially for strong anisotropy cases.^16,17 Thus, to determine the magnetic ground state of Cd₂N₃, we consider several possible configurations, including, FM, Neel antiferromagnetic (NAFM), stripe AFM (SAFM), and zigzag AFM (ZAFM), as shown in Fig. 3(a). The calculated results indicate that the FM state (ΔE_FM = E_FM − E_AFM) is energetically 14.43, 12.41, 9.75, and 9.32 meV N⁻¹ lower than the NAFM, SAFM, and ZAFM states, respectively. Furthermore, we also check the spin configuration of the coplanar 120° AFM state, and find that the Cd₂N₃ monolayer prefers energetically FM (ΔE_FM = 16.58 meV N⁻¹) states.

Fig. 3 (a) Possible spin configurations of the Kagome Cd₂N₃ lattice: FM, NAFM, SAFM, and ZAFM. (b) Angular dependence of the MAE (θ, φ) with the direction of magnetization lying on three different planes. The inset illustrates that the spin vector S on the x–y, y–z, and x–z planes is rotated with an angle θ around x, y, and z axis, respectively. (c) MC simulations of the average magnetic moment and spin susceptibility per unit cell with respect to temperature. The electron density of two spin channels for FM spin configuration is defined as 0.001 eV Å⁻³.

Magnetocrystalline anisotropy energy (MAE), defined as the required energy to rotate the magnetization from an easy axis to hard axis of a ferromagnet and determines the difficulty of spin flipping, plays a key role in establishing 2D FM ordering. For a Kagome lattice, MAE (θ, φ) at an arbitrary direction (θ, φ) follows

MAE (θ, φ) = K₁ cos² θ +K₂ cos⁴ θ+K₃ cos⁴ 3φ (2)

where K₁ and K₂ are system-dependent anisotropy constants and φ is the azimuthal angle of rotation. Fig. 3(b) shows MAE (θ, φ) as a function of the azimuth angle θ measured from the out-of-plane axis and the horizontal azimuth angle φ on the in-plane axis. We find that the energy difference MAE (θ, φ) is almost independent of in-plane azimuthal angle φ, i.e., K₃ = 0, and MAE (θ, φ) reaches its minimum in the out-of-plane axis, indicating an out-of-plane magnetization. According to the fits of angular dependence of MAE (θ, φ), both K₁ and K₂ are positive with a negligible dependence on φ, and MAE (θ, φ) reaches a maximum value of 15 μeV N⁻¹ at θ_xz = θ_yz = π/2, belonging to the family of 2D Ising magnets. Consequently, Cd₂N₃ is a 2D Ising magnet, clearly different from the cases of 1T – and 2H-VSe₂ with the in-plane magnetization.⁶³ The out-of-plane anisotropy of Cd₂N₃ could make the FM^z phase susceptible to thermal fluctuation and correspondingly feasible to realize a 2D magnet at a finite temperature.

The Curie temperature (T_c) is a key parameter for the practical application of spintronic devices. Therefore, it is necessary to understand the behavior of magnetism with temperature before implementing Cd₂N₃ into practical spintronic devices. On the basis of the 2D Heisenberg model, the spin Hamiltonian can be expressed as

where J₁ and J₂ are the nearest and next-nearest magnetic exchange interaction parameters, S_i is the spin vector of each atom, A is the anisotropy parameter, and S^z is the z component of the spin vector. A supercell of 100 × 100 with periodic boundary conditions is used here. Fig. 3(c) shows the temperature dependent magnetic moment per unit cell. The magnetic moment begins to drop dramatically at T_C = 225 K, implying the formation of the FM state. Additionally, heat capacity (C_V) has been calculated, as shown in Fig. 3(c), further confirming the FM-PM phase transition temperature at critical temperature T_C. The high T_C = 225 K indicates that the Cd₂N₃ monolayer may be a promising high-temperature spintronic material.

Having determined the magnetic ground state of the Cd₂N₃ monolayer, we present the representative band structures without SOC, as shown in Fig. 4(a), where the blue and red colors display the spin-up and spin-down channels, respectively. One can see that both the spin-up and spin-down channels are completely split away from each other. Remarkably, the spin-up channel possesses a very large band gap 2.88 eV, whereas the spin-down one did not show a gap. Thus the charge transport is dominated by the spin-down electron, and the current flow in such a system should be fully spin-polarized, which is consistent with the case in Fig. 1(a). The band character can be further confirmed by the HSE06 method, which has a larger band gap of 3.56 eV in the spin-up channel. Though the bulk gap is enhanced for HSE06, the HDS bands are not altered near the Fermi level, indicating the robustness of the band character against the computational method. As E_F lies almost in the middle of the spin gap, fully spin-polarized spin-filter efficiency can be maintained in a wide positive or negative bias range, rendering Cd₂N₃ an attractive candidate for spin-injection.

Fig. 4 The energy bands along the high-symmetry lines without (a) and with SOC (b), respectively. The spin-up and spin-down bands are colored in red and blue, respectively. The Fermi level is set to zero.

The most prominent findings on band dispersion are that in the spin-up channel, there are two several energy band crossings near E_F, as shown Fig. 5(a), which mainly originates from the representation of the D_6h point group. From the 3D band profile near the K point, accompanied by symmetry analysis, we can obtain the six Dirac-like nodes located exactly at the Fermi level. Notably, this is a typical Kagome band around the Fermi level, consisting of a completely flat band above two Dirac bands remaining at the K point. From the two linear bands in Fig. 5(a), the Fermi velocity v_F of the carriers can be evaluated using linear fitting, ћv_F ≈ dE(k)/dk. The obtained v_F = 4.3 × 10⁵ m s⁻¹ for Cd₂N₃ is comparable to that of graphene (8.2 × 10⁵ m s⁻¹),⁶⁴ and silicene (5.3 × 10⁵ m s⁻¹).⁶⁵ To understand the origin of Dirac states in Cd₂N₃, we calculate the orbital-resolved band structures, as shown in Fig. 6. One can see that the wave functions at the Fermi level are mainly from the N-2p_z orbital with a slight contribution from Cd atoms, so there is no sp² hybridization occurring on Cd₂N₃. The linear crossing Dirac states by the N-p_z orbital is consistent with the p_z Dirac state due to the nature of weak π bonds.⁶⁶ Namely, Cd₂N₃ is a superior candidate of single-spin p-band Dirac materials for high-speed spintronic devices.

Fig. 5 (a) The calculated 3D energy bands. (b) Berry curvature with SOC in 2D momentum space. The red-white-blue colors give the distribution of Berry curvature from positive to negative values, and the black dashed lines show the first BZ. (c) Distribution of the normalized Berry curvature over the BZ. (d) Chiral edge states of the Cd₂N₃ monolayer.

Fig. 6 Orbital-resolved band structures for (a) N-s (b) N-p, (c) Cd-d, and (d) Cd-s states, respectively.

The peculiar band crossings that we found can host topologically nontrivial states in the presence of effective SOC. Fig. 3(b) shows the band structure with SOC, with a gap of E_g = 5.0 meV at the Dirac-like node. The band gap is similar to the case of 2D Mn-DCA (E_g = 2.3 meV),⁴⁴ but smaller than recently predicted E_g = 20 meV in Cs₂Mn₃F₁₂.⁴³ It is noteworthy that, due to the small atomic numbers, SOC has negligible effect on the band structure of Cd₂N₃. Such band gap opening at the K point suggests a topologically nontrivial feature, which can be confirmed by the Berry curvature Ω(k) from the Kubo formula^67–69 expressed as

where the summation is over all of the occupied states, E_n is the eigenvalue of the Bloch function |Ψ_nk〉, f_n is the Fermi-Dirac distribution function, and v_x and v_y are the velocity operators. Fig. 5(b) and (c) show the Berry curvature in the BZ around the K and K′ points, which are nonzero with the same sign. By integrating the Berry curvatures over the first BZ, we calculated the Chern number (C), expressed as

One can see that C = 1 with each Dirac cone contributing 0.5, characterizing the topological nature of the gaped state. So the anomalous Hall conductivity σ_xy, σ_xy = Ce²/h, shows a quantized charge Hall plateau at a value of e²/h located in the insulating gap of the spin-up Dirac-like node. Such a nonvanishing Chern number C = 1 and quantized Hall conductivity in Cd₂N₃ characterize the QAH effect.

The two band degeneracy point and the associated band topology can be well modeled by the TB model, taking the D_6h lattice symmetry constraint into account. We introduce a Slater–Koster type TB model⁷³ with the p orbital on the Kagome lattice formed by N atoms. The energy bands near the Dirac points can be expressed by the effective Hamiltonian

H = ℏv_F(k_xτ_zσ_x + k_yσ_y) + m_Hσ_zτ_z, (7)

where v_F is the Fermi velocity, σ_x,y are the Pauli matrices, σ_z = ±1 refers to inequivalent N sublattices, and τ_z = ±1 refers to the valley pseudo-spin of the Cd₂N₃ monolayer. Wave vector k_x,y is referenced from K or K′.

SOC connects the electron spin and orbital degrees of freedom. So the crystal potential in 2D materials can be approximated by the spherical atomic potential, which gives on-site contribution to the TB Hamiltonian. By averaging the radial degree of freedom, the SOC reads

where s⃑ is the vector of the Pauli matrices representing the real spin and L⃑ is the angular momentum operator. The matrix element <⋯>_μ,ν is given on the basis of directed atomic orbitals μ,ν and ξ[small script l] is the angular momentum resolved atomic SOC strength with [small script l] = {s,p,d,…}. Obviously, the energy bands are gapped by the effective SOC, which requires nonzero s_z due to its out-of-plane magnetic moment in the Cd₂N₃ monolayer. Thus, our TB analysis provides a systematic understanding of the existing first-principles results, and furthermore, a useful guide for further material exploration.

The derived Haldane effective model describes the QAH effect,⁷⁰ and the existence of topologically protected chiral edge states is one of the most important signatures of the QAH effect. To reveal the nontrivial topological nature of the Cd₂N₃ lattice, we construct Green's functions⁷¹ for the semi-infinite boundary based on the maximally localized Wannier function method.^72,73Fig. 4(d) shows the local density of states (LDOSs) of the edge states. Obviously, the nontrivial edge states connecting the CB and VB cross the insulating gap of the spin-up Dirac cone, consistent with the Chern number C = 1. The spin-polarized Dirac-fermion mediated topological character suggests that the Cd₂N₃ monolayer is intrinsic, holding the potential QAH phase.

As is well known, the physical properties of 2D materials can be effectively tuned by strain, like the band gap engineering of 2H-TMD semiconductors.⁷⁴ Strain has also been proposed to effectively modulate dielectric properties,⁷⁵ SOC,⁷⁶ thermal conductivity⁷⁷ and interlayer coupling in vdW heterostructures (HTSs)⁷⁸ in 2D lattices. In the following, we will show that strain can induce SOC induced bulk gaps and spin-exchange interactions. Fig. 7 presents the evolution of direct gap E_g and FM exchange energy as a function of strain ε, defined as ε = (a − a₀)/a₀, where a (a₀) is the strained (equilibrium) lattice constants. Note that the nontrivial topological states remain within the strain range of 10%. This suggests that Cd₂N₃ maintains a topologically nontrivial state, which is stable against external strains. The gap E_g decreases slightly with tensile strains, becoming E_g = 4.95 meV under strain 10%. Additionally, the variation of total energies of the FM and AFM states (ΔE_FM = E_FM − E_AFM) with biaxial strains is presented. The results show that the FM ground state of Cd₂N₃ is robust against strains. The exchange energy also decreases slightly with increasing tensile strains, for example, when ε = 10%, ΔE_FM = 14.43 meV, leading to less changes of T_C. The dependency of T_C onstrain is clearly different from the cases of 2D FeCl₂ (ref. ⁵⁰) and CdSe₂,⁵¹ where T_C changes sensitively with respect to external strains. So we can infer that Cd₂N₃ may be a promising intrinsic high-temperature spintronic material. Here we must note that previously proposed strategies to realize p-band FM ordering may be hard to be experimentally controllable, because a strong external electric field or carefully selective doping is required. Our work thus reports the first intrinsic p-band HDS material, even though the experimental synthesis of Cd₂N₃ may still remain a challenge. As these magnetic materials potentially offer large spin relaxation times due to small SOC,⁷⁹ our results highlight a new promising material for experimental validation studies toward the realistic p-band spintronics applications.

Fig. 7 The variations of the global gap and exchange energy as a function of biaxial strains.

Finally, one critical point is whether the QAH effect of 2D Cd₂N₃ can remain on a substrate, since the substrates are inevitable in device applications.^80,81 As is known, the BN monolayer is chemically inert and does not easily bond strongly with other atoms, and thus may be adopted as a protective film to grow Cd₂N₃. So, the Cd₂N₃/BN HTS has been constructed, as shown in Fig. 8(a). Structural optimization indicates that the distance of Cd₂N₃ and BN layers is 3.10 Å with a binding energy of −34 meV per unit cell, a typical van der Waals structure. In this case, the main features of the QAH effect in the Cd₂N₃ lattice remain intact. Fig. 8(b) presents the calculated band structure with SOC. As expected, here there is still a SOC-induced gap at the Dirac point around the Fermi level, and the states around the Fermi level are dominantly contributed by the Kagome Cd₂N₃ band. Given that the 2D BN substrate electrically insulates the adjacent QSH layer of Cd₂N₃, protecting parallel helical edge channels from being gapped by interlayer hybridization, Cd₂N₃/BN is an ideal HTS to support the dissipationless charge/spin transport in the quantum device. This demonstrates the feasibility of constructing the quantum device with the Cd₂N₃/BN heterostructure, as illustrated in Fig. 8(c).

Fig. 8 (a) Top and side views of the epitaxial growth of the Cd₂N₃ lattice on the BN substrate. The interlayer distance is 3.10 Å. (b) The corresponding energy band structure of Cd₂N₃/BN HTS. (c) Schematic device model for proposed Cd₂N₃/h-BN heterostructure for quantum state measurement. Vertical arrows show the spin orientation of electrons in the edge states and horizontal arrows show their transport directions.

4. Conclusion

In summary, we employ first-principles calculations to demonstrate the possibility of realizing HDSs in Kagome lattices with T_C = 225 K, which is much higher than that of the reported monolayers CrI₃ (T_C = 45 K) and Cr₂Ge₂Te₆ (T_C = 20 K). The inclusion of SOC can turn Cd₂N₃ into a QAH insulator, with a sizable band gap of E_g = 5.0 meV. Notably, the nontrivial topology is robust against biaxial strain with its band gap reaching up to E_g = 4.95 meV under strain 10%. Also, the TB model is constructed to clarify the origin of the HDS and nontrivial electronic properties. Considering that Cd₂N₃ enjoys good stability and excellent flexibility, our findings would stimulate further material exploration toward the HDS matter in Kagome materials. These outstanding properties of Cd₂N₃ indicate that it is a promising 2D material for potential application in designing energy efficient spintronic devices.

Additionally, the proposed Cd₂N₃ is likely to be synthesized by molecular beam epitaxy (MBE) or by the chemical vapor deposition method. We can synthesize Cd₂N₃ with a synthesis method similar to that of Nb₂O₃.⁸² The QAH effect can be probed by scanning probe spectroscopy, ARPES, and ultimately by device fabrications and charge transport measurements. Thus, it offers a new platform for the study of the QAH effect in Kagome lattices for realizing low-dissipation spintronic devices, which is of significant fundamental importance.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was supported by the Taishan Scholar Program of Shandong Province (No. ts20190939), the National Natural Science Foundation of China (Grant No. 11804116 and 62071200) and the Natural Science Foundation of Shandong Province (No. ZR2018MA033).

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