Two-dimensional anisotropic Dirac materials PtN₄C₂ and Pt₂N₈C₆
with quantum spin and valley Hall effects

Two-dimensional topological Dirac materials ignited by graphene have attracted tremendous interests in the past decade, owing to their unique properties in the reduced dimension and promising prospects for both fundamental research and applications Wang et al. (2015a); Novoselov et al. (2004). These materials are characterized by a linear energy dispersion at the Fermi energy, exhibiting many novel phenomena and exotic physics, such as massless fermions, fractional quantum Hall effect, and quantum spin Hall effect Zhang et al. (2005); Bolotin et al. (2009); Du et al. (2009); Dean et al. (2013); Ponomarenko et al. (2013); Hunt et al. (2013); Castro Neto et al. (2009); Weiss et al. (2012); Novoselov et al. (2005); Liu et al. (2011). However, due to the various requirements associated with the crystal symmetries and orbital interactions, the available two-dimensional systems hosting Dirac cones and vanishing density of states right at the Fermi level are very rare Novoselov et al. (2005); Liu et al. (2011); Cahangirov et al. (2009); Malko et al. (2012); Huang et al. (2013); Ma et al. (2014); Xu et al. (2014); Ouyang et al. (2011); Zhou et al. (2014); Li et al. (2014); Wang et al. (2013); Gomes et al. (2012). It is still a very challenging task to searching for more novel two-dimensional Dirac materials, in which materials with topologically nontrivial properties are even rarer.

When spin-orbit coupling is included in two-dimensional Dirac materials, a global band gap opens and the materials are turned into two-dimensional topological quantum spin Hall insulators with metallic helical edge states residing in the insulating bulk gap Liu et al. (2011); Hasan and Kane (2010); Qi and Zhang (2011). Due to time-reversal symmetry, these helical edge states come in Kramers^′ pairs, so that electrons with opposite spins propagate in opposite directions. In addition, the helical edge states are protected from back-scattering, giving rise to a symmetry-protected topological phase with a quantised spin Hall conductance. However, a rather weak intrinsic spin-orbit coupling often limits the experimental realization of such a two-dimensional topological Dirac system. A sizable spin-orbit coupling is thus very crucial for the realization of a two-dimensional quantum spin Hall insulator Tsai et al. (2013); Liu et al. (2011). For example, the observation of quantum spin Hall effect in graphene has not been experimental confirmed yet, due to its negligible spin-orbit coupling and rather tiny splitting energy gap which make it easy for electrons to thermalize to the conduction bands Yao et al. (2007).

Recently, a novel class of single-atom-thick two-dimensional transition metal carbonitrides, including CrN₄C₂, CoN₄C₂, CoN₄C₁₀, Co₂N₈C₆, and Co₂N₆C₆, has been proposed based on the first-principles electronic structure calculations Liu et al. (2021a, b, c). In comparison with the prototype graphene, transition metal elements are incorporated into the new planar graphene-like structures, resulting in richer electronic properties, such as flat bands, Van-Hove singularity, and intrinsic two-dimensional ferromagnetism Liu et al. (2021a, b, c). Moreover, due to the introduction of heavy-metal atoms that have much stronger spin-orbit coupling, more prominent topological properties may be expected in this class of materials.

In this paper, based on the first-principles electronic structure calculations, we propose two new types of two-dimensional planar materials, PtN₄C₂ and Pt₂N₈C₆ (Fig. 1), which share the similar structure with CoN₄C₂ and Co₂N₈C₆ respectively Liu et al. (2021a, b), exhibiting a significant quantum spin Hall effect and holding a great promise for experimental realization. In the absence of spin-orbit coupling, they are perfect gapless Dirac semimetals. However, unlike graphene in which the Dirac points locate at the particularly symmetric K and K^′ points on the Brillouin zone boundary, the Dirac nodes of PtN₄C₂ and Pt₂N₈C₆ locate at a general k-point along the $Y$–$\Gamma$ high-symmetry path, away from the K/K^′ points. This is due to the structural distortion in comparison with the ideal honeycomb lattice. Furthermore, the linear band dispersions of PtN₄C₂ and Pt₂N₈C₆ around the Dirac nodes are anisotropic, different from the isotropic Dirac cones with the vertical axes of graphene Novoselov et al. (2005), making them suitable to realize orientation-dependent quantum devices. Once spin-orbit coupling is considered, sizable band gaps, $\sim$ 10 meV and 40 meV, are opened for PtN₄C₂ and Pt₂N₈C₆ respectively, and the materials are converted into quantum spin Hall insulators with topological index $\mathbb{Z}_{2}$=1. Topological nontrivial properties, including the Berry curvature, spin-Hall conductivity, and helical edge states are systematically studied. These quantum phenomena should be observable in experiment, especially for Pt₂N₈C₆ in which the gap is sufficiently large for practical applications at room temperature. Since PtN₄C₂ and Pt₂N₈C₆ monolayers are very similar, here we take PtN₄C₂ as an example to illustrate their electronic and topological properties. The results of Pt₂N₈C₆ are presented in the Supplementary Materials sup .

In our calculations, the plane-wave basis based method and Quantum-ESPRESSO software package were used Giannozzi et al. (2009, 2017). We adopted the generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof formula for the exchange-correlation potentials in the electronic structure simulations Perdew et al. (1996). The ultrasoft pseudopotentials were employed to model the electron-ion interactions Vanderbilt (1990). A slab geometry was applied, where we added a vacuum space of $\sim$ 20 $\AA$ in the $z$ direction to eliminate the periodic effect. The mesh of k-points grid used for sampling the Brillouin zone was 24$\times$24$\times$1, and the Marzari-Vanderbilt broadening technique was adopted Marzari et al. (1999). After full convergence tests, the kinetic energy cutoff for wavefunctions and charge densities were chosen to be 90 and 720 Ry, respectively. During the simulations, all structural geometries were fully optimized to achieve the minimum energy. Phonon band dispersions were calculated by using density functional perturbation theory based on the PHONOPY program Togo and Tanaka (2015). The edge states were studied using tight-binding methods by the combination of Wannier90 Mostofi et al. (2008) and WannierTools Wu et al. (2018) software packages.

The atomic structure of monolayer PtN₄C₂ belongs to the $Cmmm$ space group with No. $65$, which is affiliated to the $D_{2h}$ point group symmetry, as shown in Fig. 1(a). There are totally one Pt, four N, and two C atoms in the primitive cell (the solid black rhombus), in which the Pt atom is tetra-coordinated by four N atoms, forming PtN₄ complexes connected by the C atoms. All three kinds of atoms in PtN₄C₂ are coplanar after structural relaxation. The optimized lattice parameters of PtN₄C₂ are $a$ = $b$ = 4.789 $\AA$ and the angle between the two basis vectors $\alpha$ = 110.45^∘. Figure 1(c) sketches the corresponding Brillouin zone of PtN₄C₂ with the high symmetry points labeled, which is slightly distorted from the prefect honeycomb structure. The angle between the two reciprocal vectors is no longer 60^∘, and hence, the high symmetry k-points $H$ and $X$ become inequivalent $H1$ and $Y$, respectively. The dynamical stability of PtN₄C₂ can be demonstrated by calculating its phonon dispersion curve. As shown in Fig. 1(d), no considerable imaginary frequency appears in the entire Brillouin zone, indicating that the PtN₄C₂ monolayer is dynamically stable. Note that there is a tiny spoon-shaped pocket near the $\Gamma$ point as shown in the zoomed-in graph, which is due to the difficulty of achieving numerical convergence for the flexural phonon branch. It is a common issue in the first-principles electronic structure calculations on 2D materials and does not imply instability Zólyomi et al. (2014); Tran et al. (2014).

Figure 2 shows the calculated electronic band structure of PtN₄C₂. Similar to graphene, PtN₄C₂ is a perfect Dirac semimetal with vanishing density of states at the Fermi level in the absence of spin-orbit coupling, as shown in Fig. 2(a). By projecting the band structures onto different atomic orbitals, the Dirac node is found to mainly consist of the N-$p$, C-$p$, and Pt-$d$ states. More specifically, the N-$p_{z}$ and C-$p_{z}$ form a $\pi$ network together with the Pt $d_{xz}$ and $d_{yz}$ orbitals in the vicinity of the Fermi level, as shown in Fig. 2(b). However, different from those of graphene, the linearly dispersive Dirac nodes of PtN₄C₂ do not locate at the particularly high-symmetric K or K^′ points on the Brillouin zone boundary. They locate at a general k-point in the high-symmetry line $Y$–$\Gamma$, as displayed in Fig. 2(c). Therefore, there are two Dirac nodes (in the high-symmetry $Y$–$\Gamma$ and -$Y$–$\Gamma$ paths) within the first Brillouin zone, as shown in Fig. 2(d), which illustrates the energy dispersion of highest valence and lowest conduction bands of PtN₄C₂ as a function of $k$. The two bulk bands touch only at the Dirac points, forming the two Dirac cones within the first Brillouin zone. Moreover, the Dirac cones are tilted compared with that of graphene. This is due to the reduced lattice spatial symmetry $D_{2h}$ of PtN₄C₂, in comparison with the $D_{6h}$ symmetry of graphene. The distortion of primitive unit cell from an ideal honeycomb lattice causes the shift of the Dirac cones in the distorted first Brillouin zone with lower spatial symmetry Wang et al. (2015b). The linear band dispersions around the Dirac cones are therefore anisotropic, resulting in anisotropic Fermi velocities and making them possible to realize orientation-dependent quantum devices.

We further calculated the differential charge density of PtN₄C₂ to illustrate the internal bonding conditions and the charge transfer, as shown in Fig. 2(e). The red/blue colours mark an increase/decrease of the charge density. There is significant charge accumulation between the C–N, N–N, and C–C atoms, indicating that covalent bonds are formed between these atoms. The charge depletion happens mostly around the Pt atoms, supporting the transfer of electrons to the neighbor N atoms in the local PtN₄ complex. Therefore, an ionic bond is formed between Pt and N atoms.

Once spin-orbit coupling is introduced in calculations, an energy gap is opened at the Dirac point, turning the linear band dispersions into quadratic ones, as shown in Fig. 2(f). Since Pt belongs to a heavy transition-metal element, a larger energy gap of $\sim$10 meV is observed, in comparison with that of graphene on the order of $\mu$eV Yao et al. (2007). The splitting gap is experimentally visible and is capable of hosting novel quantum spin Hall states. Figure 2(g) shows the three-dimensional energy dispersion of highest valence and lowest conduction bands of PtN₄C₂ with spin-orbit coupling. Compared to the one without spin-orbit coupling (Fig. 2(c)), the quadratic feature with an open energy gap is evident. PtN₄C₂ is therefore very likely to be a quantum spin Hall insulator, possessing symmetry-protected helical metallic edge states that have myriad potential applications in electronics and spintronics.

Based on the DFT band structure with spin-orbit coupling, we construct a tight-bind Hamiltonian of PtN₄C₂ in the basis of maximally localized wannier functions through the Wannier90 program Marzari et al. (2012); Mostofi et al. (2008). The topological properties of PtN₄C₂ can then be studied with WannierTools software package based on the Green’s function method Sancho et al. (1984); Wu et al. (2018).

As shown in Fig. 3(a), the band structure given by the means of maximally localized wannier functions agrees very well with the one given by the DFT calculations, in which Pt-$s,p,d$, N-$s,p$, and C-$s,p$ orbitals are adopted as the projection orbitals. To further confirm the topological non-trivialness of PtN₄C₂, we have calculated the topological $\mathbb{Z}_{2}$ invariant. The method of Wannier charge center evolution in half Brillouin zone is adopted Yu et al. (2011); Soluyanov and Vanderbilt (2011). As shown in Fig. 3(b), it is clear from the figure that the Wannier charge center evolution curves cut an arbitrary horizontal reference line odd times, indicating a value of $\mathbb{Z}_{2}=1$. Thus the gapped PtN₄C₂ with spin-orbit coupling is indeed a two-dimensional $\mathbb{Z}_{2}$ topological insulator. Meanwhile, from Fig. 3(c), the distribution of Berry curvature shows that the Berry phases around the two Dirac points within the Brillouin zone are opposite, as determined by the time-reversal symmetry, indicating that PtN₄C₂ is also a quantum valley Hall insulator Ghaemi et al. (2010); Ding et al. (2011); Xiao et al. (2007); Marino et al. (2015).

We further investigate the helical edge states of PtN₄C₂ projected along the (100) direction, whose band dispersions are given in Fig. 4(a). The topological nontrivial edge states are visible, which thread across the bulk band gap and connect the bulk conduction and valence bands of PtN₄C₂ at the two projected Dirac points in the Brillouin zone along the high-symmetry line -$\widetilde{Y}$–$\widetilde{\Gamma}$–$\widetilde{Y}$. Figure 4(b) gives the zoomed-in view of the areas highlighted by the boxes that surround the projected Dirac nodes and the $\widetilde{\Gamma}$ point respectively in Fig. 4(a). There are two edge states separately emerging from the bulk conduction/valence bands of one Dirac node and merging into the valence/conduction continuum of the other Dirac node within the Brillouin zone. They cross each other and form a single Dirac cone inside the bulk gap at the $\widetilde{\Gamma}$ point, protected by time-reversal symmetry which induces the Kramers degeneracy, as shown in middle panel of Fig. 4(b). The spin-momenta of these states are locked and protected from backscattering. This behavior is the main characteristic of the quantum spin Hall effect, which makes PtN₄C₂ promising for novel electronic and spintronic applications Hasan and Kane (2010); Qi and Zhang (2011).

In summary, based on symmetry analysis and the first-principles electronic structure calculations, we predict that PtN₄C₂ and Pt₂N₈C₆ are two prefect two-dimensional topological Dirac materials. Due to the lattice distortion from ideal honeycomb structure, linear anisotropic band dispersions around the Dirac cones are found, making both compounds a promising platform to realize orientation-dependent quantum devices. With spin-orbit coupling included, a sizable topological nontrivial band gap opens, and the two-dimensional materials transform into a quantum spin Hall insulator as well as a quantum valley Hall insulator, characterized by an odd $\mathbb{Z}_{2}$, topological helical edge states, and symmetry-protected opposite Berry curvatures. Our work not only expands the Dirac cone material family, but also provides a new avenue to searching for more two-dimensional topological quantum spin and valley Hall insulators.

Note added: After preparation of this manuscript, we noticed a paper recently published in PRB Wang et al. (2021), proposing a PtN₄-embedded graphene PtN₄C₁₀ that shares similar structural recipes with the PtN₄C₂ and Pt₂N₈C₆ systems discussed here, also found to be a two-dimensional quantum Hall insulator.

This work was financially supported by the National Natural Science Foundation of China under Grants Nos. 12074040, 11974207, and 11974194.