Introduction

Semi-metal (SM) materials,1,2,3,4 such as Dirac semi-metal, Weyl semi-metal, and nodal line semi-metal, which have conspicuous gapless crossing points or lines near the Fermi level of the band structures, have received increasing attention for their significant physical mechanism5,6 and potential application in nanoscale devices.7,8 With the successful preparation in the laboratory of two-dimensional (2D) material, such as graphene,9 silicene,10 graphynes,11 and some transition metal borides,12,13,14 2D Dirac semi-metal (DSM) materials15,16,17,18 showed greater advantages than their three-dimensional (3D) counter-parts for their atomic-scale layer thickness and larger specific surface area. Combined with other important physical properties, such as magnetic,19,20,21,22 ferroelasticity,23 and valleytronics,24 research on 2D SM materials has driven the development of the next generation of low-energy transistors and electronic devices, and accelerated the progress of information technology advancement.25,26,27

Dirac points are protected by a certain structural symmetry, which is the key role for the study of DSMs. As a result, most studies have focused mainly on the materials with good symmetry, especially those with multiple symmetric protecting mechanisms.20,28 However, the existence of Dirac points in low-symmetric configurations still lacks research. In this kind of materials, Dirac points are protected by the only symmetric operation of the configuration, which is easier to understand and modulate. By preserving and breaking the structural symmetry, the transitions between DSMs, topological insulators, and trivial materials can be realized.

In this paper, we propose a B5ScNi monolayer as a 2D SM candidate using first-principles calculations based on density functional theory (DFT). The results of a cohesive energy and long-time molecular dynamic (MD) simulation showed a strong binding interaction and good thermodynamic stability at room temperature. Two Dirac points were observed in the Brillouin zone (BZ), protected by the only mirror symmetry of the configuration. Considering the spin–orbit coupling (SOC) effect, the non-trivial topological properties of the system have been demonstrated.

Computational Details

The calculations were performed by first-principles based on spin-polarized DFT using the Vienna Ab initio Simulation Package,29,30,31,32 choosing the suffix-plus-wave projector augmented wave33,34 pseudopotential and using the Perdew–Burke–Ernzerhof35 approximation to describe the exchange–correlation generalized functional. The DFT + U correction36 was applied to represent the strong correlation of electrons on the d-orbital in transition metal atoms, with U = 2.9 eV and 5.1 eV for Sc and Ni, respectively. The convergence of the electronic self-consist calculation was set to be less than 10−8 eV. To avoid the interaction between adjacent supercells, a vacuum layer larger than 20 Å was included and the plane wave truncation energy was set to 400 eV. Gaussian-type smearing of 10 meV was used on a k-point mesh of 15 × 23 centered around Γ for the integrations in the BZ. The atomic coordinates and the cell volume were fully optimized until the residual Hellmann–Feynman forces on each atom were smaller than 0.001 eV/Å. The SOC effect was included self-consistently. The maximum localized Wannier functions method, implemented by the WANNIER90 package,37 was used to reconstruct the energy bands close to the Fermi energy, and the iterative Green's function38,39 was used to calculate the density of edge states.

Results and Discussion

The fully optimized B5ScNi monolayer, as shown in Fig. 1a and b, has a rectangle lattice and a nearly planar configuration with Cs structural symmetry belonging to the Pm space group (No. 6). The symmetry of B5ScNi is much lower than those of other common 2D DSM materials, with only one mirror plane symmetric operator \({\widehat{M}}_{y}\). The nickel (Ni) atom localizes in the same plane of the nearby boron (B) atoms, while the scandium (Sc) atom stands out of the B skeleton for about 1.02 Å along direction c. The optimized lattice constants are a = 6.035 Å and b = 3.762 Å, respectively, which was further confirmed by scanning the energy as a function of the lattice constants. The result of biaxial strain as shown in Fig. 1c, in which ε = (a − a0)/a0, a, and a0 denote the biaxial strain and the lattice constant with and without the strain applied, respectively, and it indicates that the optimized lattice constants are indeed at the minimum in energy. The total elastic moduli are calculated to be C11 = 768.3536, C22 = 493.4744, C12 = 76.6044, and C44 = 159.5674 (all in kBar), which demonstrate the mechanical stability of B5ScNi according to the Born–Huang criteria for stability (C11, C22, C44 > 0, and C11C22–C122 > 0). In order to validate the thermodynamical stability of B5ScNi, a long-time ab initio molecular dynamics Nosé–Hoover heat-bath simulation using a 3 × 5 supercell was performed at 300 K for up to 100 ps with a time step of 1 fs, and the results are plotted in Fig. 1d. Clearly, there has been no structural failure, which indicates the thermal stability of the predicted 2D B5ScNi monolayer at room temperature.

Fig. 1
figure 1

(a) Top and (b) side view of the geometric structure of Born–Huang’s criteriaB5ScNi monolayer; (c) the energy as a function of biaxial strains; (d) the evolution of energy during the MD simulation, and the configuration at the end of the simulation (100 ps).

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Next, we investigated the electronic properties of Born–Huang’s criteriaB5ScNi monolayer by spin-polarized DFT calculations, and found that it has a nonmagnetic ground state. The band structure is plotted in Fig. 2a, from which it can be seen that the valence and conduction bands near the Fermi level cross each other, and the crossing point is located at point D (0.1, 0.5, 0.0) along R–Y. Although one of the crossing bands looks nearly flat, the Dirac point D is still of type I40. The Fermi velocity \(\hslash {v}_{\rm F}={\rm d}E(k)/{\rm d}k\) at the Dirac point is 1.460 × 105 m/s along D–R (comparable with graphene ~ 8.2 × 105 m/s41), and 2.450 × 104 m/s along D–Y. The effective mass of the electrons is close to zero, indicating the characteristic of zero-mass Dirac fermions. Furthermore, Fig. 2b illustrates the distribution of Dirac points in the whole BZ, and two Dirac points (D and D’) both located at the edge of the BZ can be observed. Figure 3a shows the orbital-projected band structure, illustrating that the crossing bands are mainly contributed by the pz orbitals of B and the d orbitals of Sc/Ni atoms.

Fig. 2
figure 2

(a) Band structure;(b) distribution of the two gapless Dirac points (D and D’) in the first Brillouin zone, as shown in the red area (Color figure online).

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Fig. 3
figure 3

(a) Orbital-resolved band structure near the Dirac point D; (b) the charge differential density of the B5ScNi monolayer.

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To understand the bonding mechanism between the B and Sc/Ni atoms, the charge differential density was performed, which is defined as

$$\Delta \rho ={\rho }_{{\mathrm{B}}_{5}\mathrm{ScNi}}-{\rho }_{{\mathrm{B}}_{5}}-{\rho }_{\mathrm{Sc}}-{\rho }_{\mathrm{Ni}},$$

in which \({\rho }_{{\mathrm{B}}_{5}\mathrm{ScNi}}\) is the total charge density of B5ScNi, \({\rho }_{{\mathrm{B}}_{5}}\) is the charge density of the B skeleton, and \({\rho }_{\mathrm{Sc}(\mathrm{Ni})}\) is the charge density of Sc (Ni) at the free atomic state. The result is plotted in Fig. 3b, from which it can be seen that the electrons transfer mainly from the B skeleton to Sc (Ni), forming an accumulation area of electrons near the transition metal atoms. The cohesive energy, defined as

$${E}_{coh}=(5{E}_{\mathrm{B}}+{E}_{\mathrm{Sc}}+{E}_{\mathrm{Ni}}-{E}_{{\mathrm{B}}_{5}\mathrm{ScNi}})/7,$$

in which \({E}_{{\mathrm{B}}_{5}\mathrm{ScNi}}\) and \({E}_{\mathrm{B}(\mathrm{Sc},\mathrm{ Ni})}\) are the energy of the B5ScNi and B (Sc, Ni) atoms, is calculated to be 3.5827 eV/atom, which is higher that of Cd4C2 reported previously42 (2.46 eV/atom), and indicates the strong binding strength between the atoms.

Since the Dirac point is protected by a certain structural symmetry, the analysis on the real part of the Kohn–Sham wavefunction near the crossing point was further performed to understand the protection mechanism, as shown in Fig. 4a. As mentioned above, the only symmetry in the system is the vertical mirror \({\widehat{M}}_{y}\), and the chirality of real-space wave functions for the mirror was carefully checked. Obviously, the two crossing bands have opposite mirror eigenvalues (± I), so that they are orthogonal mutually with any orbital hybridization forbidden by the mirror symmetry, resulting in the inevitable gapless Dirac point. As the Dirac points locate at the edge of the BZ, we further calculated the band structure of the 2 × 2 supercell to fold them inside the BZ. As shown in Fig. 4b, the Dirac point is preserved but locates along Γ–X instead, which is still clearly protected by the vertical mirror \({\widehat{M}}_{y}\) .

Fig. 4
figure 4

(a) Wave functions close to the Dirac point D, with their Eigen values of mirror symmetry; yellow and green areas show the positive and negative values of the real part of Kohn–Sham wavefunction, respectively; (b) band structure of the 2 × 2 super cell, with the mirror symmetry illustrated (Color figure online).

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All the results above are discussed without considering the SOC effect. When the SOC effect is included, the bands with spin-up and spin-down split, and the Dirac point D is slightly gapped, as shown in Fig. 5a. The band gap induced by the SOC implies the possibility of non-trivial topological properties, and the calculation of the local density of states for the semi-infinity ribbon was performed, as shown in Fig. 5b. It can be seen that there are two edge states connecting the conduction and valence band of the bulk, indicating the existence of topological edge states.

Fig. 5
figure 5

(a) Band structure with SOC considered; (b) local density of the edge states, color in scale of log10 (Color figure online).

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Conclusions

We have studied the geometric and electronic band structure of a 2D semi-metal called the B5ScNi monolayer. Our calculations show some interesting results: (1) B5ScNi has excellent thermodynamic stability at room temperature, (2) it has two Dirac points in the BZ protected by the mirror symmetry \({\widehat{M}}_{y}\), which is the only symmetry operation of the system, and (3), with the SOC considered, the Dirac points split slightly with non-trivial topological edge states. These results not only help us to under the structural protecting mechanism for configures with lower symmetry but also promote the application of 2D semi-metal materials in nanoscale electronic devices.