Abstract
Emergent Dirac fermion states underlie many intriguing properties of graphene, and the search for them constitutes one strong motivation to explore twodimensional (2D) allotropes of other elements. Phosphorene, the ultrathin layers of black phosphorous, has been a subject of intense investigations recently, and it was found that other groupVa elements could also form 2D layers with similar puckered lattice structure. Here, by a close examination of their electronic band structure evolution, we discover two types of Dirac fermion states emerging in the lowenergy spectrum. One pair of (typeI) Dirac points is sitting on highsymmetry lines, while two pairs of (typeII) Dirac points are located at generic kpoints, with different anisotropic dispersions determined by the reduced symmetries at their locations. Such fullyunpinned (typeII) 2D Dirac points are discovered for the first time. In the absence of spinorbit coupling (SOC), we find that each Dirac node is protected by the sublattice symmetry from gap opening, which is in turn ensured by any one of three point group symmetries. The SOC generally gaps the Dirac nodes, and for the typeI case, this drives the system into a quantum spin Hall insulator phase. We suggest possible ways to realise the unpinned Dirac points in strained phosphorene.
Introduction
Recent years have witnessed a surge of research interest in the study of Dirac fermions in condensed matter systems, ranging from graphene and topological insulator surfaces in twodimensions (2D) to Dirac and Weyl semimetals in 3D,^{1–4} which possess many intriguing physical properties owing to their relativistic dispersion and chiral nature. Especially, 2D Dirac fermion states have been extensively discussed in honeycomb lattices, commonly shared by groupIVa elements with graphene as the most prominent example,^{5–9} for which Dirac points are pinned at the two inequivalent highsymmetry points K and $K\prime $ of the hexagonal Brillouin zone (BZ), around which the dispersion is linear and isotropic. Later on, 2D Dirac points on highsymmetry lines were also predicted in some nanostructured materials,^{10} including graphynes^{11} and rectangular carbon and boron allotropes.^{12,13} However, the possibility of 2D Dirac points at generic kpoints has not been addressed, and such Dirac point has not been found so far.
Meanwhile, the exploration of 2D materials built of groupVa elements (P, As, Sb and Bi) has just started. Single and fewlayer black phosphorous, known as phosphorene, have been successfully fabricated, and was shown to be semiconducting with a thicknessdependent bandgap and a good mobility up to ~10^{3}cm^{2}/Vs, generating intense interest.^{14–21} While 2D allotropes with different lattice structures have been predicted and analysed for the other groupVa elements,^{22–25} we note that the puckered lattice structure similar to phosphorene has been demonstrated experimentally for Sb (refs 26–28; W. Xu et al., unpublished) and Bi^{29–32} (down to singlelayer) grown on suitable substrates, and been predicted for As as well.^{22} Motivated by these previous experimental and theoretical works, and in view of the ubiquitous presence of the Dirac fermions and the associated interesting physics, one may wonder: Is it possible to have Dirac fermion states hosted in such 2D puckered lattices? A simple consideration shows that here any possible Dirac point cannot occur at highsymmetry points. The reason is that each Dirac point at k must have a time reversal (TR) partner at −k with opposite chirality, whereas the BZ of the puckered lattice has a rectangular shape, of which all the highsymmetry points are invariant under TR. Therefore, if Dirac states indeed exist in such systems, they must be of a type distinct from those in graphene.
In this work, we address the above question by investigating the electronic structures of groupVa 2D puckered lattices. We find that Dirac fermion states not only exist, but in fact occur with two different types: one type (referred to as typeI) of (two) Dirac points are located on highsymmetry lines; while the other type (referred to as typeII) of (four) Dirac points are located at generic kpoints. Depending on their reduced symmetries, dispersions around these points exhibit different anisotropic behaviours. Points of each type can generate or annihilate in pairs of opposite chiralities, accompanying topological phase transitions from a band insulator to a 2D Dirac semimetal phase, and since they are not fixed at highsymmetry points, their locations can be moved around in the BZ. Particularly, to our best knowledge, the novel fullyunpinned (typeII) 2D Dirac points are discovered here for the first time. In the absence of spinorbit coupling (SOC), each Dirac node is protected from gap opening by a sublattice (chiral) symmetry, which can in turn be ensured by any one of three point group symmetries. The inclusion of SOC could gap the Dirac nodes, and in the case of typeI nodes it transforms the system into a quantum spin Hall (QSH) insulator phase. All these properties make the system distinct from graphene and other 2D materials. We further suggest that the novel unpinned Dirac points can be experimentally realized by the strain engineering of phosphorene. Our discovery therefore greatly advances our fundamental understanding of 2D Dirac points, and it also suggests a promising platform for exploring interesting effects with novel types of Dirac fermions.
Results
A groupVa pnictogen atom typically forms three covalent bonds with its neighbours. As shown in Figure 1 for a singlelayer phosphorene structure, the P atoms have strong sp^{3}hybridisation character hence the three P–P bonds are more close to a tetrahedral configuration. This results in two atomic planes (marked with red and blue colours) having a vertical separation of h comparable to the bond length. In each atomic plane, the bonding between atoms forms zigzag chains along ydirection. The unit cell has a fouratom basis, which we label as A_{U}, B_{U}, A_{L} and B_{L} (see Figure 1c), where U and L refer to the upper and lowerplane, respectively. The structure has a nonsymmorphic D_{2h}(7) space group which includes the following elements that will be important in our discussion: an inversion centre i, a vertical mirror plane ${\sigma}_{\mathit{\upsilon}}$ perpendicular to $\stackrel{\u02c6}{y}$, and two twofold rotational axes c_{2y} and c_{2z}. Note that due to the puckering of the layer, the mirror planes perpendicular to $\stackrel{\u02c6}{x}$ and $\stackrel{\u02c6}{z}$ are broken. With the same valence electron configuration, As, Sb and Bi possess allotropes with similar puckered lattice structures.
To study the electronic properties, we performed firstprinciples calculations based on the density functional theory (DFT). The details are described in the materials and methods. The calculated geometric parameters of groupVa 2D puckered lattices with D_{2h}(7) symmetry are summarised in the Supplementary Information. The obtained structures agree with the experiments and other theoretical calculations (refs 17,22,32; W. Xu et al., unpublished). The lattice constants a>b, reflecting that the interchain coupling is weaker than the coupling along the zigzag chains. The angle θ_{2} increases from ~70° for P to ~85° for Bi, whereas θ_{1} remains ~95°. The interplane separation h, as well as the bond lengths R_{1} and R_{3} increase by almost 1 Å; from P to Bi, while R_{2}, the distance between sites of neighbouring zigzag chains, increases only slightly, implying that the interchain coupling becomes relatively more important with increasing atomic number.
We first examine their corresponding band structures without SOC, whose effect will be discussed later. The results are shown in Figure 2. The puckered lattice of P is a semiconductor with a bandgap around Γpoint. From P to Bi, the direct bandgap at Γpoint keeps decreasing, and a drastic change occurs from Sb to Bi where linear band crossings can be clearly spotted along the ΓX_{2} line. Examination of the band dispersion around the two points (labelled as D and $D\prime $ in Figure 1d) shows that they are indeed Dirac points (see Figure 3a). Furthermore, along ΓX_{1} line, there gradually appear two sharp local band extremum points for both conduction and valence bands, where the local gap decreases from P to Sb with the two bands almost touching for Sb, yet the trend breaks for Bi. Remarkably, close examination reveals that for Sb and Bi, close to each extremum point there are actually two Dirac points on the two sides of the ΓX_{1} line (see Figures 1d and 3b). The energy dispersions around these Dirac points are shown in Figures 1e and f, clearly demonstrating the Dirac cone characters. Therefore, two types of Dirac points with distinct symmetry characters exist in this system: one pair of typeI Dirac points (D and $D\prime $) sitting on highsymmetry lines and two pairs of typeII Dirac points (near F and $F\prime $) at generic kpoints.
The band evolution around Γpoint from Sb to Bi and the appearance of typeI Dirac points in Figure 2 are reminiscent of a bandinversion process. Indeed, by checking the parity eigenvalues at Γ, one confirms that the band order is reversed for Bi around Γpoint (see Supplementary Information). For a better understanding, we construct a tightbinding model trying to capture the physics around Γpoint. Since the lowenergy bands are dominated with p_{z}orbital character (Figure 2), we take one orbital per site, and include couplings along R_{1} and R_{2} in the same atomic plane (with amplitudes t_{1} and t_{2}, respectively), as well as nearestneighbour interplane hopping along R_{3} (with amplitude t_{⊥}) (see Supplementary Information). Written in the basis of (A_{U}, A_{L}, B_{U}, B_{L}), the Hamiltonian takes the form:
where Q(k) is a 2×2 matrix of the Fourier transformed hopping terms (see Supplementary Information). The Hamiltonian (1) can be diagonalized and possible band crossings can be probed by searching for the zeroenergy modes, which exist when the condition $\mathit{\lambda}\equiv {t}_{\perp}/\left[2\left({t}_{1}+{t}_{2}\right)\right]<1$ is satisfied, with two band touching points at (0, ±k_{D}) where ${k}_{\mathrm{D}}=\left(2/b\right)\mathrm{arccos}(\mathit{\lambda})$. The direct gap at Γ can be obtained as Δ=2[t_{⊥}–2(t_{1}+t_{2})]. Hence this simple model indeed captures the emergence of two Dirac points D and $D\prime $, along with a transition as parameter λ varies: when $\mathit{\lambda}>1$, the system is a band insulator; when $\mathit{\lambda}<1$, it is a 2D Dirac semimetal. The transition occurs at the critical value ${\mathit{\lambda}}_{\mathrm{c}}=1$ when the conduction and valence bands touch at Γpoint and the band order starts to be inverted. This corresponds to a quantum (and topological) phase transition,^{33} during which there is no symmetry change of the system.
Equation (1) captures the trend observed in DFT results. The overlap betweenp_{z} orbitals is larger along the R_{3} bond, hence one expects that t_{⊥}>t_{1}>t_{2}. By fitting the DFT bands around Γpoint, one finds that from P to Bi, t_{⊥} decreases a lot, while t_{2} increases and becomes relatively more important (Supplementary Information). The result shows that (t_{⊥}, t_{1}, t_{2}) changes from (2.50, 0.77, 0.33) for Sb to (1.86, 0.63, 0.35) for Bi (units in eV). Hence λ crosses the critical value from Sb to Bi, indicating the band inversion at Γ and the appearance of two Dirac points.
The emergence of lowenergy relativistic chiral modes is the most remarkable property of Dirac points.^{33} To explicitly demonstrate this, we expand Hamiltonian (1) around each Dirac point, which leads to the lowenergy Hamiltonian
where q is the wavevector measured from each Dirac point, $\mathit{\tau}=\pm 1$ for D and $D\prime $, σ_{i}s are Pauli matrices for the subspace spanned by the two eigenstates at the Dirac point (apart from the Bloch phase factor): ${u}_{1}\u3009=(0,0,1,1)/\sqrt{2}$ and ${u}_{2}\u3009=(1,1,0,0)/\sqrt{2}$ and v_{x}=at_{⊥}(t_{1}−t_{2})/(t_{1}+t_{2}) and ${\mathit{\upsilon}}_{y}=b\sqrt{4{({t}_{1}+{t}_{2})}^{2}{t}_{\perp}^{2}}$ are the two Fermi velocities. The form of equation (2) may also be argued solely from symmetry. Compared with graphene, these typeI points are unpinned from the highsymmetry points. They can be shifted along ΓX_{2} (and even pairannihilated) by varying system parameters such as λ, although they cannot go off the line as constrained by the symmetries. In addition, different from graphene,^{6} the dispersion here is anisotropic, characterised by two different Fermi velocities.
Next, we turn to the fullyunpinned typeII Dirac points. The four typeII Dirac points start to appear for Sb in our DFT result, located close to the ΓX_{1} line. They can be more clearly seen for Bi (see Figure 3b). Again the band evolution implies a local band inversion near F and $\mathit{F}\prime $. Here F and $F\prime $ (on ΓX_{1}) are the midpoints of the lines connecting each pair of the typeII points. The lowenergy bands are mainly of p_{x}orbital character. To reproduce the fine features using a tightbinding model would require more hopping terms. Instead, we construct a lowenergy effective Hamiltonian around point F ($F\prime $) based on symmetry analysis. There the Hamiltonian is constrained by ${\sigma}_{\mathit{\upsilon}}$ , which maps inside each pair (labelled by μ=±1 for F and $F\prime $), and by i, c_{2y}, c_{2z}, and TR that map between the two pairs. Expansion to leading order in each wavevector component q_{i} gives (see Supplementary Information)
where q is measured from F (or $F\prime $), w, $w\prime $, m_{0} and m_{1} are expansion coefficients. Two Dirac points appear at (0, ±q_{0}) with ${q}_{0}=\sqrt{{m}_{0}/{m}_{1}}$ when sgn(m_{0}/m_{1})=1, corresponding to a local band inversion around q=0. Further expansion of the Hamiltonian around the Dirac point (0, νq_{0}) (ν=±1) leads to
This demonstrates that the two points at ν=±1 are of opposite chirality, as required by ${\sigma}_{\mathit{\upsilon}}$. The dispersion is highly anisotropic (at leading order, characterised by three parameters: w, q_{0} and $w\prime $) because the Dirac point is at a generic kpoint with less symmetry constraint, as compared with typeI Dirac points.
Unlike in 3D systems, Dirac nodes in 2D have a codimension of 2 hence are generally not protected from gap opening.^{33} In the absence of SOC, however, the Dirac nodes here are stable due to the protection by sublattice (chiral) symmetry between {A_{i}} and {B_{i}} (i=U, L) sites, which allows the definition of a winding number^{34,35} (that is, quantised Berry phase in units of π) along a closed loop ℓ encircling each Dirac point: ${N}_{\ell}={\oint}_{\ell}{\mathcal{A}}_{\mathit{k}}\cdot \mathrm{d}\mathit{k}/\pi =\pm 1$, where ${\mathcal{A}}_{\mathit{k}}$ is the Berry connection of the occupied valence bands. And for a 2D Dirac point, the sign of N_{ℓ} (or the ±π Berry phase) is also referred to as the chirality.^{6} Using DFT results, we numerically calculate the Berry phase for each Dirac point and indeed confirm that they are quantised as ±π. The signs are indicated in Figure 1d.
More interestingly, in the puckered lattice with a fouratom basis in a noncoplanar geometry, the sublattice symmetry can be ensured by any one of three independent point group symmetries: i, c_{2y} and c_{2z}. The resulting protection of Dirac nodes can be explicitly demonstrated in lowenergy models. For example, consider the typeI points described by equation (2). There the representations of i, c_{2y} and c_{2z} (denoted by $\mathcal{P}$, ${\mathcal{R}}_{y}$ and ${\mathcal{R}}_{z}$, respectively) are the same, which is, σ_{x}. Then the symmetry requirement ${\mathcal{R}}_{y}{H}_{\mathit{\tau}}({q}_{x},{q}_{y}){\mathcal{R}}_{y}^{1}={H}_{\mathit{\tau}}({q}_{x},{q}_{y})$ by c_{2y} directly forbids the presence of a mass term mσ_{z}. Meanwhile, since i and c_{2z} map one valley to the other, they protect the Dirac nodes when combined with TR (or ${\sigma}_{\mathit{\upsilon}}$ if it is unbroken), e.g., considering the combined symmetry of c_{2z} and TR (with representation $\mathcal{T}=K$ the complex conjugation operator): $({\mathcal{R}}_{z}\mathcal{T}){H}_{\mathit{\tau}}(\mathit{q}){({\mathcal{R}}_{z}\mathcal{T})}^{1}={H}_{\mathit{\tau}}(\mathit{q})$, which again forbids a mass generation. The underlying reason i, c_{2y} and c_{2z} each protects the Dirac node is that they each map between the two sublattices hence ensures the sublattice (chiral) symmetry. In comparison, the mirror plane ${\sigma}_{\mathit{\upsilon}}$ maps inside each sublattice, hence it alone cannot provide such protection. This reasoning is general and applies to typeII points as well. (In equation (3), i, c_{2y} and c_{2z} have representations as σ_{x} by construction, and when combined with TR, again each forbids the generation of a mass term ~mσ_{z}. See Supplementary Information.) We stress that the three symmetries i, c_{2y} and c_{2z} each protects the Dirac points independent of the other two. For example, we could disturb the system as in Figure 4 such that only one of the three symmetries survives. The corresponding DFT results confirm that the Dirac nodes still exist. Thus the crystalline symmetries actually offer multiple protections for the Dirac nodes in the current system.
SOC could break the sublattice symmetry. Hence when SOC is included, the Dirac nodes would generally be gapped.^{36} For typeI points, treating SOC as a perturbation, its leadingorder symmetryallowed form is ${H}_{\mathrm{SOC}}=\mathit{\tau}\mathrm{\Delta}{\sigma}_{z}{s}_{z}$, where s_{z} is Pauli matrix for real spin. This is similar to the intrinsic SOC term in graphene,^{37} which opens a gap of 2Δ at the Dirac points. For the typeII points, we obtain H_{SOC}=ηq_{y}σ_{z}s_{z} in equation (3) hence a gap of 2q_{0}η is also opened at these Dirac points. Gap opening by SOC is closely related to the QSH insulator phase.^{1,2,37} Here the band topology can be directly deduced from the parity analysis at the four TR invariant momenta.^{38} This means that only the band inversion at Γ between the two typeI points contributes to a nontrivial ${\mathbb{}}_{2}$ invariant; whereas that associated with typeII points does not. It follows that Sb is topologically trivial since it has only typeII Dirac points, while Bi is nontrivial since it has additional typeI points. These results are in agreement with previous studies.^{32}
Breaking all three symmetries i, c_{2y} and c_{2z} can also generate a trivial gap term mσ_{z} at the Dirac points, which competes with the SOC gap. For example, this happens when each atomic plane forms additional buckling structure.^{32} Nevertheless, as long as the trivial mass term does not close the SOCinduced gap, by adiabatic continuation the band topology will not change.
Discussion
Due to their different locations and the associated symmetries, the two types of Dirac points here exhibit properties distinct from that of graphene. With preserved sublattice symmetry and in the absence of SOC, the Dirac nodes are topologically stable—they can only disappear by pairannihilation between opposite chiralities. This is unlikely for graphene since the Dirac points are pinned at the highsymmetry points. In contrast, the two types of Dirac points here are less constrained. Pairannihilation (pairgeneration) indeed occurs during the quantum phase transition as observed from the band evolution.
It is noted that similar typeI points were also predicted in a few nanostructured materials.^{11–13} Meanwhile the typeII points discovered here are completely new. They are fullyunpinned and have highly anisotropic dispersions. With this discovery, now we can have an almost complete picture: 2D Dirac points can occur at highsymmetry points, along highsymmetry lines and also at generic kpoints.
It is possible to have Dirac points, originally sitting at highsymmetry points, to become unpinned when crystalline symmetry is reduced due to structural distortions. However, we stress that the typeII points here are distinct in that they are realized in a native crystalline structure with relatively high symmetry. Only in such a case, we can have a sharp contrast between generic kpoints where the group of wave vectors is trivial and the highsymmetry kpoints where the group is nontrivial, and accordingly the typeII point can move around (hence fullyunpinned) without any symmetrybreaking. More importantly, it is just because that typeII points occur in a state with high symmetry that the Dirac nodes can be protected (in the absence of SOC): as we discussed, the various crystalline symmetries ensure the protection of the Dirac nodes from gap opening.
It is remarkable that the two different types of Dirac points can coexist in the same 2D material. We emphasise that it is a result of the lattice structure and the valence character of the pnictogen elements. Our DFT result indeed shows that even starting from the P lattice, the two types of Dirac points can be separately tuned to appear or disappear by lattice deformations. For example, we find that the typeII Dirac points can be generated in phosphorene by applying uniaxial tensile strains along the ydirection. The DFT result in Figures 5 and 6 indeed shows the band inversion on ΓX_{1} and the formation of four typeII Dirac points around a strain of 16%. Since phosphorene has excellent mechanical properties and a critical strains >25% has been predicted,^{39} it is promising that the novel straininduced topological phase transitions and the appearance of typeII Dirac points can be directly observed in strained phosphorene. The lattice deformation that produces typeI points is discussed in the Supplementary Information. Similar scenarios occur for other groupVa elements as well.
So far, singlelayer As, Sb and Bi in their freestanding form have not been realized yet. Nevertheless, in view of the rapid progress in experimental techniques, we expect that these materials could be fabricated in the near future. Especially, for Sb and Bi, the puckered structures have been demonstrated by PVD growth down to singlelayer thickness on suitable substrates (refs 26–32; W. Xu et al., unpublished). Besides the topological properties, the presence of Dirac states is expected to endow these 2D materials with many intriguing properties for applications, such as the very high mobility, the halfquantised quantum Hall effect,^{40} the universal optical absorption^{41} and etc. Due to the highly anisotropic dispersions of these new Dirac points, the electronic transport properties such as the conductivities would show strong direction dependence. In addition, since there is no symmetry connection between the two types of Dirac points, when they are both present, it is possible to independently shift each type of points relative to the Fermi level, e.g., by strain engineering, leading to selfdoping and even the interesting scenario with both electronlike and holelike Dirac fermions in the same system. With the multiple Dirac points with different chiralities, it is possible to further control the carriers near different Dirac points for valleytronic applications.
In conclusion, based on firstprinciples calculations of 2D allotropes of groupVa elements with puckered lattice structure, we predict the coexistence of two different types of Dirac points: Dirac points on highsymmetry lines and at generic kpoints. In particular, the 2D Dirac points at generic kpoints are fullyunpinned, have highly anisotropic dispersions and are discovered here for the first time. Combined with lowenergy effective modelling, we unveil the lowenergy properties of these Dirac points. We show that their appearance is associated with the bandinversion process corresponding to a topological phase transition. The topology/symmetry protection of the Dirac nodes is analysed in detail. Interestingly, because of the unique lattice structure, there is a tripleprotection of the nodes by three independent point group symmetries. This also implies versatile methods to control the locations, as well as the dispersions around the Dirac points. When SOC is strong, the Dirac nodes are gapped, and in the case of typeI points (such as for Bi) this drives the system into a QSH insulator phase. We further show that the topological phase transition and the novel unpinned Dirac points can be realized in strained phosphorene. Our work represents a significant conceptual advance in our fundamental understanding of 2D Dirac points. The result also suggests a new platform to explore novel types of 2D Dirac fermions both for their fascinating fundamental properties and for their promising electronic and valleytronic applications.
Materials and methods
Firstprinciples calculations
Our firstprinciples calculations are based on the DFT implemented in the Vienna ab initio simulation package.^{42} The projector augmented wave pseudopotential method is employed to model ionic potentials.^{43} Kinetic energy cutoff is set to 400 eV and kpoint sampling on the rectangular BZ is with a mesh size 20×20. The minimum vacuum layer thickness is >20 Å; which is large enough to avoid artificial interactions with system images. The structure optimisation process is performed including SOC with the local density approximation for the exchangecorrelation energy^{44} and with van der Waals corrections in the Grimme implementation.^{45} The force convergence criteria is set to be 0.01 eV/Å. Hybrid functional (HSE06)^{46} is used for the band structure calculations.
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Acknowledgements
The authors thank D.L. Deng and Shengli Zhang for helpful discussions. This work was supported by NSFC (Grant No. 11374009, 61574123 and 21373184), the National Key Basic Research Program of China (2012CB825700), SUTDSRGEPD2013062, Singapore MOE Academic Research Fund Tier 1 (SUTDT12015004), A*STAR SERC 122PSF0017 and AcRF R144000310112. H.L. acknowledges support by Singapore National Research Foundation under NRF Award No. NRFNRFF201303. Y.L. acknowledges Special Program for Applied Research on Super Computation of the NSFCGuangdong Joint Fund (the second phase). The authors gratefully acknowledge support from SR16000 supercomputing resources of the Center for Computational Materials Science, Tohoku University.
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Y.L. and D.Z. performed the firstprinciples calculations. G.C. and S.G. helped with the data analysis and model fitting. H.L. and S.A.Y. did the analytical modelling and symmetry/topology analysis. W.C., Y.J., J.J. and X.s.W. participated in the discussion and analysis. X.s.W., Y.P.F., Y.K., Y.L., S.A.Y. and H.L. supervised the work. Y.L., S.G. and S.A.Y. prepared the manuscript. All authors reviewed the manuscript.
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Correspondence to Yunhao Lu or Shengyuan A Yang or Hsin Lin.
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Supplementary Information accompanies the paper on the npj Computational Materials website (http://www.nature.com/npjcompumats)
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Lu, Y., Zhou, D., Chang, G. et al. Multiple unpinned Dirac points in groupVa singlelayers with phosphorene structure. npj Comput Mater 2, 16011 (2016). https://doi.org/10.1038/npjcompumats.2016.11
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