Amine
Benhaij
*^{a} and
Omar
Mounkachi
*^{ab}
^{a}Laboratory of Condensed Matter and Interdisciplinary Sciences (LaMCScI), Faculty of Sciences, Mohammed V University of Rabat, BP 1014, RP Rabat, Morocco. E-mail: amine_benhaij@um5.ac.ma; omar.mounkachi@um5.ac.ma
^{b}College of Computing, Mohammed VI Polytechnic University, Lot 660, Hay Moulay Rachid, Ben Guerir, 43150, Morocco

Received
13th May 2024
, Accepted 19th August 2024

First published on 19th August 2024

This study presents a theoretical examination of the electronic band structure of AA (AB) stacked bilayer blue phosphorus system within the fifth intralayer (5NN) and second interlayer nearest-neighbor (2NN) multi-orbital tight-binding (MOTB) approach. The variation of energy levels has been investigated through the symmetrical tensile strain of the low-buckled honeycomb lattice. Here, the primary objective is to examine the existence of Dirac electronic features in hexagonal stacked bilayer geometry. Our theoretical calculations predict that the AA bilayer is a new hexagonal two-dimensional material with p_{x,y}-orbital Dirac-like states at the high-symmetry K point. Consequently, these systems can host massless (massive) Dirac fermions. In particular, the AA bilayer exhibits zero-gap Dirac-like properties and manifests distinguishable Dirac-like cones in the presence of weak spin–orbit coupling when a modest stretch of 2.30% is achieved with a remarkably high Fermi velocity of approximately v_{f} ≈ 0.12 × 10^{5} m s^{−1}. The behavior of the dispersion bands aligns reasonably well with recent experimental observations. Moreover, a stretch of 7.17% breaks some of the sublattice equivalence and enhances the spin–orbit interaction, resulting in the emergence of an electronic band gap of approximately ≈ 0.27 eV in the proximity of the high-symmetry K point. Furthermore, the tiny gap induced by the spin–orbit interaction implies topological nontriviality in the electronic state (quantum anomalous Hall state) of the honeycomb lattice. These findings categorize the AA bilayer as a rare two-dimensional Dirac-like material. This work provides, to the extent of our knowledge, a pioneering investigation into the existence of Dirac electronic properties in bilayer blue phosphorus. In addition, we present the first derivation of the MOTB model. However, the identified electronic characteristics designate this two-dimensional system as an ideal candidate for high-performance nanoelectronic devices.

Other mono-elemental 2D materials within the C group comprise silicon (Si, Z = 14), germanium (Ge, Z = 32), and tin (Sn, Z = 50) atoms, characterized by a low-buckled honeycomb formation in hybridized sp^{2} (sp^{3}) mode, which have already been successfully fabricated.^{15–17} Transitioning to the B family, borophene emerges as an intriguing 2D boron compound, which also exhibits Dirac cones with a Fermi velocity (v_{f}, characterizes the speed at which electrons move near the Fermi level) approximately two-fold that of graphene along the high-symmetry Γ–X (Y–M) direction.^{18–22} Nevertheless, these configurations do have particular restrictions. For instance, the disparity between their on- and off-current states (on/off ratios) is insufficient for effective electronic use.^{23} Consequently, attaining both an adequate on- and off-current state ratio alongside superb carrier mobility in these substances presents a crucial challenge.

Nonetheless, the N column presents a compelling alternative for synthesizing 2D black phosphorus (puk-P), arsenene (As, Z = 33), antimonene (Sb, Z = 51), and bismuthene (Bi, Z = 83).^{24–27} In contrast, much research has concentrated on multilayer structures made solely of elemental phosphorus (P, Z = 15) among these materials.^{28–30} Furthermore, optical examinations^{29,31} have revealed that the energy band gap (E_{g}) of puk-P varies from 0.5 to 2 eV, contingent upon the number of sheets present.^{32–34} This gap is notably higher compared to its bulk counterpart, exceeding it by approximately ≈ 0.3 eV.^{24,32,35} In addition, the puk-P-based FET displays an adequate on/off ratio of 10^{5} and excellent carrier mobility of approximately ≈ 10^{3} cm^{2} V^{−1} s^{−1}.^{24,36,37} Despite its potential as a 2D semiconductor, puk-P rapidly oxidizes in the presence of light, air, and humidity.^{38} However, previous research has confirmed that its stability can be preserved using thin films of Al_{2}O_{3}.^{39}

The distinctive characteristics of monolayer puk-P have propelled it into a rapidly evolving domain within condensed matter and materials science, with particular emphasis on its novel allotrope, blue phosphorene (buk-P).^{40} Initially foreseen through first-principles (FP) calculations, buk-P has been successfully fabricated on an Au(111) substrate via the MBE approach.^{41} Furthermore, it adopts a structure akin to that of graphene, germanene, arsenene, stanene, and silicene,^{17,23,25,42–44} characterized by a low-buckled honeycomb geometry. Notably, the synthesis of these compounds in group-IV often involves strong interfacial interactions with substrates, which can disrupt crucial p_{z}-orbital Dirac excitations. However, previous research has demonstrated the existence of robust Dirac states arising from the p_{x} and p_{y} orbitals (p_{x,y}-orbitals) in silicene grown on a 4H-SiC(0001) substrate.^{45} These states exhibit exceptional tunability and a greater degree of control over their electronic properties. Moreover, the stronger spin–orbit coupling (SOC) observed in heavier group-IV elements suggests the potential of realizing an exotic QSHE at room temperature. This result is expected to be applicable to germanene, arsenene, and stanene. Consequently, buk-P holds promise for fulfilling the expectations established for the aforementioned materials, especially regarding the existence of massless Dirac excitations.

Numerous examinations have been carried out to elucidate its mechanical, electrical, and optical properties.^{41,46–50} According to these theoretical analyses, the monolayer buk-P demonstrates an indirect E_{g} of approximately ≈ 2.0 eV. In addition, the carrier mobility is estimated to be approximately ≈ 4.6 × 10^{2} cm^{2} V^{−1} s^{−1} for the armchair axis and ≈ 4.7 × 10^{1} cm^{2} V^{−1} s^{−1} for the zigzag direction. Moreover, its E_{g} lies within the range of the visible light spectrum (1.65–3.1 eV), suggesting its promise for optoelectronic applications as a 2D system.^{41,48,51–53} Furthermore, recent experimental studies have delved into determining the E_{g} of a monolayer utilizing a set of DFT, ARPES, and STS techniques. The reported results (0.8–1.1 eV)^{41,51,54,55} diverge from the theoretical FP predictions.^{40,56} In addition, the presence of Dirac electronic features at the K point of the first Brillouin zone (BZ) has been experimentally demonstrated,^{57} aligning with theoretical predictions indicating a Fermi velocity of approximately v_{f} ≈ 0.85 × 10^{6} m s^{−1},^{58–61} which broadens the potential uses of this system.

The emergence of Dirac excitations predominantly results from the contributions of the P atom s- and p_{x,y}-orbitals at the K point. In addition, it has been confirmed that they exhibit exceptional adaptability along the Γ–K direction while maintaining their degeneracy.^{61} Notably, the in-plane orientation of the p_{x,y}-orbitals minimizes their interaction with the substrate. Consequently, these Dirac states exhibit reduced sensitivity to the substrate, potentially leading to enhanced robustness for practical applications. Recently, numerous 2D configurations, including X-hydride/halide (X = N–Bi),^{62} functionalized germanene,^{63} hydrogenated and halogenated buk-P,^{59} hydrogenated borophane,^{21} specific organic substances,^{64–66} and fluoridated tin films,^{16,67} featuring Dirac fermions originating from p_{x,y}-orbitals have been demonstrated. These studies underscore the diversity of Dirac excitations and their prevalence across various material platforms. Despite considerable advancements in the fabrication of these compounds, the experimental validation of 2D Dirac systems has remained limited.^{68,69}

The remarkable progress achieved in monolayer buk-P research necessitates a thorough investigation of its bilayer counterpart. The latter configuration presents a unique opportunity to explore potentially novel properties that may not be present in monolayer form. Nonetheless, the present surge of interest and anticipated importance attributed to the p_{x,y}-orbital Dirac excitations in the honeycomb configuration demand a thorough and profound understanding of these phenomena. Moreover, carrying out an in-depth investigation and attaining a profound perception of the nature of these Dirac electronic features is crucial for thoroughly elucidating their underlying physical principles and harnessing their promising attributes across a diverse spectrum of technological uses. Therefore, the emergence of Dirac-like excitations in the AA (hexagonal) stacked bilayer system within the 5NN in- and 2NN out-of-plane coupling MOTB approach has been explored. Here, we aim to improve upon the few recently presented interpretations and quantitatively enhance the multilayer buk-P overall TB picture through the incorporation of more distant interacting sites.^{70} The incorporation of distant neighbors beyond the 2NN in-plane and NN interlayer couplings is crucial for representing the energy dispersion throughout the BZ. Thus, the quantitative deviation of the 2NN intralayer and NN out-of-plane coupling MOTB calculations can be enhanced. Note that our models for the unstrained bilayer (monolayer) closely align with the most effective self-energy calculations based on the GW method. In addition, this effective MOTB Hamiltonian provides an intuitive understanding of the p_{x,y}-orbital Dirac states and distant transfer integral processes both in- and out-of-plane. However, for the first time, the AA (AB) stacked bilayer electronic band structure has been reported utilizing the aforementioned approximations.

This article is structured as follows: in Section II, we explore the structural configuration of the monosheet and derive its 5NN intralayer interaction MOTB dispersion bands to verify the applicability of our 5NN in- and 2NN out-of-plane coupling model. We dedicate Section III entirely to the discussion of the electronic MOTB Hamiltonian and the resulting energy dispersion representing the valence orbital electrons of the bilayer (monolayer) buk-P system. This section considers various parameters coupling atomic orbitals (AOs), distant transfer integrals, and the neglected overlap. Furthermore, the presence of Dirac electronic features in the AA bilayer configuration has been investigated through an applied symmetrical tensile strain of the low-buckled honeycomb formation. In addition, the spin–orbit interaction in the aforementioned system has been discussed. Finally, in Section IV, we draw our conclusions and final remarks. The details of the calculations conducted within the MOTB approximations are provided in the ESI.†

(1) |

The valence AOs s, x, y, and z are represented by n, while m denotes A_{1j} (an atom in sublattice A at a given site) and B_{0} from the two different sublattices. In addition, N represents the number of unit cells, _{m} represents the lattice translation vector, represents Bloch's wave vector, represents the phase factor, and ϕ_{n}( − _{m}) represents an atomic wave function. Moreover, Bloch's molecular orbital (MO) wave function is constructed by summing up the individual Bloch's wave functions. This summation incorporates contributions from various Bloch sums, which account for the periodic nature of the lattice. Mathematically, Bloch's MO wave function can be written as:

(2) |

We must first determine the values of E(k) and the coefficients c_{n,m} in order to solve eqn (3). The variational method is one way to achieve this, as demonstrated in eqn (4):

(3) |

(4) |

The system of equations resulting from multiplying both sides of eqn (3) yields eight equations that depend on c_{n,m} and E(k), which can be solved in matrix form as a generalized eigenvalue problem. Note that solving the Hamiltonian matrix can be accomplished using Matlab, Python, Mathematica, GNU Octave, Fortran, or Julia. In this paper, the MOTB computations are all performed employing self-developed Matlab scripts, and the resulting energy dispersion levels are illustrated in [Fig. 2 and 4–6]. Therefore, the following can be used to represent the effective Hamiltonian matrix:

(5) |

To derive the 4 × 4 matrices M_{0} and M_{1} within the NN coupling MOTB model, we consider the three A_{1j}-NNs and sum over their lattice displacement d_{1j}, as shown in [Fig. 1 upper panel, top view]. Note that superscript t in matrix M_{1} represents the transpose sign or operation, and the asterisk denotes the complex conjugate. The Hamiltonian in eqn (5) is represented by the 8 × 8 matrix with B_{3s}, B_{3x}, B_{3y}, B_{3z}, A_{3s}, A_{3x}, A_{3y}, and A_{3z} as the basis.

(6) |

(7) |

Moreover, the following can be used to define the geometrical and phase factors:

(8) |

The diagonal entries in eqn (6) represent the on-site energy levels of the valence electrons. Here, we adopt the values reported in recent research.^{61,70,74,75} Conversely, the hopping integrals of the NN coupling matrix between the A_{1j} and B_{0} positions are represented by the off-diagonal elements in eqn (7). The electronic band structure of the monosheet can be derived by diagonalizing the resulting 8 × 8 secular equation as follows:

(9) |

However, by utilizing the hopping integral parameters provided by the TB-DFT approximation, we establish our 5NN intralayer interaction MOTB calculations. The selection of 5NN is based on achieving a balance between computational efficiency and accuracy. It captures the necessary interactions within the layer without introducing excessive computational complexity that comes with higher-order in-plane interactions. Therefore, the quantitative accuracy of the model depends on the next-NN hopping energies, which are intricately linked to the atomic arrangement and the underlying crystal lattice structure of the material. Note that the hopping integral parameters exponentially approach zero as the distance between the NN sites increases. Consequently, the contribution of higher-order integrals becomes negligible (6NN ≈ 0 eV).

Hence, within the framework of the 2NN interaction MOTB description, the Hamiltonian in eqn (5) can be formulated as follows:

(10) |

The new 4 × 4 matrix M_{2} represents the 2NN interactions between the B_{0} and B_{2j} substances. Note that from here onward, we consider that the transfer parameter V_{ll′q} exponentially depends on the vector positions d_{nj}:^{74,76,77}

(11) |

(12) |

The newest 4 × 4 matrices M_{3}, M_{4}, and M_{5} can be considered the third-, fourth-, and fifth-NN in-plane interactions between the central and A_{3j}, A_{4j}, and B_{5j} atoms, consecutively. However, the primitive reciprocal lattice vectors are represented by b_{1} and b_{2}, with their components relative to the rectangular coordinate system (k_{x}, k_{y}) provided by 2π/a (1, 1/√3) and 2π/a (1, −1/√3), respectively. The corresponding BZ is a Bravais honeycomb lattice, as depicted in the highlighted inset [Fig. 2] with the high-symmetry points located at Γ = (0, 0), M = 2π/a (0, 1/√3), and K = 2π/a (1/3, 1/√3).

This section outlines the MOTB description of the bilayer buk-P systems. To do so, the previously discussed model in eqn (12) is adopted. The unit cell depicted in [Fig. 3 (red rhombus)] includes four atoms, the A_{1j} (B_{0}) from the upper layer and A_{1j+} (B_{0+}) from the lower sheet. The primitive lattice vectors a_{1} and a_{2}, as well as the lattice constant a, are identical to those of the monosheet (the lattice constant has been preserved, independently of the stacking mode, alongside the buckling height). Thus, the reciprocal lattice and first BZ are identical to those of the monosheet. Moreover, the AA and AB interlayer distances Δ_{int} are 3.24 and 3.21 Å, respectively.^{70}

1. AA stacking.
In a basis with the elements A_{1j} (B_{0}) and A_{1j+} (B_{0+}), the transfer integral parameter matrices can be derived as follows:

(13) |

The top-left and bottom-right 8 × 8 blocks H_{mono}(k) represent the behavior within the top A_{nj} (B_{nj}) and bottom A_{nj+} (B_{nj+}) sheets, consecutively. On the other hand, the top-right and bottom-left 8 × 8 blocks of the Hamiltonian describe interlayer interactions, where H_{l} is given by:

(14) |

In this stacking configuration, the new 4 × 4 matrices M_{2′}, M_{3′}, M_{zeros}, and M_{5′} in eqn (14) can be regarded as the NN and 2NN out-of-plane interactions between B_{0} (B_{0+}), B_{0} (A_{1j+}), A_{1j} (B_{0+}), and A_{1j} (A_{1j+}), consecutively. Note that M_{zeros}, as indicated, is a matrix of zeros, and M_{2′} describes the NN interlayer coupling (as well as M_{5′}). Furthermore, the resulting energy levels are illustrated in [Fig. 4(a)]. The overall electronic band levels are similar to those of the monosheet, with each monolayer energy level divided into two bands by an energy that is approximately equal to the out-of-plane couplings. A comparison with preceding theoretical calculations and experimental studies shows good matching with the findings.^{46,71,80,81} This approach offers a computationally efficient alternative to more complex methods.

Fig. 4 Electronic band structure (a) of the AA bilayer system and (b) the AB stacked formation within the 5NN intra- and 2NN interlayer coupling model along the Γ–M–K axis. |

Despite being significantly weaker than the in-plane couplings, the out-of-plane interactions present profound differences between the electronic and transport characteristics of monolayer and bilayer buk-P systems. However, the numerous VBM (CBM) with comparable energy values indicate various potentials of other indirect electronic transitions. These transitions can have a significant influence on optical and electrical features, leading to interesting phenomena (excitonic effects, photovoltaic responses, and nonlinear optical behavior). Thus, further studies are necessary to provide deeper insights. Note that the NN interlayer couplings qualitatively show good overall matching of the electronic band structure. Consequently, the incorporation of the 2NN interlayer couplings considerably enhances the overall VBM (CBM) picture with FP computations.^{46,70,71,80,81} Furthermore, the VBM and CBM energies occur at the Γ–K and Γ–M regions consecutively, with an E_{g} of approximately ≈ 1.44 eV. The incorporation of distant neighbors beyond the 2NN in-plane and NN interlayer couplings is crucial for representing the energy dispersion throughout the BZ.

2. AB stacking.
In our AB stacking form, where the lower sheet stacks with the shift a/2 (1, −√3) in the x–y plane with respect to the first one, only atoms A_{1j} of the upper layer and atoms B_{0+} of the lower layer are directly connected by tunneling [as shown in Fig. 3(b), side view with solid vertical black line].

(15) |

The new 4 × 4 matrices M_{zeros}, M_{2′}, M_{4′}, and M_{5′} in eqn (15) can be regarded as the NN and 2NN out-of-plane interactions between B_{0} (A_{1j+}), B_{0} (B_{0+}), A_{1j} (B_{0+}), and A_{1j} (A_{1j+}), consecutively. Moreover, the resulting energy dispersion band is shown in [Fig. 4(b)]. Furthermore, the VBM and CBM energies occur at Γ–K and Γ–M regions consecutively, with an E_{g} of approximately ≈ 1.48 eV.

The overall electronic band levels are similar to those of a monolayer, where each energy level is divided into two bands. Thus, the split of the energy levels originates from the atomic positions. Note that E_{g} in the AA (AB) stacked bilayer reduces because of the weakening of the confinement effect. In a single-layer 2D material, the electrons are strongly confined to move in only two dimensions, leading to a large E_{g} and unique electronic properties. However, as more layers are added, the confinement becomes weaker, and electrons are able to spread out in the third dimension (reduced Coulomb repulsion). This increases the overlap between the electronic wave functions in adjacent layers, which leads to the formation of new electronic states that modify the band structure. Specifically, it modifies the effective mass of the electrons, which in turn affects the electronic band structure. These effects combined to produce a narrower E_{g} for multilayer 2D materials compared to their single-layer counterparts. Thus, the interlayer interactions in the AA (AB) stacking bilayer form can modify the electronic structure and contribute to the reduction of E_{g}. Note that the overall AA features resemble the AB stacking case with some differences due to the specific atomic arrangements, and the E_{g} in the latter (≈ 1.48 eV) is slightly larger compared to the AA stacking form (≈ 1.44 eV). Furthermore, we demonstrate that, for all examined stacking forms, bilayer buk-P retains its indirect E_{g} semiconducting feature, as depicted in [Fig. 4], which is, unfortunately, a serious limitation to the use of this 2D material, especially in the solar cell industry, due to the poor efficiency that characterizes indirect E_{g} semiconductors.^{49,84} In contrast, it is a suitable candidate for optoelectronic devices.^{41,51,85}

Our findings suggest that the electronic characteristics of all the systems presented are outstanding for various applications due to other possible electronic transitions; however, we observed the existence of unique aspects in the energy dispersion, suggesting that quasi-direct electronic transitions and Dirac electronic features can occur. Therefore, the use of these systems in various electronic devices is applicable. In addition, this demonstrates the suitability of the present 5NN intra- and 2NN interlayer coupling MOTB descriptions for quantitative analysis.

However, it was found that the AA is energetically more stable than the AB stacking sequence.^{46,71,80,81,83} Furthermore, no experimental details regarding their stacking patterns have been reported to date. This presents a significant knowledge gap that future research should address. Therefore, we propose the AA stacked bilayer buk-P configuration as a more realistic and pertinent reference for investigating the presence of Dirac electronic features in this system. In addition, we anticipate that this concept can be extended to AB and AC configurations. Our MOTB calculations are established by employing the SKH hopping integrals (yet we do not expect accurate findings for some potential reasons, which we shall mention afterward; although this model might not be perfect for presenting highly accurate quantitative descriptions, the adopted interpretation can easily capture qualitative aspects of the energy dispersion). The resulting energy dispersion bands are illustrated in [Fig. 5(a)–(f)]. Although the theoretical introduction of substantial strain into a bilayer buk-P appears feasible, a thorough examination (validation) is paramount to establishing its practical viability. The successful convergence of these approaches will ensure the practical applicability of the proposed strained materials.

3. Dirac-like cones.
The manifestation of Dirac electronic characteristics in 2D materials is contingent on stringent criteria pertaining to specific symmetries, parameters, Fermi level, and band overlap conditions.^{61,68,69} Therefore, the realization of Dirac systems is comparatively rare and is confined to a limited set of lattice configurations.^{86} Typically, the honeycomb emerges as the predominant platform for accommodating these distinctive features.^{15,42} For instance, in a periodic honeycomb arrangement, the occurrence of Dirac electronic features in the material within the NNs depends on the capability of the three transfer parameters to establish a triangular formation. These hopping integrals are subject to the influence of the atomic structure, notably the bond length, which is closely related to the crystal lattice. Nevertheless, the emergence of Dirac electronic features in monosheet buk-P can be comprehensively elucidated through our effective MOTB model, the intricacies of which are thoroughly expounded in the (ESI†).^{61}

The application of biaxial tensile strain induces a significant alteration in the geometrical characteristics of the bilayer system (lattice symmetry, electronic interactions, and the potential experienced by electrons). Consequently, shifting the positions of the VBM and CBM. This results in a strain-induced band gap tuning, with the VBM shifting upwards and the CBM downwards, leading to a reduction in the overall band gap. It is necessary to adjust intralayer and interlayer interactions, thereby inducing Dirac-like cones.^{12,53,59,82} Overall, the theoretical basis lies in the interplay between strain-induced symmetry breaking and band splitting. Compared to graphene, with its honeycomb lattice structure, which inherently possesses the necessary conditions for Dirac cones, strain-induced Dirac cones can be more tunable but potentially less stable.^{3,6,8,77} However, this transformation manifests as a notable elongation of bond length b, accompanied by a substantial decrease in buckling height h_{BA} and interlayer distance Δ_{int}. As a result of these modifications, the a demonstrate an approximate augmentation of ≈ 21.95%, 28.05%, 29.88%, 32.32%, 40.24%, and 43.29% relative to the pristine form. Moreover, under increasing tensile strain, the bilayer material adopts a quasi-planar configuration, whereby the system angle approaches approximately ≈ 120°, closely resembles the characteristic bonding angle observed in sp^{2} hybridized configurations. However, this transition from a buckled form to a quasi-planar hexagonal lattice profoundly affects the electronic structure (the energy levels of the s and p_{x,y,z} orbitals can be shifted). This critical distinction leads to the emergence of Dirac-like cones that exhibit distinct characteristics for in-plane σ and out-of-plane π bonds.

Our findings, as illustrated in Fig. 5, reveal a key characteristic of the energy band structure of the AA stacking mode under tensile strain. Notably, zero E_{g} (≈ 1 meV) emerges as a cone-shaped dispersion (linear crossing slightly anisotropic within the model) centered around the Γ–K direction upon reaching a tensile strain of approximately ≈ 2.30%, which aligns reasonably well with recent experimental observations,^{57} as illustrated in Fig. 5(a). This behavior of the energy dispersion bands can be mathematically described by the equation E = ħ·v_{f}·k. Furthermore, the Dirac-like point is located approximately ≈ 2.46 eV above E_{f}. This upward shift can be attributed to the inherent electron deficiency within the bilayer system. Interestingly, the Dirac-like intersection in Fig. 5(b) shifts to a new position closer to its original location. Moreover, the increased a induces a downward energetic shift of the crossing point, which brings it closer to the Fermi line, as demonstrated in Fig. 5(c). This observation underscores the tunability of the Dirac-like cones along the Γ–K direction with respect to E_{f} by varying the hopping integral parameters. In addition, the deposition of the bilayer system on a substrate enables the modulation of the energy positions of the Dirac points and compensates for the electron deficiency within it. Note that we provide compelling evidence that a tunable E_{g} can be realized through intralayer strain without compromising the underlying crystallographic symmetry. This translates into the ability to manipulate the energy range in which massless Dirac fermions emerge.

Nevertheless, the energy dispersion bands near the contact points of the CBM and VBM [as illustrated in Fig. 5(b)–(f)] exhibit a parabolic shape (quadratic dependence on Bloch's wave vector k). This implies that electrons residing in the vicinity of these points effectively possess features akin to massless Dirac fermions. Consequently, these points can be considered analogous to Dirac cones. Note that the energy disparity between the CBM and VBM is quantified to be approximately ≈ 0.27–0.4 eV, as demonstrated in the insets of Fig. 5(b)–(f). The energy separation can be modulated through the application of both an electric field and doping.

Clearly, a undergoes a significant increase of 43.29%, transitioning from a = 3.28 Å in the pristine bilayer to a = 4.7 Å observed in our model. However, b experiences a simpler elongation from 2.26 Å to 2.71 Å in the strained representation (a = 4.7 Å), which corresponds to a stretch of approximately ≈ 19.58%. Furthermore, the Δ_{int} undergoes a significant decrease of approximately ≈ −33.95%, transitioning from Δ_{int} = 3.24 Å to Δ_{int} = 2.14 Å observed in the strained model a = 4.7 Å. The Dirac electronic features can be directly attributed to the two equivalent sublattices forming the hexagonal lattice geometry. Through spatial inversion, the sublattices interchange. Importantly, the emergence of a Dirac-like cone is achievable even with a relatively low (2.30%) elongation of the P–P bond. The key is to separate the energy levels of the in-plane s- and p_{x,y}-orbitals, and out-of-plane p_{z}-orbitals through the sp^{2} configuration, yielding respective σ- and π-character Dirac-like cones. Hence, it is evident that the emergence of the distinctive Dirac-like cone spectrum is not solely attributed to strain but is significantly influenced by the geometrical phase transition from a buckled form to quasi-planar geometry. This highlights the critical role of structural modifications in shaping electronic band structures. Note that a further increase in a induces a downward energetic shift and a wider gap. This trend continues until strong correlation effects dominate (the inter-P hopping parameter becomes negligible), which leads to the disappearance of the Dirac electronic features.^{58} Furthermore, we unveil a key distinction between the p_{x,y}- and p_{z}-orbital Dirac-like states. Whereas out-of-plane π Dirac fermions typically exhibit only two linearly dispersed bands near the Dirac intersection, the in-plane σ counterparts possess additional bands. Interestingly, this observation aligns with the established trends reported in prior studies on p_{x,y}-orbital Dirac states.^{59,61,87,88}

In strained buk-P, the electronic states near E_{f} (1.58–2.46 eV) predominantly originate from the p_{x,y}-orbitals. Consequently, these states present exceptional tunability and a greater degree of control over their electronic properties. In addition, v_{f} is more sensitive not only to the transfer integral parameters through the σ (π) bonding but also to the alterations of a, h_{BA}, Δ_{int}, and buckling angles induced by the applied tensile strain. In bilayer buk-P, the Fermi velocity is generally lower than that in the monolayer. This is due to the weakening of the confinement effect and reduced Coulomb repulsion. Our calculations revealed a remarkably high Fermi velocity of approximately v_{f} ≈ 0.12 × 10^{5} m s^{−1}. This is consistent with the findings of previous experimental and theoretical studies that investigated analogous geometry,^{89–92} thereby affirming the reliability of the methods utilized in our calculations. However, the key factor governing the formation of Dirac-like cones is the geometrical phase shift from buckled to quasi-planar geometry.^{58,59,87,88} Thus, bilayer buk-P can be transformed from an indirect E_{g} configuration into a 2D Dirac-like system by applying tensile strain, an electrical field, doping, or covalent functionalization.

The realization of Dirac cones in experimental settings poses significant challenges. In some cases, the substantial strain values predicted theoretically may not be readily achievable within the limits of practical laboratory techniques.^{58,59,86} The 3D VBM and CBM bands are further depicted in Fig. 6, where the crucial features of the Dirac cone are prominently displayed along the high-symmetry Γ–K direction of the BZ.

4. Spin–orbit coupling.
Within the framework of the MOTB approximation, the manifestation of Dirac electronic features relies on specific combinations of the hopping integral parameters.^{93,94} Therefore, the quantitative accuracy of the model depends on the hopping energies, which are intricately linked to the atomic arrangement and the underlying crystal lattice structure of the material. In the case of bilayer buk-P, the application of tensile strain disrupts some of the honeycomb lattice symmetry, leading to enhanced SOC. This slight symmetry breaking at the sublattice level triggers the opening of a tiny E_{g} of approximately ≈ 0.27 eV and enables tuning of the inversion asymmetry, transforming the massless Dirac fermions into massive particles.^{62,95} Furthermore, crystal symmetry permits the existence of an intrinsic (on-site) SOC.^{58,96} Similar observations have been reported in silicene and germanene,^{42,45,97} where buckling enhances the SOC because of the direct hybridization between the out-of-plane π and in-plane σ orbitals. Moreover, the dominance of the p_{x,y}-orbitals over their p_{z} counterparts at low energy levels is predicted to significantly influence the emergence of a large intrinsic SOC. This enhanced overlap can lead to a stronger interaction between the electron's spin and its orbital motion, contributing to the SOC. The p_{x,y}-orbitals are more sensitive to relativistic effects, which are the core reason behind SOC. This makes them more susceptible to spin–orbit interactions compared to p_{z}-orbitals. Despite the presence of an E_{g}, SOC, and tunable inversion asymmetry in the buckled hexagonal structures, the potential to realize intriguing anomalous Hall effects remains.^{62}

Intriguingly, the bilayer exhibits a zero-gap Dirac-like character and presents a distinguishable Dirac-like cone in the absence of a strong SOC, as depicted in Fig. 5(a). However, upon the incorporation of different a, h_{BA}, and Δ_{int} values, the Dirac-like cone can be gapped, and a new tiny E_{g} of approximately ≈ 0.27 eV emerges at the Dirac-like points, as demonstrated in Fig. 5(b). This trend continues with the increased a until the SOC becomes strong, at which point an E_{g} of approximately ≈ 0.4 eV emerges, as depicted in Fig. 5(f). Notably, variations in buckling strength are expected to exert a significant influence on the magnitude of the SOC. It is important to acknowledge that these variations may also lead to subtle modifications in other parameters within the model.^{97} The gap opening by the SOC implies the topological nontriviality of the electronic state (anomalous QH state) of the honeycomb lattice,^{58,96,98} so the configuration can be effectively classified as a quantum spin Hall insulator, expanding the potential applications of this 2D system, particularly in electronic and spintronic devices.^{99}

Although this MOTB interpretation provides a qualitative understanding of the behavior of a dispersion band under tensile strain, its predictions align reasonably well with recent experimental observations.^{57,100} Therefore, it serves as a valuable foundation for more complex models of advanced physical phenomena (proximity effect and spin relaxation). Note that a perfect free-standing monosheet cannot exist in reality (intrinsic structural instability). Consequently, the ideal Dirac cone-shape observed in the monolayer, as predicted by the representation and supported by the current experiment,^{57} can only be a theoretical construct. Realistically, the cone will inevitably distort due to symmetry changes. In simpler terms, the cone-like levels demonstrated in the experiment are not true Dirac cones because they do not correspond to massless fermions.^{95}

In contrast to its monolayer counterpart, the bilayer buk-P exhibits greater synthetic accessibility. This, coupled with the presence of theoretically identified Dirac-like electronic features and a near-zero band gap (≈ 1 meV), positions bilayer buk-P as a promising candidate for the next generation of high-performance nanoelectronic devices. Notably, its E_{g} properties offer a potential solution to overcome the limitations of graphene, which lacks a native band gap. The intriguing behavior observed in bilayer buk-P raises the question of whether similar Dirac-like electronic features can be achieved in few-layer buk-P (e.g., three, four, or five sheets) with reduced strain requirements (flexible substrates). Exploring this possibility through future experimental studies could provide valuable insights into the manipulation of Dirac fermions and potentially lead to novel material properties.

We have conducted a comprehensive examination of the valence states of bilayer (monolayer) buk-P configurations within MOTB interpretations. Initially, a preliminary MOTB model was employed, considering only NN couplings. Subsequently, interactions up to the 5NN were incorporated, transforming the unstrained monolayer description from a qualitative to a quantitative interpretation, enabling more rigorous calculations. To validate the reliability of the approximation, the energy dispersion bands reported in previous GW-based calculations were used as a reference. The findings exhibited excellent agreement with the established approach. Incorporating distant neighbors beyond the 2NN intralayer and NN interlayer coupling is crucial for representing the energy dispersion of the configurations. Nonetheless, it has already been demonstrated the nonnegligible influence of these couplings in bilayer (monolayer) systems. Thus, the quantitative deviation of the 2NN intralayer and NN out-of-plane interaction MOTB calculations can be enhanced by incorporating 5NN in-plane and 2NN interlayer interactions. As anticipated, however, the hopping parameters diminish exponentially as the distance between the origin and the NN interacting sites increases. Note that there is always an underestimation when the transfer parameters decrease exponentially as the distance expands. Furthermore, the shift in the interlayer distance due to the stacking sequences is critical for determining the E_{g}. The behavior of the energy levels for all stacking forms is extremely connected to the coupling of the interlayer orbitals of each sheet. For instance, in the AB stacking mode, the Δ_{int} is smaller in contrast to the AA, and consequently, the coupling between the sheets is maximized. Thus, the influence of confinement is stronger in this stacking mode than in the AA. This effect directly impacts the evolution of E_{g} of the configurations along with the NN couplings.

The AA bilayer exhibits a zero-gap Dirac-like character (E_{g} ≈ 1 meV) and presents a distinguishable Dirac-like cone in the absence of a strong SOC, demonstrating an approximately linear conical dispersion near the CBM and VBM, centered around the Γ–K direction, with a v_{f} ≈ 0.12 × 10^{5} m s^{−1}, upon reaching a tensile strain of approximately ≈ 2.30%. The increased application of symmetrical tensile strain in bilayer buk-P disrupts some of the honeycomb lattice symmetry, leading to enhanced SOC. Furthermore, the tuning of inversion asymmetry at the sublattice level triggers the opening of a tiny E_{g} of approximately ≈ 0.27–0.4 eV. However, the dominance of the p_{x,y}-orbitals instead of their p_{z} counterparts within low energy levels and variations in the buckling strength are predicted to be significant factors in the emergence of a large intrinsic SOC. Note that the gap opening by the SOC implies the topological nontriviality of the electronic state (quantum anomalous Hall state) of the honeycomb lattice. In addition, the key factor governing the formation of Dirac-like cones is the geometrical phase shift from a buckled form to quasi-planar geometry. Thus, bilayer buk-P can be transformed from an indirect E_{g} configuration into a 2D Dirac-like system by applying tensile strain, an electrical field, doping, or covalent functionalization.

Finally, understanding the formation mechanism and electronic features of a Dirac-like cone offers valuable insights into the design principles of novel Dirac-like materials. In addition, our comprehensive method, combined with experimental findings and FP calculations, holds substantial potential for elucidating the inherent features of Dirac-like configurations beyond bilayer buk-P.

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## Footnote |

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d4cp01988a |

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