Discovery of an Ideal Quadratic Nodal Point in Two-Dimensional Binary Compound C₅P₁

Abstract

Recent years have witnessed a surge in research focus on topological states within two-dimensional materials, emerging as a pivotal complement to studies on three-dimensional systems. This work presents a comprehensive investigation into the exceptional topological characteristics of the C₅P₁ monolayera binary compound whose electronic structure and topological behavior have been systematically explored through computational analysis. The material exhibits a simple electronic configuration featuring merely two bands proximate to the Fermi energy, with their crossing phenomena giving rise to a distinct quadratic nodal point, a hallmark topological attribute. Detailed investigations into the band formation mechanisms and dispersion characteristics were undertaken to clarify the fundamental origins of the observed topological phenomena. Of particular significance are two distinct edge states emerging from the nodal point and extending continuously toward the nodal line, exhibiting pronounced spatial delocalization along the material’s interface. The C₅P₁ monolayer’s incorporation of lightweight elements contributes decisively to the exceptional stability of both its topological features and corresponding edge states upon inclusion of spin–orbit coupling effects. Furthermore, to assess the potential for practical application, the strain-dependent modulation of these topological bands was systematically probed, alongside a thorough characterization of the material’s subtle mechanical anisotropy. These analyses provide critical insights into how external perturbations modulate topological behavior, offering guidance for experimental synthesis and device integration. Taken together, the comprehensive analysis of this prototypical topological phase, along with the demonstrated intrinsic stability of the material, lays a robust foundation for forthcoming experimental verification. This study significantly advances the dynamic field of two-dimensional topological materials, opening new pathways for innovation in the design of quantum devices and deepening our understanding of topological phenomena.

Introduction

Topological materials have emerged as a groundbreaking paradigm in the realms of condensed matter physics and solid-state research, with their seminal discoveries − catalyzing an extraordinary surge of scholarly exploration and propelling profound progress in understanding topological phenomena over recent decades. The swift progression within this domain is largely driven by advancements in topological band theory, − which provides a comprehensive framework linking the intrinsic symmetries of crystalline materials to their topological properties and the governing principles of band topology. Since its conceptualization, this theoretical framework has found extensive application in investigating and characterizing the topological characteristics of diverse material systems through a synergistic combination of theoretical modeling and experimental validation. While early investigations were primarily focused on topological insulators, contemporary research efforts have progressively shifted toward metallic and semimetallic systems. − Concurrently with advancements in three-dimensional topological systems, two-dimensional (2D) materials have risen to prominence as a unique and captivating research frontier in recent years. − These atomically thin systems have garnered significant attention from the scientific community owing to their extraordinary physical phenomena and substantial technological potential. Two-dimensional materials offer transformative potential through their unique planar architectures, which significantly streamline technological integration by minimizing compatibility constraints and enabling sophisticated device engineering. While sharing fundamental topological characteristics with their three-dimensional counterparts, these atomically thin systems manifest a distinctive transformationwherein bulk surface states transition to confined edge statesthus presenting unprecedented opportunities for experimental exploration and fundamental scientific inquiry.

Within these systems, a wide array of topological states has been identified, which are generally classified into three main categories: − nodal points, − nodal lines, − and nodal surfaces. , The emergence of these topological properties stems from linear intersections in the electronic band structure, where energy bands converge to form distinctive geometrical patterns. Notably, isolated crossings manifest as nodal points, whereas extended intersections form nodal lines or surfaces. These band crossings play a pivotal role in defining the intrinsic topological properties of the material’s bulk electronic structure, while simultaneously giving rise to exotic surface states that mirror the Dirac cones characteristic of topological insulators. − Acting as essential indicators, such localized surface modes facilitate the characterization of topological metals and semimetals, thereby forging a crucial connection between conceptual frameworks and empirical observations. − In particular, the presence of Fermi arcs links isolated nodal points, whereas drumhead-like surface states encircle extended nodal lines, underscoring a fundamental interplay between the material’s interior band topology and its boundary manifestations. From a conceptual standpoint, nodal lines may be regarded as extended chains formed by interconnected nodal points, with the associated drumhead states arising from the coalescence of numerous Fermi arcs. Rooted in the symmetries of the crystal lattice and governed by topological principles, these protected boundary phenomena hold substantial potential for propelling innovations in quantum devices and advanced material systems.

With the ongoing evolution of topological physics, investigations into topological effects have progressed beyond conventional linear band-crossing models, extending into regimes characterized by higher-order dispersions, including quadratic and cubic variants. , In contrast to linear dispersions, these higher-order topological phases display distinctive attributes, such as varied pseudospin arrangements, anomalous charge transport properties, and unconventional responses to magnetic fields. − Moreover, they accommodate unconventional topological invariants and intricate manifold geometries, manifesting particularly robust topological characteristics within two-dimensional (2D) platforms. The ability to control and engineer the associated boundary modes holds profound promise for the design of advanced quantum technologies, thereby establishing 2D platforms as frontrunners in emerging technological paradigms. Although enthusiasm for higher-order topological frameworks has intensified, scholarly efforts have largely prioritized linear dispersion contexts, especially in three-dimensional architectures, leaving their 2D counterparts underexplored. This gap in the literature highlights an urgent imperative to identify and analyze novel candidates exhibiting higher-order band behaviors, particularly those with streamlined architectures amenable to experimental scrutiny. Advancing this domain is essential for revealing unprecedented physical principles and accelerating the creation of cutting-edge quantum systems.

Driven by these needs, we present the C₅P₁ monolayer, a 2D material with exceptional topological properties. The monolayer adopts a tetragonal crystallographic structure with a distinctive pentagonal lattice motif, exhibiting exceptional thermal, dynamic, and mechanical robustness, similar as other pentagonal monolayers, like penta-SiC₂, penta-SiH, penta-SnHF and penta-HpGeH/FpGeF. Featuring a streamlined electronic configuration with merely two isolated energy bands, the material displays metallic characteristics characterized by sharp band intersections, which give rise to compelling topological phenomena. Notably, a quadratic nodal point is identified, with its formation mechanisms and dispersion relationships rigorously examined through theoretical modeling. Employing a locally constructed tight-binding Hamiltonian, we further elucidate the associated edge states, which manifest as two distinct edge arcs emanating from the nodal point. Significantly, upon inclusion of spin–orbit coupling (SOC) effects, the key topological signatures persist with minimal alteration, underscoring the material’s inherent stability. To advance practical applicability, strain-induced modifications to the topological bands and the material’s subtle mechanical anisotropy are additionally characterized. These findings offer critical insights to inform future experimental endeavors and technological implementations, especially in the context of structural integration and quantum device engineering. Collectively, these findings highlight the exceptional potential of the C₅P₁ monolayer as an outstanding platform for investigating exotic topological properties. By harmonizing structural simplicity, inherent stability, and distinctive topological attributes, this material emerges as an exemplary platform for deepening insights into two-dimensional systems and unlocking the possibilities of innovative topological phenomena.

Computational Methods

Ab initio calculations were performed within the formalism of density functional theory (DFT), , utilizing the Vienna Ab Initio Simulation Package (VASP) , as the computational platform. The projector augmented wave (PAW) method , was employed to describe electron–ion interactions, with exchange–correlation effects approximated by the Perdew–Burke–Ernzerhof (PBE) functional within the generalized gradient approximation (GGA). To enhance the precision of electronic structure computations, the nonlocal hybrid Heyd-Scuseria-Ernzerhof (HSE06) functional − was additionally implemented. Following rigorous convergence testing, computational parameters were optimized to a 500 eV plane-wave cutoff energy and an 8 × 8 × 1 Monkhorst–Pack k-point grid to ensure numerical accuracy. Structural relaxations were performed until residual atomic forces were minimized below 1 × 10^–4 eV/Å, with self-consistent field cycles converging to an energy tolerance of 1 × 10^–6 eV. A 15 Å vacuum gap was introduced to mitigate artificial interlayer interactions, while long-range van der Waals forces were accounted for using the DFT-D3 correction scheme. , Thermodynamic stability was evaluated via ab initio molecular dynamics (AIMD) simulations, utilizing a 5 × 5 × 1 supercell and the Nosé-Hoover thermostat within the canonical NVT ensemble. − These simulations, conducted at discrete temperatures of 100, 200, and 300 K with time step of 1femtosecond and total duration of 3000 steps, provided critical insights into thermal robustness under dynamic conditions. The vibrational properties and structural stability were examined by calculating phonon dispersion spectra through the density functional perturbation theory (DFPT), , utilizing the computational capabilities of the Phonopy package. , The mechanical robustness of the system was quantified through the stress–strain methodology, , providing precise measures of elastic properties. For the topological characterization, maximally localized Wannier functions were generated employing the WANNIER90 package, , with surface state projections computed using the WANNIERTOOLS code. Data extraction and postprocessing workflows were streamlined using the VASPKIT package, ensuring efficient compilation and analysis of simulation results. Regarding the three-dimensional representation of the crystal structure and electronic band surface, the Vesta software and the Matplotlib library have been employed.

Results and Discussions

The two-dimensional binary compound C₅P₁ monolayer exhibits a tetragonal crystal structure belonging to the space group P4̅ (No. 81). As shown in Figure a, the primitive cell encompasses a total of six atoms, including one phosphorus (P) atom located at the 1b Wyckoff site and five carbon (C) atoms allocated to the 4h (four atoms) and 1d (one atom) Wyckoff positions. When observed from the overhead perspective, this atomic ensemble configures into a notable pentagonal motifemphasized by the purple shadingthat interlinks to produce a structural motif reminiscent of an origami windmill design. By comparison, the side-on projection unveils a stratified architecture akin to a sandwich arrangement in the atomic layering of the crystal lattice. After thorough geometric optimization, the equilibrated lattice constants were established as a = b = 3.892 Å, demonstrating strong agreement with findings documented in earlier literature. , The interatomic bond lengths were measured at 1.758 Å for P–C bonds and 1.598 Å for C–C bonds, while bond angle analysis revealed a C–P–C angle of 135.45°notably larger than the 122.68 °C–C–C angleindicating inherent asymmetry in the bonding geometry. Comprehensive crystallographic information, encompassing complete lattice parameters and the precise atomic positions, is presented in Supporting Information Table S1. The distinctive structural framework of the C₅P₁ monolayermarked by pentagonal motifs and anisotropic bonding patternshighlights its intricate geometric arrangement, which fundamentally underpins its remarkable physical and chemical behavior.

1.

Structural and stability characterization of the C₅P₁ monolayer. (a) Top and side views depicting the atomic configuration of the C₅P₁ binary monolayer, with the primitive unit cell outlined by a square boundary. The six-atom motif forms a distinctive pentagonal arrangement, emphasized by a purple-shaded region. (b) Time-dependent variations in the overall potential energy and the phonon dispersion profile (c) for the C₅P₁ monolayer. To probe thermal robustness, ab initio molecular dynamics (AIMD) computations were executed along a 3000 fs pathway at thermal conditions of 100, 200, and 300 K, incorporating a 5 × 5 × 1 supercell arrangement.

Following exhaustive optimization of the C₅P₁ monolayer binary compound’s structural architecture, rigorous assessment of its thermal and dynamic stability emerged as a critical next step. To evaluate thermal resilience, first-principles molecular dynamics (AIMD) simulations were executed using a 5 × 5 × 1 supercell configuration across temperature gradients of 100, 200, and 300 K. These simulations encompassed 3000 iterations with a 1 fs time step, yielding stability profiles illustrated in Figure b. The resultant energy fluctuation patterns demonstrate minimal deviation throughout the simulation period, with no appreciable structural degradation observed across the tested temperature rangevalidating the material’s exceptional thermal robustness even under near-ambient conditions (300 K). Supporting Information Figure S1 provides detailed structural snapshots at varying temperatures, further confirming that while atomic vibrational amplitudes increase marginally with thermal excitation, the crystalline lattice maintains its integrity and periodicity. Dynamic stability was interrogated through phonon dispersion calculations of the C₅P₁ monolayer. As illustrated in Figure c, the complete lack of imaginary frequency components throughout the Brillouin zone substantiates the material’s resilience to structural distortions, thereby affirming its dynamical robustness. The high-symmetry momentum trajectory utilized in these assessments was formulated in accordance with the substance’s crystallographic attributes, as further expounded upon in Supporting Information Figure S2. To further ensure the reliability of these results and to exclude potential artifacts arising from finite-size effects or sampling mesh choices, additional phonon dispersion calculations were conducted with varied supercell sizes and k-point meshes. As summarized in Figures S3 and S4, the phonon spectra remain virtually unchanged regardless of these parameters, and no imaginary frequencies were found under any tested condition. These complementary analyses collectively affirm the C₅P₁ monolayer’s robust stability under both thermal and dynamic perturbations, providing strong theoretical justification for experimental synthesis endeavors. The multifaceted validation of structural integrity not only substantiates its potential for practical implementation but also positions the C₅P₁ monolayer as a compelling candidate for advanced two-dimensional material research, offering promising avenues for technological innovation in diverse applications.

Leveraging the optimized crystalline framework of the C₅P₁ monolayer binary compound, electronic band structure computations were conducted, with results visualized in Figure . The Fermi energy was designated as the reference 0 eV. Considering the comparatively low atomic masses of the constituent elements, (SOC) effects were purposefully omitted from the preliminary analysis; this aspect will be thoroughly examined in subsequent investigations. For the computation of electronic band structures, both the (PBE) functional and the hybrid Heyd-Scuseria-Ernzerhof (HSE06) method were employed, with the resulting band structures from these two methodologies overlaid for comparative analysis in Figure . Electronic-structure calculations identify the C₅P₁ monolayer as metallic, evidenced by bands that intersect the Fermi energy. Two bands situated immediately around the Fermi level are cleanly separated from higher- and lower-lying states, and within this pair a pronounced crossing is observed at the Γ point. Notably, the overall dispersion and topology obtained with the HSE06 hybrid functional closely mirror those computed using the PBE approximation. This close correspondence indicates that PBE reliably captures the essential features of the C₅P₁ band structure. Given its substantially lower computational cost together with its faithful reproduction of the salient trends, PBE is adopted as the primary framework for the analyses that follow.

2.

Electronic band dispersion profiles for the binary C₅P₁ monolayer in proximity to the Fermi level, derived through calculations employing both PBE and HSE functionals. The lower panel provides a zoomed-in depiction of two distinct band sections aligned near the Fermi energy, emphasizing the subtle intersection patterns within the electronic spectrum, with particular focus on the quadratic nodal degeneracy located at the Γ symmetry point.

The symmetry of C₅P₁, defined in bulk by space group P4̅ (No. 81), reduces upon monolayer formation to the associated layer group P4̅ (No. 50). Under this layer-group symmetry, the electronic bands intersect at Γ to produce a quadratic nodal feature, with the associated little cogroup identified as S₄ and generators corresponding to the symmetry operation S_4z . In the basis of the irreducible representation {¹E,²E}, the generator and the time-reversal Τ can be represented by

$${\mathrm{S}}_{4\mathrm{z}}={\mathrm{i}\sigma }_z,\mathrm{{\rm T}}={{\sigma }_x}^{\mathrm{{\rm K}}}$$

where σ_i(i = x, y, z) are the Pauli matrices, and ^Κ denotes the complex conjugate. The symmetry constraints can be expressed as

$${\mathrm{S}}_{4z}{\mathrm{{\rm H}}}_{\mathrm{f}}^{\mathrm{e}\mathrm{f}}(k){\mathrm{S}}^{4z-1}={\mathrm{{\rm H}}}_{\mathrm{f}}^{\mathrm{e}\mathrm{f}}(-k_y,{\mathrm{k}}_x),\mathrm{{\rm T}}{\mathrm{{\rm H}}}_{\mathrm{f}}^{\mathrm{e}\mathrm{f}}(k){\mathrm{{\rm T}}}^{-1}={\mathrm{{\rm H}}}_{\mathrm{f}}^{\mathrm{e}\mathrm{f}}(-k_x,-k_y)$$

the effective Hamiltonian (up to the second order) reads

$${\mathrm{{\rm H}}}_{\mathrm{f}}^{\mathrm{e}\mathrm{f}}(k)=ϵ(k)+[(\alpha {k^2}_{+}+\beta k_{-}^2){\sigma }_{+}+\mathrm{H}.\mathrm{c}.]$$

where ϵ­(k) = ω₀ + ω₁(k _x + k _y ) with ω_i being real parameters, k _± = k _x ± ik _y , σ₊ = (σ _x + iσ _y )/2,and α, β are complex parameters. This effective model identifies the Γ-point crossing as a quadratic node, characterized by parabolic dispersion and quadratic splitting for small in-plane k. With this compact two-band basis defining the low-energy sector around the Fermi level, we extended the analysis across the entire two-dimensional Brillouin zone. The global evaluation allows precise testing of topological criteria and reveals the microscopic, symmetry-based mechanisms responsible for the observed topological responses. The three-dimensional band dispersion profiles within the k _x –k _y plane, presented in Figure , reveal that the two bands intersect exclusively at the central Γ point while maintaining distinct separation across other momentum regionsconsistent with the earlier layer group symmetry analysis. Notably, such a simplified yet well-defined band structure featuring a quadratic nodal point has been rarely reported in existing literature, with even fewer precedents documented in contemporary two-dimensional material systems. Furthermore, considering the dimensional distinctions between our system and three-dimensional systems, a Z-valued topological charge can be defined for the quadratic nodal point in two dimensions and it can be derived by computing a 1D winding number on a closed loop encircling the node as follow

$${\mathrm{{\rm N}}}_C=\frac{1}{4\pi \mathrm{i}}\oint \mathrm{T}\mathrm{r}{\sigma }_z{\mathrm{{\rm H}}}_{\mathrm{f}}^{\mathrm{e}\mathrm{f}}-1(k){\mathrm{\nabla }}_k{\mathrm{{\rm H}}}_{\mathrm{f}}^{\mathrm{e}\mathrm{f}}(k)·\mathrm{d}k$$

3.

Three-dimensional (3D) band structure dispersions of the C₅P₁ monolayer binary compound, computed along the high-symmetry paths X–Γ–X and M–Γ–M within the Brillouin zone. To emphasize topological features, only the two primary topological bands are visualized, highlighting their dispersive behavior across the momentum space.

Our calculation yields |Ν _C | = 2 for the present quadratic nodal point, which indicates there should be even number of topological edge state.

To gain deeper insight into the finite energy fluctuations surrounding the quadratic nodal point and enable intuitive visualization of diverse band dispersion behaviors, we conducted a detailed analysis of band spectra along multiple k-path segments intersecting the central Γ point within the k _x –k _y plane. The coordinate system defining these path segments is provided in Supporting Information Figure S2, with corresponding band dispersion results presented in Figure . At Γ, the quadratic band touching exhibits isotropic second-order dispersion, implying equal in-plane effective masses along all azimuthal directions. Away from the node, the dispersion range grows appreciably between the Γ–X and Γ–M lines, with Γ–M showing a more pronounced energy spread. Such behavior aligns with general expectations for topological band structures, where symmetry dictates the spatial distribution of nodal features but does not rigidly determine their energetic bandwidths. The crossing sits in immediate proximity to the Fermi energymost notably, the lower member of the pair cuts through the Fermi energyso the metallic response is dominated by carriers from these topological states. This constellation of properties underscores the material’s nontrivial character and its potential utility across diverse device contexts.

4.

Calculated local electronic band structures of the C₅P₁ monolayer binary compound are depicted along selected high-symmetry paths intersecting the quadratic nodal point located at the Γ point. The identical pair of bands previously visualized in the band surface analysis of Figure are reproduced herein, with consistent color coding preserved to enable direct comparative analysis of their dispersive behavior across momentum space.

In three-dimensional material systems, the manifestation of topological phases is intrinsically linked to the presence of nontrivial surface states, which transition to edge states in two-dimensional configurations. To investigate the topologically relevant bands in our study, we conducted a rigorous analysis of their orbital contributions, revealing that carbon p-orbitalsparticularly p _z orbitalsdominate the correlated topological two bands. Leveraging these orbital components, a maximally localized Wannier tight-binding Hamiltonian was formulated to enable detailed exploration of correlated topological edge states. Validation of this Hamiltonian model was achieved through comparative analysis of band structures derived from DFT calculations and Wannier fitting, with Supporting Information Figure S5 demonstrating perfect overlapconfirming the model’s efficacy. Employing this validated framework, we computed the edge states of the C₅P₁ monolayer binary compound, as presented in Figure , utilizing two projected edge paths: X̅–Γ̅–X̅ and M̅–Γ̅–M̅. Figure a illustrates the scenario without SOC, while Figure b incorporates SOC effects. Both visualizations reveal prominent edge states emanating from the Γ̅ nodal point and extending toward the Brillouin zone boundaries. Significantly, the quadratic dispersion characteristics give rise to two distinct edge states, consistent with the above Z-valued topological charge of 2 and departing from the conventional single edge state observed in linear dispersion systems. This extended spatial distribution, spanning from the central Γ̅ point to the zone edges, enhances prospects for experimental detection and practical implementation. Notably, the incorporation of SOC induces minimal perturbations to the edge states, preserving their structural integrity. This observation reinforces our earlier conclusion that SOC effects may be safely neglected in subsequent analyses. Compared to prior research, the robustness of these topological features under SOC conditionscoupled with the system’s clean, simplified topological configurationpresents compelling opportunities for further investigation and technological application. The salient features of the edge states significantly enhance their detectability in experiments and confer substantial flexibility for incorporation into applied settings.

5.

Projected edge band structures of monolayer C₅P₁ along the X̅–Γ̅–X̅ and M̅–Γ̅–M̅ high-symmetry directions. Panel (a) presents results obtained in the absence of SOC, whereas panel (b) includes SOC effects.

To facilitate future experimental exploration and inform potential technological applications, an exhaustive investigation into the mechanical properties of the C₅P₁ monolayer binary compound was undertaken. A comprehensive computational approach based on stress–strain analysis , has been applied to determine key mechanical properties of the C₅P₁ monolayer. The investigation yielded values for three fundamental elastic constantsC₁₁, C₁₂, and C₆₆alongside derived quantities such as shear modulus, Young’s modulus, and Poisson’s ratio. These results are systematically compiled in Supporting Information Table S2 for reference. To assess the mechanical integrity of the monolayer, we employed the established elastic stability criteria pertinent to tetragonal systems, requiring that C₁₁ > 0, C₆₆ > 0, and C₁₁ > |C₁₂| hold true. The data satisfy these conditions unequivocally, thereby affirming the C₅P₁ monolayer’s mechanical stability and robustness. This outcome strongly supports the practical feasibility of synthesizing the material and integrating it into device architectures. Beyond confirming stability, we evaluated the degree of mechanical anisotropy to elucidate directional dependence in elastic responses. The anisotropy analysisillustrated in Figure where radial scales are nonzero to accurately represent magnitude variationsindicates that the C₅P₁ monolayer exhibits moderate anisotropic behavior. Quantitatively, the anisotropy indices calculated are 1.089 for Young’s modulus and 1.158 for the shear modulus, reflecting subtle directional variation. Notably, the maximum Young’s modulus aligns with the crystallographic (110) direction, whereas the shear modulus attains its highest value along the (100) axis. These findings provide critical insights into how the material’s mechanical properties vary with orientation, underscoring its versatility for applications that require tailored mechanical performance or precise directional control, such as heterostructure assembly or strain engineering.

6.

Theoretically calculated angular-dependent characteristics of Young’s modulus and shear modulus for the monolayer binary compound C₅P₁. To accommodate the distinct magnitude ranges of the two mechanical parameters, separate scaling intervals are implemented, with these calibration ranges explicitly labeled on the left vertical axis for quantitative clarity.

To address critical challenges in two-dimensional material applicationswhere heterojunction integration or substrate-supported fabrication often encounters lattice mismatch-induced strain in both in-plane and out-of-plane directionsthis study investigates how such strain modulates physical properties, particularly topological electronic characteristics. Strain-induced lattice distortions are known to alter nodal structure winding configurations and band crossing conditions, making their systematic analysis essential for material design. This study investigates how uniformly applied strains, both within the plane and perpendicular to itspanning from −10% compressive to +10% tensileimpact the topological electronic characteristics of the C₅P₁ monolayer binary compound. Such large strain range is purposefully selected to delineate the theoretical limits of structural response rather than immediately realizable experimental conditions. This broad scope provides essential insights into the strain-dependent evolution of the electronic structure and informs future experimental and theoretical efforts. The findings, illustrated in Figure , specifically emphasize the behavior of the two bands integral to the system’s topological features. The quadratic nodal point located at the Γ high-symmetry position remains intact throughout the entire strain interval, and this is attributed to the preservation of lattice symmetry under both compressive and tensile conditions. Additionally, a comparative evaluation of the band dispersions and associated energy shifts reveals that in-plane strain exerts a more pronounced influence on the electronic structure than out-of-plane deformation. This disparity arises primarily from the material’s atomic-scale thickness, which constrains vertical lattice distortion even under equivalent strain magnitudes, thereby mitigating out-of-plane electronic perturbations. Importantly, under +10% in-plane tensile strain, the quadratic nodal point undergoes a favorable energy shift toward the Fermi energy level, unlocking promising avenues for fundamental research and technological exploitation. Such strain-tunable topological stability distinguishes the C₅P₁ monolayer as a versatile platform for next-generation electronic applications. Moreover, our analysis extends to the edge states under identical strain conditions, as detailed in Figures S6 and S7. The continued presence of prominent edge states even under substantial strain indicates that the topological properties of the C₅P₁ monolayer are well maintained, further underscoring its potential for robust, strain-engineered topological devices.

7.

Electronic band structure of the C₅P₁ monolayer binary compound was systematically calculated under the influence of uniform strain applied in two configurations: (a) within the plane of the material and (b) perpendicular to it. The magnitude of each strain condition is clearly indicated by color-coded labels, enabling a precise quantitative analysis of how applied mechanical deformation modulates the electronic properties.

To advance the mechanistic understanding of strain effects, supplementary mechanical characterizations were performed under the aforementioned deformation regimens, with outcomes visualized in Figure . Owing to the material’s anisotropic nature, Hill averaging was implemented to enable consistent cross-directional comparisons. Data analysis reveals a monotonic reduction in both Young’s modulus and shear modulus with increasing tensile strain magnitude, under both in-plane and out-of-plane loading conditions. Consistent with earlier findings, in-plane strain induced more pronounced mechanical perturbations relative to out-of-plane deformation, aligning with the greater lattice distortion observed in planar configurations. Comprehensive mechanical parameters under specific strain states are detailed in Supporting Information Table S3. Critical stability assessments confirmed the C₅P₁ monolayer retains robust mechanical integrity across the full −10% to +10% strain spectrum, with all tetragonal elastic stability criteria rigorously satisfied throughout the deformation range. Collectively, these results not only validate the exceptional mechanical resilience of the C₅P₁ monolayer under extreme strain conditions but also establish a quantitative framework to guide experimental validation of its topological electronic properties, bridging computational predictions with future laboratory investigations.

8.

Variations in the averaged Young’s modulus and shear modulus of the C₅P₁ monolayer binary compound under uniform in-plane and out-of-plane strain conditions. The equilibrium state has been set to the base point with zero variation and directional stress perturbations across different loading strains have been overlapped.

Conclusions

This research presents a comprehensive computational study of the C₅P₁ monolayer binary compound, employing first-principles calculations to systematically investigate its structural and topological properties. The material exhibits a distinctive atomic configuration, wherein the six atoms per formula unit interconnect to form a pentagonal pattern. Leveraging this unique configuration, we conducted a multifaceted stability assessment encompassing thermal, dynamic, and mechanical criteria. The comprehensive analysis underscores the exceptional stability of the C₅P₁ monolayer across multiple evaluated parameters, substantiating its suitability for both experimental realization and potential technological deployment. Structurally robust, this material exhibits a metallic electronic band structure characterized by the dominance of just two bands in close proximity to the Fermi energy, a simplified band topology consistently confirmed through calculations employing both the PBE functional and the hybrid HSE06 method. This concise electronic landscape facilitates prominent topological phenomena, notably the emergence of a quadratic nodal point precisely situated at the Γ point within the Brillouin zone. To elucidate the fundamental origins of this distinct topological attribute, detailed three-dimensional band structure analyses were undertaken, which identified the p _z orbitals of carbon atoms as the principal contributors underpinning the nodal behavior. To deepen the investigation of its topological characteristics, a low-energy tight-binding Hamiltonian was formulated, grounded in dominant orbital interactions. Utilizing this model, edge states were computed, revealing two distinct modes emerging from the quadratic nodal point. These states propagate toward the Brillouin zone boundary, forming spatially expansive distributionsfeatures that enhance experimental detectability and device integration potential. Importantly, incorporating SOC effects induces only minimal modifications to both the electronic band structure and the associated topological features. This inherent robustness can be attributed to the presence of low atomic mass elements within the material, which significantly suppress SOC-related perturbations. To further assess its suitability for practical use, the material’s behavior under mechanical strainboth in-plane and out-of-planewas systematically evaluated. Even under strain magnitudes reaching ±10%, the electronic band topology remains largely unaltered, with the quadratic nodal point maintaining its distinctive character. Concurrently, mechanical property characterization reveals mild anisotropy, with elastic constants varying moderately across crystallographic directionsattributes that enhance engineering adaptability, particularly for precision heterojunction designs and structural integration. Collectively, these properties demonstrate that the C₅P₁ monolayer couples a streamlined and resilient topological electronic framework with a clean band structure, positioning it as a promising candidate for both experimental validation and future technological deployment. Its verified stability across thermal, dynamical, and mechanical domains not only affirms its practical viability but also provides foundational guidance for successful synthesis efforts. This work contributes to advancing the design principles of 2D topological materials, paving the way for next-generation quantum device development.

The data is available throughout the manuscript and supporting files.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acsomega.5c09374.

• Detailed lattice parameters and atomic positions of the C₅P₁ monolayer binary compound in the optimized structural configuration. The final atomic configurations after AIMD simulations across the temperature range of 100, 200, and 300 K. Comparative analysis of the band structure from DFT calculations and Wannier function fittings. Computed elastic constants (C₁₁, C₁₂, and C₆₆), Young’s modulus (E), shear modulus (G), and Poisson’s ratio (ν) of the C₅P₁ monolayer and under strain conditions (PDF)

The author declares no competing financial interest.