Design of polar XC₃ (X = P, As, Sb, Bi) monolayers with coupled bandgap, polarization, and optical responses

Abstract

Bandgap engineering and polarization control in graphene-based systems are crucial for developing high-performance two-dimensional (2D) semiconductors. However, simultaneously achieving a sizable bandgap, intrinsic polarity, and strong light–matter interaction remains challenging. Here, we propose a new class of carbon-based polar semiconductors, monolayer XC₃ (X = P, As, Sb, Bi), designed by substituting group-V elements into graphene to break its sublattice symmetry. This symmetry breaking not only opens wide bandgaps (2.23–3.11 eV) but also induces spontaneous out-of-plane (OOP) electric polarization (−3.1–8.1 pC m⁻¹) and an internal electric field, stabilizing polar phases and facilitating photocarrier separation. The resulting electronic structures exhibit a distinctive Mexican-hat-shaped valence band and strong band nesting, leading to intense visible-to-near-ultraviolet optical absorption (>10⁵ cm⁻¹). Moreover, XC₃ monolayers possess large and anisotropic carrier mobilities and exhibit band-edge alignments suitable for photocatalytic water splitting across a wide pH range (0–10). These findings establish a general route to 2D polar semiconductors that integrate coupled electronic, optical, and catalytic functionalities, offering a promising platform for graphene-derived optoelectronic and energy applications.

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1. Introduction

Graphene, a 2D material composed of sp²-hybridized carbon atoms, exhibits exceptional electrical, mechanical, and thermal properties. However, its intrinsic zero bandgap severely restricts its applications in semiconductor and optoelectronic devices.^1–3 In contrast, 2D pnictogen monolayers—such as phosphorene, arsenene, antimonene, and bismuthene-consisting of sp³-hybridized atoms, possess intrinsic semiconducting behavior with tunable band gaps and high on/off ratios.^4–10 Nevertheless, the high chemical reactivity of phosphorene leads to poor environmental stability, which greatly limits its practical device applications.^11–13

2D carbon–pnictogen compounds have attracted increasing attention due to their structural tunability and versatile electronic properties, enabled by the combination of robust sp² C–C bonding and the flexible coordination chemistry of group-V elements.^14,15 First-principles studies have revealed diverse structural phases with intriguing chemical bonding and band dispersions, demonstrating the promise of carbon–pnictogen hybrids for applications in nanoelectronics and optoelectronics.^16,17 In particular, phosphorus carbide (CP) monolayers have been extensively explored. Wang et al. predicted three stable CP allotropes (α-, β-, and γ-CP) exhibiting high carrier mobilities and Dirac semimetallic states,¹⁶ while Guan et al. reported additional polymorphs spanning metallic to semiconducting behaviors,¹⁸ illustrating the structural richness of P–C bonding networks.

Theoretical progress has been supported by experimental validation. Tan et al. synthesized few-layer α-CP via carbon doping of black phosphorus,¹⁹ and the measured hole mobility (1995 cm² V⁻¹ s⁻¹) agrees well with theoretical predictions.²⁰ This breakthrough confirms the feasibility of stabilizing 2D carbon–pnictogen frameworks experimentally. Moreover, other carbon–pnictogen monolayers, including AsC, C₃P, C₆P, CP₂, and CP₃, have been theoretically proposed^21–23 and exhibit useful functionalities such as high carrier mobility, strong visible-light absorption, and appropriate band edges for photocatalytic water splitting.¹³ Chemical substitution has further broadened their tunability in stability and band structure engineering.^24,25

Despite these advances, most reported carbon–pnictogen monolayers possess centrosymmetric crystal structures,²⁶ which fundamentally limits their multifunctionality. Inversion symmetry suppresses spin–orbit-related effects,²⁷ prohibits electric spontaneous polarization,^28,29 and eliminates second-order nonlinear optical responses,³⁰ restricting their potential in spintronics, ferroelectric devices, and nonlinear photonics. Therefore, designing non-centrosymmetric carbon–pnictogen monolayers with intrinsic polarity and tunable symmetry-breaking effects remains an open challenge.

In this work we introduce an intrinsically polar family of carbon–pnictogen monolayers, XC₃ (X = P, As, Sb, Bi), obtained by substitutional doping of graphene with group-V atoms. First-principles calculations show that every XC₃ sheet hosts spontaneous OOP electric polarization (−3.1–8.1 pC m⁻¹) arising from buckled polar X–C bonds. Ab initio molecular-dynamics runs at 300 K and vibrational analysis confirm thermal and dynamic stability. The materials are indirect-gap semiconductors (2.13–3.11 eV) that absorb strongly across the visible range. Their Mexican-hat-shaped valence bands induce pronounced band nesting, resulting in optical absorption coefficients exceeding 10⁵ cm⁻¹ near the band edge. Importantly, the band edges of every XC₃ monolayer straddle the water-splitting redox levels throughout pH 0–10, enabling concurrent H₂ and O₂ evolution. XC₃ thus defines a hitherto overlooked family of polar 2D semiconductors that unite robustness, intrinsic polarity, and strong visible-light response, promising for next-generation optoelectronic and energy-conversion devices.

2. Computational methods

First-principles calculations were conducted using density functional theory (DFT) as implemented in the Vienna Ab initio Simulation Package (VASP).^31–34 The generalized gradient approximation (GGA) in the Perdew–Burke–Ernzerhof (PBE) formulation was employed to account for exchange–correlation interactions.³⁵ For more precise electronic structure calculations, the Heyd–Scuseria–Ernzerhof (HSE06) screened hybrid functional was utilized.³⁶ An energy cutoff of 500 eV was applied, and full atomic relaxation was performed, with convergence criteria set to 10⁻⁵ eV for total energy and 10⁻³ eV Å⁻¹ for atomic forces.³⁷ A vacuum spacing of 20 Å was included along the OOP direction to prevent interlayer interactions. Brillouin zone (BZ) sampling was performed using a 12 × 12 × 1 Monkhorst–Pack grid.

3. Geometric structure and stability

As shown in Fig. 1(a and b), monolayer XC₃ (X = P, As, Sb, Bi) can be constructed by symmetrically incorporating two X atoms into a 2 × 2 graphene supercell. Each unit cell contains eight atoms: six carbon atoms forming an inner planar hexagonal lattice and two pnictogen atoms occupying the outer hexagonal sites. This high-symmetry substitution preserves the overall crystalline motif of graphene while introducing chemical modulation through the dopant atoms. A structurally analogous compound, NC₃, has already been experimentally synthesized,³⁸ indicating that XC₃ can be regarded as a family of NC₃ derivatives obtained by replacing nitrogen with heavier group-V elements.³⁹

Fig. 1 (a) Top and side views of the α-XC₃. (b) Corresponding views of the β-XC₃. (c) The spontaneous electric polarization (P_s) and buckling height (h) of α-XC₃. (d) Energy difference (ΔE) between the α and β phases.

Although XC₃ shares a similar lattice topology with NC₃, its local bonding environment is distinctly different. The electron localization function (ELF) analysis in Fig. S3 in the SI⁴⁰ reveals that while NC₃ maintains a planar sp² hybridization, the heavier pnictogen atoms in XC₃ exhibit a pronounced tendency toward sp³-like hybridization with neighboring carbon atoms. This hybridization character breaks planarity and drives a spontaneous OOP buckling of the lattice, forming a polar trigonal–pyramidal configuration around each pnictogen atom. Interestingly, the coexistence of two such X-centered pyramidal units per unit cell allows two possible arrangements, giving rise to the α- and β-phases.

As depicted in Fig. 1(a), the α-phase displaces both pnictogen atoms to the same side of the carbon sheet, generating a globally polar structure (space group P6mm) with a buckling height h. The trigonal–pyramidal units tilt cooperatively, producing a net OOP P_s. Both the h (0.67–0.94 Å) and P_s (−3.1–8.1 pC m⁻¹) increase monotonically with the pnictogen atomic number with both the h and P_s increase monotonically with the pnictogen atomic number [Fig. 1(c)]; notably, PC₃ points host opposite polarization direction to the other three monolayers. In contrast, the β-phase in Fig. 1(b) shifts the two pnictogen atoms in opposite directions, yielding a centrosymmetric, non-polar lattice with space group P3̄ and point group S₆. Comparing the two phases thus demonstrates that the polarity of XC₃ is dictated solely by the sense of pnictogen displacement.

To determine the thermodynamically stable phase, we performed total-energy calculations for both α- and β-phases. As summarized in Fig. 1(d), the energy difference, defined as ΔE = E_β − E_α, is negative for all four XC₃ compounds, indicating that the α-phase is energetically favored and thus the ground state. Therefore, in the following sections, we focus on the polar α-phase of XC₃, which represents the most stable configuration with OOP P_s. Furthermore, the mechanical, dynamical, and thermodynamic stability of the α-phase is verified by phonon dispersion, abinitio molecular dynamics (AIMD) simulations, and the calculations of binding and formation energies, as shown in Fig. S1 and S2.

4. Electronic structures

HSE06 band structures and densities of states (DOS) for four carbon–pnictogen monolayers are displayed in Fig. 2. PC₃ and AsC₃ [Fig. 2(a and b)] are indirect-gap semiconductors with the valence-band maximum (VBM) at M and the conduction-band minimum (CBM) at Γ, and the gap increases from 2.13 eV (PC₃) to 3.05 eV (AsC₃). For SbC₃ and BiC₃ [Fig. 2(c and d)] the CBM moves to K while the VBM lies along the Γ–K path, and the gap narrows to 3.11 eV (SbC₃) and 2.87 eV (BiC₃).

Fig. 2 (a)–(d) Band structures density of states (DOS) of the XC₃ monolayers. (e) 3D view of the two LCB below the Fermi level in PC₃. (f) 3D view of the two HVB above the Fermi level in PC₃.

Projected DOS reveals the orbital character behind this evolution. For PC₃ and AsC₃ [Fig. 2(a and b)] the highest valence band (HVB) is built from C and X p_z states, whereas the lowest conduction band (LCB) is dominated by C p_z. Moving to SbC₃ [Fig. 2(c)] the HVB acquires strong Sb p_x,y weight and the LCB mixes Sb p_z and C p states. In BiC₃ [Fig. 2(d)] the HVB is mostly Bi p_z and the LCB is composed of Bi p_x,y plus C p. Thus, with increasing atomic number of X the p_x,y contribution to the conduction states grows at the expense of p_z, underscoring the strengthening of in-plane orbital hybridization.

Most importantly, all XC₃ display unique and distinctive band structure features: (i) the two HVBs cross at the K point, forming Dirac points akin to those observed in pristine graphene. However, due to the substitution of two X atoms in place of C atoms in XC₃, two additional electrons are introduced, leading to fully occupied Dirac bands, thereby inducing semiconducting behavior. Thus, the band gap opening in XC₃, which maintains the system's intrinsic high symmetry, fundamentally arises from alterations in electron filling. This methodology diverges from conventional band engineering techniques in graphene, which typically disrupt the system's symmetry via adsorption or the imposition of external fields. (ii) Of particular interest is the observation that HVBs of PC₃ exhibit a characteristic “Mexican hat” profile, as shown in Fig. 2(e and f). This feature is most pronounced in the band structures of PC₃ and AsC₃, while it becomes progressively less distinct in SbC₃ and BiC₃. This unique band structure significantly enhances the DOS of the HVB near the band edges. The conduction bands of PC₃ and AsC₃ also feature Dirac-like structures, as indicated by the arrows in Fig. 2(a and b), with the LCB presenting a saddle point at the M point, contributing to a pronounced DOS peak.

To further clarify the relationship between crystal structure and electronic properties, the main structural parameters and electronic characteristics—including the lattice constant (a), buckling height (h), C–X bond length (d_C–X), indirect band gap (E_g), and the bandwidth of the highest valence band (ΔE_HVB)—are summarized in Table S1 of the SI. These results provide a quantitative comparison of how the electronic structures evolve with the underlying structural parameters. Overall, except for BiC₃, the XC₃ compounds exhibit clear systematic trends with increasing atomic size of X. From PC₃ to SbC₃, the lattice constant, buckling height, and C–X bond length gradually increase. Correspondingly, the band gap increases, while the bandwidth of the HVB decreases. BiC₃ shows a slight deviation from this trend, which can be attributed to its more complex orbital hybridization and modified electronic interactions. These results reveal a clear structure–property correlation and help elucidate the structural origin of the characteristic electronic structures in the XC₃ family.

5. Optical property

Materials possessing unconventional electronic band features, such as Van Hove singularities or Mexican-hat-like dispersions, have been shown to strongly affect charge transport and optical responses, thus offering potential for next-generation device applications. In this section, we examine the optical properties of XC₃ compounds as intrinsic semiconductors and show that their characteristic band structures play a decisive role in shaping the light absorption spectrum.

The optical absorption coefficient α(ω) was obtained from the complex dielectric function calculated using first-principles methods.^41,42 Fig. 3(a–d) present the optical absorption spectra of XC₃ compounds evaluated with the HSE06 functional. For comparison, we also computed the absorption spectra of two widely studied 2D semiconductors, monolayer MoS₂ and black phosphorene.⁴³

Fig. 3 (a)–(d) Optical absorption spectra of XC₃ monolayers. For comparison, the absorption coefficients of monolayer MoS₂ and black phosphorus (BP) are also shown. The arrows mark the first band-edge absorption peak. (e) Enlarged view of the PC₃ band structure near the Fermi level, where the shaded pink region highlights the band nesting feature. (f) Side view of the 3D distribution of the local band gap function for PC₃. (g) Transition dipole moment (TDM) between the highest valence band (HVB) and the lowest conduction band (LCB) of PC₃ (left axis), plotted together with the corresponding local band gap (right axis). (h) Partial joint density of states (JDOS) between the HVB and LCB of PC₃. (i) Top view of the 3D distribution of the local band gap function for PC₃.

All XC₃ exhibit pronounced band-edge absorption peaks exceeding 10⁵ cm⁻¹, which are stronger than those of MoS₂ and black phosphorene, as indicated by the arrows in Fig. 3(a–d). Among them, PC₃ shows the highest absorption intensity in the visible region, with its first band-edge peak located at 3.45 eV and reaching 3.5 × 10⁵ cm⁻¹. From PC₃ to AsC₃, the peak absorption intensities slightly decrease and the peak positions exhibit a systematic blue shift, moving from the blue-violet region (3.45 eV) to the near-ultraviolet region (4.17 eV), consistent with the increase in band gap. Conversely, from SbC₃ to BiC₃, a red-shift trend is observed with decreasing band gap.

To elucidate the mechanisms underlying the ultrahigh optical absorption in the XC₃ series, we analyze PC₃ as a representative example, focusing on its unique electronic band structure and interband transition probability . For semiconductors, the absorption coefficient for direct transitions is directly related to the transition probability, which can be approximated using Fermi's golden rule.⁴⁴ This rule gives the probability per unit time that an electron undergoes a transition from an initial state |v〉 in the valence band to a final state |c〉 in the conduction band at the vector in the Brillouin zone:

where and denote the energies of the conduction and valence bands at , respectively, and H′ is the perturbative Hamiltonian describing the coupling between the electromagnetic field and the electronic states. The first term represents the transition dipole moment (TDM) between the initial and final states, denoted as M_if,⁴⁵ while the second term corresponds to the joint density of states (JDOS), J_cv(ℏω):Here, denotes the local band-gap function that links the conduction and valence bands. This formulation captures the JDOS by integrating over isoenergy surfaces in momentum space. Thus, according to Fermi's golden rule, the optical absorption coefficient on a microscopic level is jointly determined by both the TDM and the JDOS.

As illustrated in Fig. 3(f and i), the 3D distribution of E_cv exhibits a nearly flat dispersion near the M point (highlighted in green), implying that . This behavior arises from the similar dispersions of the conduction and valence bands along the Γ–M direction, where , resulting in a pronounced band-nesting effect in the energy range of 3.40–3.47 eV, as indicated by the pink region in Fig. 3(e). Analogous to the divergent density of states (DOS) induced by flat electronic bands, the flatness of E_cv in the nesting region gives rise to a large JDOS contribution. In addition, the M point (3.42 eV) serves as a saddle point in the band structure, further enhancing the JDOS. This is confirmed by our DFT calculations, which reveal a pronounced JDOS peak at approximately 3.45 eV, as shown in Fig. 3(h). Therefore, the strong absorption peak originates from the combined effects of band nesting and the saddle-point singularity in PC₃.

In Fig. 3(g), the TDM between the HVB and LCB, along with the local bandgap distribution, are presented along the entire high-symmetry path near the Fermi level for PC₃. A notable finding is the presence of significantly high allowable transition probabilities near the M point. Along the M–Γ direction, the transition probability gradually decreases, reaching zero at the Γ point, where transitions are forbidden. Conversely, along the M–K direction, the probability first increases and then sharply decreases, reaching a maximum near the M point. Therefore, we infer that it is the combination of TDM and JDOS contributes to the prominent band-edge absorption, as demonstrated in Fig. 3(a). A similar analysis of AsC₃ is provided in the SI.

6. Carrier mobility

The high light absorption coefficient ensures that the 2D polar semiconductor XC₃ can effectively capture photons and generate a large number of photogenerated carriers. Whether these carriers can be efficiently collected and utilized also depends on their mobility within the material. Based on the deformation potential theory, the carrier mobility (µ_x,y) can be expressed as:⁴⁶where e is the elementary charge, ℏ is the reduced Planck constant, k_B is the Boltzmann constant, T is the room temperature, is the effective mass, C_x,y is the elastic modulus and E_x,yⁱ is the deformation potential constant for the ith band. The subscripts x and y directions correspond respectively to the armchair and zigzag transport directions.

Table 1 summarizes the relevant parameters, and the carrier mobility values for the XC₃ were derived using eqn (3). Unlike the isotropic and symmetric ultrahigh carrier mobility observed in graphene for both electron and hole, the XC₃ family display a wide range of carrier mobility, spanning from 10 to 10⁵ cm² V⁻¹ s⁻¹. More importantly, they exhibit pronounced anisotropy and significant asymmetry between electron and hole.

Table 1 The carrier mobility with relevant parameters for XC₃ materials, including C_x,y (J m⁻²), (m₀), E_x,y (eV), and µ_x,y (10³ cm² V⁻¹ s⁻¹). The m₀ is the rest mass of electron. The VA refers to different group VA elements and e/h stands for electron/hole carrier

VA	e/h	Cx	Cy			Ex	Ey	µx	µy
PC3	e	182.78	182.75	1.163	1.156	1.210	1.030	1.97	2.74
PC3	h	182.78	182.75	0.492	1.665	0.290	1.380	104.12	1.36
AsC3	e	142.95	142.19	2.238	2.012	1.011	0.970	0.63	0.76
AsC3	h	142.95	142.19	0.510	0.864	1.010	2.914	8.64	0.62
SbC3	e	106.63	106.61	0.393	0.473	3.010	2.310	1.48	2.09
SbC3	h	106.63	106.61	0.661	1.158	1.730	5.124	1.31	0.09
BiC3	e	86.28	86.30	0.341	0.341	6.370	6.430	0.39	0.38
BiC3	h	86.28	86.30	0.513	0.878	0.850	2.850	14.53	0.39

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Taking PC₃ as an example, the hole mobility exhibits a strong directional dependence: it reaches 1.04 × 10⁵ cm² V⁻¹ s⁻¹ along the x direction, whereas along the y direction it is only 1.4 × 10³ cm² V⁻¹ s⁻¹, differing by nearly two orders of magnitude. This pronounced anisotropy primarily originates from the directional variation of the carrier effective mass and deformation potential, as summarized in Table 2. In contrast, the electron mobility of PC₃ is relatively isotropic, with comparable values of 1.97 × 10³ and 2.74 × 10³ cm² V⁻¹ s⁻¹ along the x and y directions, respectively. Furthermore, comparison of electron and hole mobilities shows that along the x direction, holes (1.04 × 10⁵ cm² V⁻¹ s⁻¹) have significantly higher mobility than electrons (1.97 × 10³ cm² V⁻¹ s⁻¹), whereas the opposite trend occurs along the y direction, where electrons (2.74 × 10³ cm² V⁻¹ s⁻¹) are more mobile than holes (1.36 × 10³ cm² V⁻¹ s⁻¹). The other three XC₃ compounds show similar transport behavior to PC₃. Such pronounced anisotropy and electron–hole asymmetry in carrier mobility are beneficial for the spatial separation of photogenerated carriers, suggesting that these materials hold strong potential for high-performance optoelectronic applications.

Table 2 The bandgap E_g (eV), first band-edge absorption peak E_abs (eV), maximum carrier mobility µ_max (10³ cm² V⁻¹ s⁻¹), spontaneous electric polarization P_s (pC m⁻¹), intrinsic electric field E_in (V Å⁻¹), electrostatic potential difference between the two surfaces Δϕ (V) of XC₃

XC3	Eg	Eabs	µmax	Ps	Ein	Δϕ
PC3	2.13	3.45	104.12	−3.1	−0.52	−0.35
AsC3	3.05	4.17	8.64	1.3	0.19	0.15
SbC3	3.11	3.59	2.09	3.9	0.49	0.44
BiC3	2.87	3.61	14.53	8.1	1.04	1.14

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7. Photocatalytic water splitting properties

In photocatalytic water splitting, the built-in electric field in 2D polar materials can spatially separate photogenerated electrons and holes, thereby suppressing their recombination. Meanwhile, the field-induced band bending enables the hydrogen evolution reaction (HER) and oxygen evolution reaction (OER) to occur spontaneously at different active sites. Specifically, HER predominantly occurs at the CBM of the electron-rich surface, while OER occurs at the VBM of the hole-rich surface.^47–49

To further clarify the relationship between the intrinsic polarization and carrier separation capability, we quantitatively evaluated P_s, the intrinsic electric field E_in, and the electrostatic potential difference between the two surfaces Δϕ. The calculated results are summarized in Table 2 and Fig. S5. It can be clearly seen that a larger polarization strength leads to a stronger built-in electric field and a larger surface potential difference. Such an enhanced internal electric field promotes the spatial separation of photogenerated carriers by driving electrons and holes toward opposite surfaces, thereby improving the carrier separation efficiency.

For XC₃ materials, the carbon atomic layer and the X atomic layer are defined as the upper and lower surfaces, respectively. Notably, PC₃ exhibits an opposite surface polarity compared with the other three compounds (AsC₃, SbC₃, and BiC₃). As shown in Fig. 4, the calculated VBM and CBM energy levels of all XC₃ compounds straddle the redox potentials of water over a wide pH range (0–10), indicating that they are thermodynamically capable of overall photocatalytic water splitting. Under neutral conditions (pH = 7), the electron reduction potential (U_e) follows the order BiC₃ (1.53 eV) > SbC₃ (1.06 eV) > AsC₃ (0.87 eV) > PC₃ (0.43 eV), suggesting that BiC₃ provides the strongest driving force for HER. In contrast, the hole oxidation potential (U_h) follows the order SbC₃ (1.21 eV) > BiC₃ (1.20 eV) > AsC₃ (1.07 eV) > PC₃ (0.81 eV), implying that SbC₃ is favorable for OER.

Fig. 4 Electronic band edge alignment of XC₃ with respect to the water redox potentials. The hydrogen and oxygen evolution potentials are shown as dashed lines.

The XC₃ materials possess several advantages as photocatalysts for water splitting. Their high optical absorption coefficients, reaching up to 10⁵ cm⁻¹, enable efficient light harvesting in the ultraviolet-visible region, while their high carrier mobilities promote rapid transport of photogenerated carriers. Moreover, the intrinsic polarization generates a built-in electric field that effectively drives spatial charge separation and suppresses electron–hole recombination. These synergistic properties demonstrate the strong potential of XC₃ materials for efficient photocatalytic water splitting.

8. Discussion and conclusion

In summary, we have theoretically proposed a new family of polar semiconductors, XC₃ (X = P, As, Sb, Bi), formed by symmetric substitution of group-V elements into graphene. As summarized in Table 2, these monolayers exhibit excellent structural stability and outstanding optoelectronic performance, characterized by strong visible-to-near-UV absorption coefficients exceeding 10⁵ cm⁻¹ near the band edges. The enhanced optical absorption originates from pronounced band nesting and large transition dipole moments. Moreover, the spontaneous polar structure induces an intrinsic OOP electric field, while anisotropic carrier mobilities promote efficient separation and transport of photogenerated carriers. Combined with their favorable band-edge alignment for photocatalytic reactions, XC₃ monolayers hold great promise for solar-driven energy conversion. This work not only deepens the understanding of bandgap and polarity engineering in graphene-derived systems but also establishes XC₃ as a versatile platform for next-generation optoelectronic and photocatalytic applications.

Conflicts of interest

There are no conflicts of interest to declare.

Data availability

The data that support the findings of this study are available from the corresponding author upon reasonable request. All key results are reproducible based on the “Computational methods” in the main text.

Supplementary information (SI): structural stability analyses of XC₃, optical properties of AsC₃, electrostatic potential differences of XC₃, and the electronic band structures and optical absorption spectra of SbC₃ and BiC₃ with spin–orbit coupling (SOC) taken into account. See DOI: https://doi.org/10.1039/d6ra00272b.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (NSFC, Grant No. 12204330). Dr Botao Fu also thanks the Sichuan Normal University for financial support (Grant No. 341829001). The numerical computations were performed at the Hefei advanced computing center.

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Footnote

† These authors contributed equally to this work.

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