Monolayer CS as a metal-free photocatalyst with high carrier mobility and tunable band structure: a first-principles study

All first-principles calculations are performed based on the norm-conserving pseudopotential [27] and Perdew–Burke–Ernzerh (PBE) exchange–correlation functional [28] as implemented in the Cambridge Serial Total Energy Package [29]. Because the PBE functional always underestimates the band gap of semiconductors, we also use the Heyd–Scuseria–Ernzerhof screened hybrid functional (HSE06) [30] to correct the band structure. A cutoff energy of 1500 eV is used for the plane-wave basis set. The k-points of the Brillouin zone are sampled with 0.02 (0.01) Å⁻¹ spacing for the geometry optimization (properties calculation). A large vacuum space of 20 Å is chosen to avoid the interlayer interaction between layers (along the c-axis). Considering the effect of dispersion interaction between layers, the van der Waals calculations based on the Tkatchenko–Scheffler approach is performed [31]. The tolerances of energy and force are 5  ×  10⁻⁷ eV/atom and 0.001 eVÅ⁻¹, respectively. The linear response method is used for the phonon dispersion calculations [32].

Furthermore, to clarify the thermal stability of α-CS, the ab initio molecular dynamics simulations are performed at temperatures of 800, 1000, and 1200 K in the canonical ensemble. To explore the stability of α-CS in liquid water, a conductor-like screening model (COSMO) [33] is used to simulate a H₂O solvent environment at 300 K. In this canonical ensemble, a 3  ×  3 supercell (36 atoms) is used, and the total simulation time is 10 ps with a time step of 1 fs.

The carrier mobility of 2D materials based on deformation theory (DP) theory proposed by Bardeen and Shockley can be expressed as the following formula [34]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \mu =\tfrac{e{{\hbar }^{{\rm 3}}}{{C}_{2{\rm D}}}}{{{K}_{{\rm B}}}T{{m}_{i}}*{{m}_{d}}{{(E_{1}^{i})}^{2}}}.\nonumber \end{align}$$

In this equation, e, K_B, T, C_2D, E₁, and m^* are electron charge, Boltzmann constant, temperature (300 K), in-plane stiffness, deformation potential constant, and effective mass, respectively. The C_2D is defined as $(E-{{E}_{0}})/{{S}_{0}}={{C}_{2{\rm D}}}{{(\Delta l/{{l}_{0}})}^{2}}/2$, where E₀ is the total energy of cell without dilation and S₀ is the equilibrium lattice area. The strain step is 0.5%. The average effective mass m_d is determined by ${{m}_{d}}=\sqrt{m_{x}^{*}m_{y}^{*}}$. The term E₁ is defined by $\tfrac{\Delta V}{{\hspace{0pt}}^{\Delta l}\diagup{\hspace{0pt}}_{{{l}_{0}}}\;}$, which represents the energy of CBM for electrons and the VBM for holes along the transport (a and b) direction. ΔV is the conduction band or valence band energy change under certain strain. ${{l}_{0}}$ is the lattice constant along the transport direction, and $\Delta l$ is the deformation of ${{l}_{0}}$· $m_{i}^{*}$(i  =  h for holes, i  =  e for electrons) is based on the effective mass approximation ${{m}^{*}}={{\hbar }^{2}}{{({{\partial }^{2}}E/\partial {{K}^{2}})}^{-1}}$.

We first design five possible 2D configurations of monolayer CS (α, β, γ, δ, and ε) (shown in the ESI). The α-CS is the most energetically stable (figures 1(a) and (b)), which is similar to the α-carbon monosulfide nanostructure [35]. In the primitive cell of α-CS, one C (S) atom binds to two neighboring S (C) atoms. The structure properties of α-CS are listed in table 1. The D₀ (1.79 Å) of α-CS is about 0.43 Å shorter than that of black phosphorene [36]. The lattice constants are smaller than those of α-SiS, and GeS [37, 38]. To evalute the stability of α-CS, we first calculate the binding energy (E_b), which is defined by the following formula: E_b  =  (nE_C  +  nE_S  −  E_α-CS)/2n, where the E_C (E_S) and E_α-CS represent the single point energies of the C (S) atom and α-CS, respectively. n is the number of atoms in the primitive cell. The E_b of α-CS (6.11 eV/atom) is comparable with those of black phosphorene (5.36 eV/atom), α-PN (6.76 eV/atom) and Pmma-CO (7.72 eV/atom) [39–41], indicating that the C–S bonds of α-CS are robust.

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Then, to explore the kinetic stability of α-CS, the calculated vibrational phonon spectrum shows no apparent imaginary modes existing in the acoustic branch frequencies (figure 1(c)). Moreover, the highest optical frequency is up to 848 cm⁻¹, which is higher than those of black phosphorus (~450 cm⁻¹), MoS₂ (~500 cm⁻¹), and silicene (580 cm⁻¹) [42–44], suggesting the strong bonding nature of α-CS. It is well known that the Raman spectrum reflects the vibrational modes of atoms and relates to the phonon dispersion. As shown in the Raman spectra (figure 1(d)), the two higher peaks at about 757 and 770 cm⁻¹ are attributed to the in-plane optical vibrational modes mainly arising from the vibration of C atoms. The two lower peaks (356 and 420 cm⁻¹) are corresponding to the out-of-plane acoustic vibrational modes. In addition, at the same calculation level, the Raman spectra of graphene are calculated. The well-known G band appears at 1563 cm⁻¹, which agrees with the experimental value (1582 cm⁻¹) [45]. Our results will provide a fingerprint of verifying the conformation of α-CS in the future experiment.

Furthermore, to clarify the thermal stability of dynamically stable α-CS under vacuum condition, the ab initio molecular dynamics simulations are performed. The α-CS is thermally stable at 1000 K. However, at 1200 K, the structure has been destroyed with the bond broken and geometry reconstructed (figures 2(a)–(c)). Hence, α-CS may melt at a temperature between 1000 K and 1200 K. The possible melting point of α-CS is much higher than those of reported p-SiC (600 K) and GeS (500 K) [6, 46]. Furthermore, α-CS as a photocatalyst also needs to be insoluble in water. We further check the stability of α-CS in the water solvent environment at 300 K (figure 2(d)). α-CS can maintain the structure stability in water. The average bond length slightly changes and is nearly 1.19% larger than that under vacuum condition at 0 K. Therefore, our result confirms that the α-CS as a photocatalyst has a good thermodynamic stability.

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To verify the mechanical stability of α-CS, we calculate its linear elastic constants. The necessary and sufficient elastic stability condition of monolayer α-CS can be defined by the following formula [47]:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle {{C}_{11}}>0, {{C}_{{\rm 11}}}{{C}_{{\rm 22}}}>C_{{\rm 12}}^{{\rm 2}}, {{C}_{{\rm 66}}}>{\rm 0}{\rm .}\nonumber \end{align}$$

The values of elastic constants matrix components C₁₁, C₂₂, C₆₆, and C₁₂ are 21.06, 113.14, 0.41, and 14.49 GPa, respectively. Clearly, the elastic constants satisfy the above-mentioned conditions, suggesting that α-CS is mechanically stable. We next calculate the in-plane stiffness (C_2D) of α-CS (for more details see next section). The value of C_2D is 39.84 (221.76) N m⁻¹, which is about two times than that of black phosphorene (24.26 (103.28) N m⁻¹) along the armchair (zigzag) direction [39]. Therefore, the C–S bonds of the α-CS are robust.

The band structures from PBE and HSE06 show α-CS is an indirect gap semiconductor (figure 3(a)). The CBM lies in the Γ point, whereas the VBM locates on the Y–Γ path. The PBE and HSE06 computations give band gaps of 1.11 and 1.96 eV, respectively. Notably, a direct band gap (2.02 eV at the HSE06 level) of α-CS at the Γ point is quite close to the indirect band gap. As recently reported of perovskites, the weakly indirect band gap may be an advantage for photovoltaic materials with strong light absorption and long carrier lifetime [48]. Compared with the band gaps of Cmmm-SiS (1.01 eV), Pma2-SiS (1.22 eV), and α-AsP (0.90 eV) [49, 50], the band-gap of α-CS is more suitable for the visible light absorption.

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At the near Γ point, the energy of the states of the top valence bands is suppressed, resulting in the shifting of VBM from Γ to a near k point. The top valence band shows a Mexican-hat-like dispersion around the Γ point. As presented in figure 3(b), this unique feature of band structure leads to both a high density of states (DOS) and an almost one-dimensional-like van Hove singularity. The projected DOS reveals that the Mexican-hat-like valence bands near the Fermi level are predominantly contributed by C-2p orbitals. In detail, the valence bands below the Fermi level consist of zone I and zone II. The zone I is mainly contributed by the 2p states of C atoms and partly by the 2p states of S atoms, exhibiting a π-type antibonding character (shown in the inset of zone I). At the energy region of  −4 to  −7 eV (zone II), the p orbitals of C and S atoms are hybridized, which indicates that the in-plane σ-type covalent bonding character between C and S atoms (the inset in zone II). Furthermore, the σ-bonding strength between C and S atoms along the b (zigzag) direction is stronger than that along the a (armchair) direction, leading to the difference in average length of C–S bonds along the a and b direction. For more information, we also compute the charge densities associated with the CBM and VBM (figures 3(c) and (d)). The CBM locates on C and S atoms, while VBM mainly distributes on C atoms, consistent with the PDOS result.

To obtain more insight into the nature of chemical bonding of α-CS, we also calculate the charge density difference (CDD) shown in figure 3(e). There are charges accumulated on the C atoms and C–S bonds, while the charges are dissipated on the S atoms. According to the Hirshfeld charge analysis, the C atoms partly transfer the 2s electrons to the S 3p orbitals. Simultaneously, S atoms back-donate some electrons to the C 2p orbitals. Therefore, there is a small charge (0.20 e) transfer from S atoms to C atoms, which agrees with the analysis of CDD. On the basis of the overlap population analysis of α-CS, the values of C–S bond are 1.13 (C₁–S₁, C₂–S₂) and 0.36 (C₁–S₂, C₂–S₁), suggesting that the bonds connected C₁ (C₂) and S₁ (S₂) are covalent, while C₁ (C₂) and S₂ (S₁) are partially covalent/ionic bonds.

For applications in the high-performance photocatalyst, the 2D material needs to possess not only a moderate band gap but also high carrier mobility. Thus, we calculate the room-temperature carrier mobility of α-CS to explore its electrical conductivity. The carrier mobility μ lies on C_2D, E₁, and m^*. To obtain the C_2D and the deformation potential constant E₁, we further calculate the total energy shift E–E₀ (figure 4(a)) and the band edge positions of VBM and CBM with respect to the dilation $\Delta l/{{l}_{0}}$, respectively (figure 4(b)). All computed values of C_2D, E₁, m^*, and μ are summarized in table 2.

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As mentioned above, the difference between the C–S bond strength in armchair and zigzag directions will lead to anisotropic features of the 2D elastic modulus and the deformation-potential constant. As shown in figure 4(b), E₁ is more insensitive to the strain along a (armchair) than that along b (zigzag) direction. In addition, the effective mass of holes evidently depends on the direction (table 2). The above results can be understood from the charge density of VBM (figure 3(d)). The range of charge delocalization along the armchair direction is wider compared with that along the zigzag direction, indicating that the effective mass of holes along the armchair direction is much smaller than that along the zigzag direction. In contrast, the effective mass of electrons shows a rather weak direction-dependence. This result can be well explained by the CBM exhibiting both covalent bonds and anti-bonds along the armchair and zigzag directions (figure 3(c)). Thus, the effective masses of electrons along the armchair (0.38 m₀) and zigzag (0.32 m₀) direction are approximately equal.

The predicted carrier mobility of α-CS shows an anisotropic characteristic (table 2). The hole and electron mobilities along the armchair direction are 1580.41 cm² V⁻¹ s⁻¹ and 8453.19 cm² V⁻¹ s⁻¹, respectively, about three times higher than that along the zigzag direction (436.87 and 3034.46 cm² V⁻¹ s⁻¹), suggesting that the armchair direction is more favorable for the carrier conduction. The carrier mobility of α-CS is much higher than those of monolayer MoS₂ (~60–200 cm² V⁻¹ s⁻¹) [51]. The high carrier mobility and large difference in hole/electron mobilities will efficiently help migrate and separate photo-generated carriers. The main reasons for the high hole mobility of α-CS are discussed in the following aspects. On the one hand, the valence band along the Γ–Z path (armchair direction) is more dispersive than the conduction band (figure 3(a)), resulting in the larger (smaller) effective mass of electron (hole) of 0.38 m₀ (0.23 m₀). Therefore, the high hole mobility along the armchair direction benefits from the smaller effective mass of hole. On the other hand, for the case of the zigzag direction, the effective mass of electron is smaller than that of hole, but the deformation potential E₁ of the CBM is greatly larger than that of the VBM. The CBM is mainly contributed from the delocalized states of C and S atoms, while VBM is partially contributed from the localized states of C atoms (figures 3(c) and (d)). Therefore, the E₁ of the CBM is more sensitive to strain compared with that of the VBM along the zigzag direction, resulting in the large difference of E₁ between CBM (−9.78 eV) and VBM (−1.69 eV), where the latter leads to the high hole mobility. Therefore, the high hole mobility attributes to the small effective mass and deformation potential.

Applying external strain is a common way to tune the band structures of 2D materials. We have performed calculations on the relation between the band gap and strain for α-CS (figure 5(a)). In the b direction, the band gap exhibits a linear decrease under tensile strain and then drops to zero when tensile strain is up to  +0.15. In addition, the band gap decreases and arrives at zero at compressive strain of  −0.18. When the a-direction strain is applied, the band gap monotonically decreases with increasing the tensile strain and becomes zero at  +0.22. However, for compressive strain, the band gap is initially increased up to the maximal value (1.16 eV) at  −0.03 and then decreases linearly with further increased compression. As shown in figure 5(b), the D₀ along the a direction is more sensitive to the strain than that along the b direction, which agrees with the analysis of the in-plane stiffnesses. In addition, for the a direction, the strength of C₁–S₁ bonds gradually weakens with increasing the compressive and tensile strain, while the strength of C₁–S₂ bonds enhances (weakens) under the compressive (tensile) strain. In the case of b direction, the strength of C₁–S₁ bonds can be enhanced (weakened) by increasing the compressive (tensile) strain. In addition, the strength of C₁–S₂ bonds can be enhanced from partially covalent/ionic bonds to covalent bonds, by increasing the tensile strain along the b direction.

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To shed more lights on the effect of band gap with strain, we calculate the band structures of α-CS at different uniaxial strains along the b direction (shown in figure 5(g)). The CBM shifts downward with increasing the stain strength and crosses the Fermi level at ε = -0.18. On the other hand, at ε  =  +0.15, α-CS also experiences a transition from semiconductor to metal due to its VBM crossing the Fermi level. Its CBM (2p states of C atoms) moves downward and away from the Γ point to a near k point locating on the Γ–Z path. The strength of C₁–S₁ bond weakens greatly along the zigzag (b) direction at ε  =  +0.15, resulting in the charge density around CBM (the charge density around CBM not shown) favouring the armchair (a) direction. The corresponding position of CBM transfers from Γ point to a point between Γ and Z (armchair direction).

To check the stability of the strain induced system, we calculate the phonon spectra under these certain strains. As shown in figures 5(c)–(f), α-CS will be stable with the axial tensile strain of  +0.15 in the b direction and  +0.22 in the a direction. For the compressive strain of –0.18 along the a (b) direction, α-CS has apparent imaginary frequencies of the acoustic phonon branch. However, α-CS is still dynamical stable with the compressive strain of  −0.12 in the a (b) direction. Therefore, the band gap of α-CS can be flexibly modulated through the applied strain in a relatively wide range, which may be useful for applications in nanoelectronic devices.

Very recently, many theoretical and experimental works have demonstrated that applying external E-field can efficiently modify the electronic structures of 2D materials [52–54]. We then explore the relationship of the band gap of α-CS with the perpendicular E-field. As shown in figure 6(a), the band gap does not depend on the E-field until 5 V Å⁻¹. However, the band gap sharply decreases with further increase of the E-field. Specially, when the E-field is up to 0.75 V Å⁻¹, the band gap closes. As shown in figure 6(b), the D₀ and C–S bond lengths decrease only 0.12% compared with those of the pristine α-CS, indicating that the geometry of α-CS is insensitive to E-field. To explore the stability of this structure at 0.75 V Å⁻¹, we also calculate the corresponding phonon spectra (figure 6(c)). It is stable with well separated positive acoustical and optical frequencies.

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To better understand this phenomenon, the band structures of α-CS with different E-fields are calculated (figures 6(d)–(f)). The position of VBM is almost the same (with respect to the Fermi level), while the CBM moves downward with increasing the E-field and then crosses the Fermi level at the E-field of 0.75 V Å⁻¹. As presented in figure 6(d), the conduction bands (C-2p and S-3p states) near the Fermi level are split (locating at Γ–Z and Γ–Y path) and partly degenerate (locating on Z–T–Y path). With respect to the C 2p orbital energy, the 3p orbital energy of S atom is higher. Thus, the S-3p orbitals are obviously affected under the perpendicular E-field in comparison with the C-2p orbitals. Therefore, the splitting of corresponding CB (blue bands composed by S-3p states) is further enhanced with increasing the E-field strength. Correspondingly, this certain band moves downward to the Fermi level (figures 6(e) and (f)). In contrast, the CB (red bands composed by C-2p states) is insensitive to the E-field.

For the α-CS bilayer, we design six possible stacking conformations, namely, AA-, AA'-, AB-, AB'-, AC-, and AC'-stacking (figure S2). The layer-to-layer space of six stacking patterns differs slightly ranging from 3.09 Å (AC-) to 3.47 Å (AA'-). Furthermore, the phonon dispersion calculations confirm that these bilayers are kinetically stable (figure 7). The AC'-stacking is the most favorable conformation for all the bilayer α-CS. The calculated band gaps (at the HSE06 level) of bilayers range from 1.60 to 1.92 eV, which are all smaller than that of single-layer α-CS (1.96 eV). As shown in figure 7, the feature of indirect band gap for bilayer α-CS is retained regardless of the stacking pattern. According to the band structures of six stacking patterns, the VBM locates on Γ point, while the CBM lies on the Γ–Z path. The positions of VBM with respect to the Fermi level are almost the same, while the CBM of the AB'- and AC-stacked bilayers are shifted upward compared with those of the AA-, AA'-, AB-, and AC'-stacked bilayers. To get more insights into the difference of the CBM, we further calculate the charge density corresponding with the CBM and VBM of six stacking patterns (figure 7). The VBM is contributed from delocalized states of the C atoms, while the CBM is mainly contributed from localized states of C and S atoms. Moreover, the wavefunction overlap indicates that weak 'quasi-bonds' exist in the interlayer regions, which results in the decrease of the band gap from monolayer to bilayer. Therefore, different stacking patterns play significant roles in electronic structures of bilayers.

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We further explore a suitable substrate for the α-CS growth. Graphite is usually used as a kind of common substrate for the nanoscale-material growth [55–57]. We choose the graphite (0 0 1) surface because its lattice constants are close to those of α-CS. The unit cell of heterostructure is composed by 2  ×  2 unit cells for both monolayer α-CS and four graphite layers (figure 8(a)). The average values of the two separate lattice parameters of the new heterostructure are a  =  b  =  4.90 Å, which are chosen to be the co-periodic lattice parameters. A lattice mismatch (δ) exists in heterostructure, which defined by this formula: δ  =  [(a₁ – a₂)/a₂]  ×  100%. The a₁ and a₂ represent the lattice constants of monolayer α-CS on the graphite (0 0 1) surface and individual α-CS monolayer, respectively. The lattice mismatch is only 0.03% for our system. The optimum space (3.50 Å) between monolayer α-CS and graphite layers is slightly larger than that of the graphite (3.40 Å).

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We next calculate the binding energy of α-CS  +  graphite heterostructure. It can be defined as: E_b  =  E_total  −  E_graphite  −  E_α-CS. The E_total, E_graphite, and E_α-CS are the energies of the composites, graphite substrate, and α-CS, respectively. The E_b is  −40.28 meV, which is comparable to that of SiGe/BN heterostructure (−65.2 meV) [58], indicating that α-CS is more easily exfoliated from the substrate surface. Importantly, the DOS of α-CS has not been affected by the graphite substrate, where the most attractive electronic properties are well preserved (figure 8(b)). Therefore, it is possible for the α-CS grown on the graphite substrate.

The appropriate band edge positions for straddling the water redox potentials are fundamental. It is noted that the water redox potentials depend on the pH value of medium [59–61]. Figure 9(a) shows the band edges of VBM and CBM (with respect to the vacuum level) for monolayer and bilayer α-CS compared with the water redox potentials at pH  =  0 and pH  =  7. For monolayer α-CS, the band alignments can straddle the water redox potentials at pH  =  7. Although the α-CS is not suitable as an intrinsic photocatalyst in a pH  =  0 solution, its VBM energy (−5.42 eV) is very close to the oxidation potential of H₂O/O₂ (−5.76 eV). To precede the water splitting, the small bias potential should be required of 0.34 eV. Notably, the CBM energy of α-CS (−3.37 eV) is much higher than the reduction potential of H⁺/H₂ (−4.44 eV), suggesting that the force of driving water reduction reaction is strong. On the other hand, for all α-CS bilayers, the band edges of VBM and CBM are suitable for photocatalysis at pH  =  7. Moreover, the band alignments of AB- and AB'- stacking bilayers locate at favorable positions for water splitting at pH  =  0, which implies that different stacking orders can efficiently adjust the band edge of α-CS.

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We also investigate the light harvesting performance of α-CS by calculating the absorption coefficient in- and out-of-plane with the HSE06 functional, as shown in figure 9(b). The out-of-plane and in-plane absorption coefficient correspond to the polarization vectors perpendicular and parallel to the surface of α-CS, respectively. The prominent peaks of absorption coefficients occur at ~150 nm, and the highest value is up to 10⁵ cm⁻¹, suggesting that α-CS displays strong absorption strength in the ultraviolet region. Notably, the in-plane absorption coefficient is larger than that of the out-of-plane at the range of 200–800 nm, suggesting that the parallel polarized light exhibits stronger harvest ability in the ultraviolet and visible light regions. Thus, monolayer α-CS has a potential application in optical and photo-electronic devices.

Apart from the structural stability of α-CS in implicit water at room temperature (figure 2(d)), we have also explored the properties of water adsorbed on α-CS. The double-side adsorption of 6 water molecules on the 3  ×  3 supercell of α-CS is employed (figure 10(a)). The optimum space between water molecules and α-CS is about 1.74 Å, and the structural integrity of α-CS is not destroyed. The H–O bond length (H–O–H bond angle) of adsorbed H₂O increases 0.02 Å (~2°) in comparison with that of the isolated water molecule, showing a small structural change during the whole adsorption process. The water adsorption energy of 0.37 eV is slightly higher than that of monolayer MoS₂ (0.23 eV) [62], suggesting the adsorption energy lying between the physisorbed and chemisorbed states. Here, we present further discussions on the PDOS of the H₂O  +  α-CS system (figure 10(b)). Clearly, the H-s (O-p) and C-sp (S-p) orbitals are hybridized in the energy range from  −2 to  −1 eV. The Hirshfeld charge analysis shows that each water molecule obtains 0.1 e from α-CS. Therefore, monolayer α-CS could be favorable for water adsorption.

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