Enhancing hydrogen evolution of MoS₂ basal planes by combining single-boron catalyst and compressive strain

Abstract

MoS₂ is a promising candidate for hydrogen evolution reaction (HER), while its active sites are mainly distributed on the edge sites rather than the basal plane sites. Herein, a strategy to overcome the inertness of the MoS₂ basal surface and achieve high HER activity by combining single-boron catalyst and compressive strain was reported through density functional theory (DFT) computations. The ab initio molecular dynamics (AIMD) simulation on B@MoS₂ suggests high thermodynamic and kinetic stability. We found that the rather strong adsorption of hydrogen by B@MoS₂ can be alleviated by stress engineering. The optimal stress of -7% can achieve a nearly zero value of \text{Δ}{G}_{\text{H}} (-0.084 eV), which is close to that of the ideal Pt–SACs for HER. The novel HER activity is attributed to (i) the B–doping brings the active site to the basal plane of MoS₂ and reduces the band-gap, thereby increasing the conductivity; (ii) the compressive stress regulates the number of charge transfer between (H)–(B)–(MoS₂), weakening the adsorption energy of hydrogen on B@MoS₂. Moreover, we constructed a SiN/B@MoS₂ heterojunction, which introduces an 8.6% compressive stress for B@MoS₂ and yields an ideal \text{Δ}{G}_{\text{H}}. This work provides an effective means to achieve high intrinsic HER activity for MoS₂.

Graphical abstract

Keywords

MoS₂ / stress engineering / single-atom catalyst / HER / charge transfer / DFT / heterojunctions

1 Introduction

With the increasing depletion of fossil fuels and the deterioration of the atmospheric environment, the production of hydrogen fuel through environment-friendly electrochemical and solar-driven photoelectrochemical (PEC) water splitting has become a promising pathway to achieve sustainable energy development [1-7]. The water-splitting reaction (H₂O → H₂+ 1/2O₂) is divided into two half-reactions: the oxygen evolution reaction (OER) and the hydrogen evolution reaction (HER), which occur on the anode and cathode, respectively [1, 8]. Taking HER as an example, the mechanism of reaction in an acidic environment involves three steps: (i) Generating an adsorbed hydrogen atom on the cathode (the Volmer or discharge reaction: H⁺ + e^- + * → H*); (ii) the combination of two adsorbed hydrogen to produce H₂ (the Tafel reaction: 2H* → H₂ + 2*); (iii) another electron reacts with the adsorbed hydrogen and another proton in the solution to form H₂ (the Heyrovsky reaction: H* + H⁺ + e^- → H₂ + *) [8-11]. The high activation energies required to form the reaction intermediates on the cathode surface will inevitably create the large kinetic barriers [12]. The resulting overpotential (the potential difference between theoretical thermodynamic potential and the experimental potential) required to drive the reaction has a significant effect on HER performance and must be taken into account [13]. Therefore, how to effectively reduce the overpotential to the greatest extent has become the most critical issue for the commercialization of the HER.

Fortunately, numerous related reports have shown that the introduction of suitable catalysts can decrease the overpotential and speed HER up, maximizing the energy conversion efficiencies [14-27]. As a traditional industrial hydrodesulfurization catalyst, MoS₂ has attracted widespread attention in recent years because of its performance tunability, low cost, and good stability [28-33]. MoS₂ has three common structures, 1T-, 3R-, and 2H-MoS₂. Among them, the semiconductor 2H–MoS₂ has the best thermodynamic stability [1, 34], which will be hereinafter referred to as MoS₂. Previous studies have shown that the basal plane of MoS₂ is usually inert for H adsorption, and only a few active sites are distributed on the edge [35, 36]. Therefore, these defects should be overcome before the large-scale application of MoS₂. So far, the initial MoS₂ optimization can be divided into two directions, one strategy to change MoS₂ monolayer nanostructures, such as vertical nanoflake [37], nanoparticles [38], nanowires [39], mesoporous films [40] or defect-rich films [41], maximum exposure edge active sites, another strategy to enhance intrinsic activity of the edge sites by atomic scale modifications, such as construction defects [36, 40] and chemical doping at edge sites [42]. Note that the basal planes constitute the vast majority of bulk materials [36]. A strategy to activate the basal plane is beneficial for maximizing the commercialization of MoS₂. Unfortunately, the relevant research is not complete.

Recently, single-atom catalysts (SACs) become a rising star in the field of catalysts, which perfectly inherit the advantages of both heterogeneous and homogeneous catalysts [43-45]. The extreme atomic utilization efficiency enables SACs to exhibit excellent catalytic performance even under low loss conditions, which is conducive to achieving the atomic economy [46]. Secondly, the active center sites of SACs are completely exposed [47]. The charge transfer effect between the active center atom and the substrate and their low coordination environment ensures the activity of SACs in various reactions [48-50]. In addition, structural simplicity facilitates precise characterization, which, in turn, helps to achieve rational SACs design at the atomic scale [44, 51]. Importantly, we noticed that MoS₂ with stable physicochemical properties is suitable as a substrate for SACs to avoid single-atom agglomeration [52]. Currently, the vast majority of SACs that have been developed are transition metal-based, such as Pt SACs [53, 54], Ru SACs [55, 56], Mo SACs [57, 58], Cu SACs [59, 60], Fe SACs [61, 62], Co SACs [63, 64] and our previous study on V SACs [52]. However, the deficiency of metal SACs including low selectivity, low faradaic efficiency, low production rate, and excessive dependence on pH must be overcome before they are widely used [65, 66]. Fortunately, Braunschweig et al. pioneered a new strategy for N₂ fixation based on metal-free center in 2018 [67]. However, to our knowledge, only a few metal-free SACs have been reported, such as single-boron catalyzed N₂ reduction reaction (NRR) [68-70], and single-iodine catalyzed HER [71]. Therefore, the exploitation of highly efficient metal-free SACs and their application in electrocatalytic HER will become a research hotspot in this field. Importantly, Liu et al. [72] emphasized that the single boron atom catalysts (B SAC) embedded in the defective MoS₂ monolayer with S vacancies may be more inclined to promote HER in their study on the SACs for NRR.

In addition, strain engineering has been proven to be a powerful strategy for regulating the electronic structure of the two-dimensional (2D) nanomaterial and optimizing the catalytic performance in recent years [73-76], which can restore the hope of many initially unpromising catalysts [77-83]. As early as 1998, Mavrikakis et al. [84] regulated the surface electron structure and the energy of reaction intermediates by adjusting the surface strain. Since then, many researchers have focused on adjusting surface stress to improve catalyst efficiency [82, 85-89]. Regarding a famous case of oxygen reduction reaction (ORR) on platinum, previous studies have shown that a 1% compressive strain can increase its activity by more than 300% [82, 86]. Importantly, Li et al. [36] pointed out in their work on activation and optimization of monolayer MoS₂ for HER by introducing S vacancies and strain that appropriate strain can significantly improve the properties of MoS₂ for HER. Similarly, Pan et al. [90] successfully adjusted the electronic and magnetic properties of MoS₂ nanoribbons with strain engineering, and predicted the potential applications to spintronics and photovoltaic cells.

In this work, we successfully reported for the first time a strategy to achieve high HER activity of MoS₂ basal planes by combining strain and B SAC through density functional theory (DFT) calculations. The ab initio molecular dynamics (AIMD) simulation on B@MoS₂ (single boron atom embedded in MoS₂ basal plane) suggests high thermodynamic stability. Through applying tunable stress, the optimal stress to obtain ideal hydrogen adsorption free energy (\text{Δ}{G}_{\text{H}}) was determined. The underlying mechanism for the novel HER performance was explored. Moreover, the IV–V/B@MoS₂heterojunction (heterojunction of the group IV–V compounds and B@MoS₂) was constructed to simulate the effect of compressive stress on the HER performance of B@MoS₂. The methodology of combining SACs and stress could be a general strategy of optimizing catalytic performance for materials beyond MoS₂.

2 Calculation methods

Our DFT calculations were performed using the generalized gradient approximation of the Perdew–Burke–Ernzerhof functional (GGA/PBE), implemented in the Vienna Ab Initio Simulation Package (VASP) [91-93]. The projector augmented wave (PAW) [91] potential was used to describe the ion-electron interactions with the valence state 2s and 2p for B, 3s and 3p for S, 4d and 5s for Mo, respectively. The thickness of the vacuum layer was fixed at 15 Å in the z-direction to prevent periodic layer-to-layer interactions. A Γ-centered 3×3×1 (geometry optimization) and 5×5×1 (self-consistent energy) Monkhorst–Pack grids for the Brillouin zone sampling and cutoff energy of 500 eV for plane wave expansion were used in the calculations. In the geometry optimization, all atoms were allowed to relax in the cell until the maximal residual force and energy on each atom was less than 0.01 eV/Å and 1.0 × 10^-4 eV/atom, respectively. The long-range van der Waals (vdW) interaction often has a great influence on the structure of 2D materials [94], MoS₂ being one of them [17]. In this work, Grimme’s scheme (D3) [95] was used to modify the long-range dispersion interactions.

The binding energy of the B atom (E_b(B)) embedding (substitution or adsorption) on MoS₂ was calculated by the following equation:

\begin{array}{}\text{(1)}& E_{\text{b}}(\text{B})=E_{\text{B}@\text{Mo}{\text{S}}_2}-\left(E_{{\text{MoS}}_2}+E_{\text{B}}-n{E}_{\text{S}}\right)\mathrm{,}\end{array}

where E_{{\text{B@MoS}}_2} and E_{{\text{MoS}}_2}are the total self-consistent energy of the B@MoS₂, MoS₂. E_{\text{B}} and E_{\text{S}} are the cohesive energy of the B and S atoms [96, 97]. n = 1 if B atom substitutes for S atom, otherwise n = 0. In general, the negative E_b(B) corresponds to a stable catalyst system.

Regardless of the acid solubility and alkalinity of the reaction, the overpotential of the HER is always determined by the Gibbs free energy change of the Volmer reaction (H⁺ + e^- + * → H*) [8, 46]. Thus, the absolute value of \text{Δ}{G}_{\text{H}} is the energy barrier to be overcome for the HER [8, 98, 99]. \text{Δ}{G}_{\text{H}} is calculated by the following equation:

\begin{array}{}\text{(2)}& \begin{array}{cc}\text{Δ}{G}_{\text{H}}=& \left[E_{\text{H}*}+E_{\text{Z}\text{P}\text{E}\left(\text{H}*\right)}-T{S}_{\left(\text{H}*\right)}\right]-E_{(*)}\\ & -\frac{1}{2}\left[E_{\left({\text{H}}_2\right)}+E_{\text{Z}\text{P}\text{E}\left({\text{H}}_2\right)}-T{S}_{\left({\text{H}}_2\right)}\right]\mathrm{,}\end{array}\end{array}

where E_{\text{H}*}, E_{(*)} and E_{\left({\text{H}}_2\right)} are the total self-consistent energy of the H* (B@MoS₂ adsorbs one proton), * (B@MoS₂), and free H₂ molecule, respectively. E_{\text{Z}\text{P}\text{E}} and S are the zero-point energy (ZPE) and entropy, respectively. According to the method developed by Norskov and colleagues [100, 101], the free energy of (H⁺ + e^-) has been replaced by half the chemical potential of a hydrogen molecule. For H₂, ZPE, and S can be obtained from NIST-JANAF thermochemical tables [102]. It should be noted here that the pH is assumed to be zero, which means the relationship between the overpotential and the pH is ignored in this work. According to the Sabatier principle, the \text{Δ}{G}_{\text{H}} of an ideal catalyst should be infinitely close to zero, where hydrogen is bound neither too strongly nor too weakly [103].

In addition, if the ZPE and entropy are not included in the Eq. (2), we can also obtain the adsorption energy of H⁺ (E_ads(H⁺)) on the surface of B@MoS₂ by the following equation:

\begin{array}{}\text{(3)}& E_{ads}\left({\text{H}}^{+}\right)=E_{\text{H}*}-E_{(*)}-\frac{1}{2}{E}_{\left({\text{H}}_2\right)}\mathrm{.}\end{array}

3 Results and discussion

3.1 Stability and electronic structure of B@MoS₂

Based on other similar studies [104, 105], we constructed a 3 × 3 × 1 MoS₂ supercell, including 9 Mo and 18 S atoms, and optimized by GGA/PBE + D3 method which has been proved to accurately describe the geometry and electronic structure of MoS₂ [42], as shown in Fig. 1(a). Note that we also have carried out a similar study on a 4 × 4 × 1 MoS₂ supercell, and obtained consistent structural parameters and E_b(B), proving that 3 × 3 × 1 MoS₂ supercell is large enough. The lattice parameter of the primitive MoS₂ is 3.168 Å, the thickness of the monolayer is 3.2 Å, and the bond length of Mo–S bond is about 2.41 Å, which are in good agreement with the experimental report [106] and other theoretical studies [97, 107-109]. High stability is a precondition for catalysts. Strong bonding between the substrate and the supported atom is necessary to avoid the aggregation of the single atom. For the embedded B atom, we considered four typical structures, including a B atom substituting one S atom and a B atom adsorbed on top of Mo atom, S atom, and vacancy, as shown by the circle in Fig. 1(a). The most stable B@MoS₂ structure with the lowest E_b(B) of -1.607 eV is obtained, as shown in Fig. 1(b). A negative E_b(B) means that the B atom on the B@MoS₂ surface is energetically more stable than that in the bulk form [52]. The other three metastable structures and E_b(B)s are summarized in Fig. S1. Therefore, the following B@MoS₂ refers specifically to the structure of B atom adsorbing on top of Mo atom Fig. 1(b). Furthermore, the AIMD simulations on at 300 K for 10 ps with a time step of 2 fs is performed to check the thermal stability of B@MoS₂ structures [69, 96]. As shown in Fig. 2, no obvious geometry buckling of B@MoS₂ was observed during the AIMD simulations. Therefore, B@MoS₂ is thermodynamically stable.

The calculated band structure and the DOS (total density of states (TDOS) and partial density of states (PDOS)) of MoS₂ and B@MoS₂ are shown in Fig. 1(c) and Fig. 1(d), respectively. The MoS₂ behaves as a direct gap semiconductor with the band-gap energy (E_g) of 1.73 eV, which is close to the theoretical result obtained by Pan et al. (1.72 eV) [90]. The B-doping introduces an impurity level in the forbidden gap, which provides a new platform for electronic transitions and improves the conductivity of the catalyst. Both the conduction band minimum (CBM) and the valence band maximum (VBM) of MoS₂ and B@MoS₂ are mainly contributed by Mo-d and S-p orbitals, of which the Mo-d is the main component. The impurity states near the center of the forbidden band of B@MoS₂ are generated by the hybridization of B-p orbitals and Mo-d orbitals, as seen in Fig. 1(d) and Fig. 4(c).

3.2 The \text{Δ}{G}_{\text{H}} of B@MoS₂ under stress activation and optimization

The dispersed B atoms as active sites make the inert basal plane of MoS₂ active for H⁺ adsorption. According to Eq. (2), the \text{Δ}{G}_{\text{H}} of B@MoS₂ is calculated to be -0.451 eV, which is far from the ideal value of zero. Interestingly, previous studies showed that the process of adsorbing H⁺ on the surface of MoS₂ is an endothermic reaction [36], which is converted into an exothermic spontaneous reaction after doping with B atoms (refer to Fig. S3). The loading of B SAC on MoS₂ makes it a strong absorbent for H⁺, leading to the large barrier for H₂ release. We note that the electronic structure of 2D materials is sensitive to strain [13]. So, it is productive to adjust its adsorption ability towards adsorbates by strain engineering. In this work, we attempt to use this method to weaken the adsorption ability of B@MoS₂ for H⁺ to activate and optimize the capacity of B@MoS₂ for HER.

The biaxially adjustable stress is simulated by changing the lattice size of the B@MoS₂, as shown in Fig. 3(a). According to the direction of stress, the strain can be divided into tensile strain and compressive strain. The stress magnitude is determined by the following equation:

\begin{array}{}\text{(4)}& \epsilon \text{=}\left(\frac{L-L_0}{L_0}\right)\times 100\text{\%}\mathrm{,}\end{array}

where L₀ and L are the calculated supercell sizes before and after the stress application, respectively [90]. Therefore, ε is positive for tensile stress, whereas negative for compressive stress.

Since there are very stable atomic bonds in the 2D MoS₂ plane, the breaking strength is up to 18 N·m^-1, which can sustain 11% elastic strain [110, 111]. Therefore, within the tolerable range of MoS₂, we investigated the \text{Δ}{G}_{\text{H}} under tunable strain (-11% ≤ ε ≤ 6%), and the result is shown in Figs. 2(b) and (c). The lattice parameter and \text{Δ}{G}_{\text{H}} data of B@MoS₂ under different stress are summarized in Tables S1 and S2. In the range of the applied strain, the adsorption of H⁺ is always the spontaneous exothermic process, indicating the strong adsorption capability of B@MoS₂ for H⁺. As the compressive stress increases, the adsorption capacity of B@MoS₂ for H⁺ gradually decreases until ε = -7%. We see that the \text{Δ}{G}_{\text{H}} corresponding to ε = -7% is closest to zero, about -0.084 eV, which is very close to that of the ideal Pt–SACs for HER (~ -0.07 eV) [53, 54]. On the contrary, as the tensile stress increases, the adsorption capacity of B@MoS₂ for H⁺ continues to increase until saturation (ε = 5%). Therefore, ε = -7% is the most suitable stress for B@MoS₂ to catalyze HER. In the strained B@MoS₂ with ε = -7% (-7% B@MoS₂), the Mo–S bonds are shortened under the compressive stress, but the reduction is only about 1.66%, indicating that the B@MoS₂ keeps the lattice structures [90]. In addition, the E_b(B) of -7% B@MoS₂ is determined to be -2.24 eV, which means that the -7% B@MoS₂ has certain structural stability while having the best catalytic performance.

3.3 Origins of the high HER activity on -7% B@MoS₂

We further discuss the origin of high hydrogen desorption activity of B@MoS₂ under proper compressive stress. The band structure and DOS of -7% B@MoS₂ were calculated, as shown in Fig. 4(a). The E_g of B@MoS₂ is further reduced under the action of compressive stress. The CBM and VBM are still mainly contributed by the Mo-d and S-p orbitals. Comparing Fig. 4(c) and Fig. 4(d), the contribution of B-p orbitals to the hybridization of Mo-d and B-p orbitals further increases in range of -0.75—0.5 eV. At the same time, it is interesting that the extraneous B atom moves the Fermi energy level into the conduction band, greatly improving the conductivity of the original MoS₂, and ensuring the effective transfer of electrons in the HER process.

To associate the \text{Δ}{G}_{\text{H}} with the interaction between the substrate and adsorbed hydrogen, we illustrate the stress-dependent E_ads(H⁺) by Eq. (3) in Fig. S2. Obviously, E_ads(H⁺) and \text{Δ}{G}_{\text{H}} have the same trend with the applied stress. The ε = -7% corresponds to the weakest adsorption for H. One knows that E_ads(H⁺) is closely related to the charge transfer between the adsorbed hydrogen and the substrate. The charge density differences of B@MoS₂ and -7% B@MoS₂ before and after adsorbed hydrogen are shown in Fig. 5. Positive (electron deficit) and negative (electron excess) lobes are shown in light-blue and crimson isosurfaces, respectively. It is clear that charge redistribution occurs between the adsorbed hydrogen, fixed B atom, and MoS₂. According to the calculation results of Bader charge transfer (Table S3), we draw a schematic diagram of the charge transfer path under special stress (ε = 0, -7%, -11%), as shown in Fig. 6. We represent the number of transferred electrons by the red number. For all the models, B atom donates electrons to the adsorbed hydrogen and the substrate simultaneously. Compared with the stress-free B@MoS₂, the -7% compressive stress enhances the Mo-d and B-p hybridization. So, the charge transfer between B and MoS₂ rapidly increases from 0.592e to 1.01e, and the electrons obtained by adsorbed hydrogen decreases from 0.533e to 0.346e, leading to the weakened adsorption for H⁺ (Fig. S2). However, as the compressive stress increases continually (ε > -7%), we find that the charge transfer between B atom and MoS₂ is basically unchanged, suggesting the hybridization saturates. At the same time, the charge exchange between adsorbed hydrogen and B atom increases, and the E_ads(H⁺) is enhanced. The effect of stress on the d orbitals of the metal atom can explain this phenomenon. In general, the sp bands of metal are broad and structureless, while the d-band are narrow. The interaction between d states and adsorbate states can be significantly changed by even small stress changes. The compressive strain causes the Mo-d orbitals to overlap, reducing the d-band center energy [74]. Therefore, more unoccupied antibonding states are pushed below the Fermi level, allowing B to interact with them, and promoting charge transfer between B and MoS₂ [74, 112]. When the overlap of Mo-d orbitals reaches saturation, this promotion is interrupted. Thus, the charge transfer between B atom and adsorbed hydrogen is restored.

3.4 The realization of suitable stress

The current mainstream stress realization method can be summarized as manufacturing lattice mismatch, such as generating strain through lattice mismatches of 2D material heterojunction or homojunction [13]. The idea of using the unique electronic structure of the heterojunction to reduce the overpotential of the HER has been verified by a large number of studies [113-116]. Typically, Singh et al. [115] carried out a study on graphene-supported TiO₂ to promote carrier photocatalytic water splitting, and the ultrathin MoSe₂ nanosheets coated on hollow carbon spheres synthesized by Yuan et al. [116] was used as high-efficiency catalysts for HER, etc. Inspired by depositing a monolayer MoS₂ grown by chemical vapor deposition (CVD) on a SiO₂ substrate with microcavities [117], we envision the growth of MoS₂ on a 2D material surface with a suitable lattice parameter. In this section, two monolayer group IV–V compounds (SiN and GeN) are examined as substrates, which have slightly smaller lattice parameters than that of B@ MoS₂ (SiN: 2.895 Å, GeN: 3.091 Å) and high theoretical stability [118]. As shown in Fig. 7, the IV–V/B@MoS₂ nano-heterojunction (GeN/B@MoS₂ and SiN/B@MoS₂) are constructed, which include 4 × 4 IV–V supercell and 4 × 4 B@MoS₂ supercell.

After full relaxation, the equilibrium distance between the IV–V layer and the B@MoS₂ layer is 3.5 Å, and the cross-sectional dimensions of two heterojunction supercells are 12.364 Å × 12.364 Å (GeN/B@MoS₂) and 11.580 Å × 11.580 Å (SiN/B@MoS₂), respectively. To evaluate the stability of the IV–V/B@MoS₂ heterojunction, the interface adhesion energy (E_adh) is defined according to the following equation:

\begin{array}{}\text{(5)}& E_{adh}=E_{\text{heter}}-E_{\text{IV}-\text{V}}-E_{\text{B@Mo}{\text{S}}_2}\mathrm{,}\end{array}

where E_{\text{heter}}, E_{\text{IV}-\text{V}}, and E_{{\text{B@MoS}}_{\text{2}}} are the total self-consistent energy of the IV–V/B@MoS₂ heterojunction, the pure IV–V, and the B@MoS₂ monolayer, respectively. The E_{adh} of IV–V/B@MoS₂ is as low as -3.376 eV (GeN/B@MoS₂) and -3.262 eV (SiN/B@MoS₂), respectively, which are stronger than the known GaS/GaSe (-1.947 eV) and graphene/MoS₂ (-0.023 eV) [119, 120]. Therefore, GeN/B@MoS₂ and SiN/B@MoS₂ are considered feasible. According to Eq. (4), the GeN/B@MoS₂ heterojunction is equivalent to applying a 2.4% compressive stress to B@MoS₂ and yields a \text{Δ}{G}_{\text{H}} of -0.281 eV, which is close to the predicted value in Fig. 3(c). Excitingly, the SiN/B@MoS₂ heterojunction is equivalent to apply an 8.6% compressive stress to B@MoS₂ and gives a \text{Δ}{G}_{\text{H}} of -0.098 eV. It provides a theoretical basis for B@MoS₂ as an alternative to precious metal Pt–SACs for HER.

4 Conclusions

In summary, based on DFT and AIMD calculations, we designed a stable metal-free B SAC on MoS₂ (B@MoS₂) for HER. The anchored B atoms make the MoS₂ basal surface to be active for H adsorption. By optimizing stress, a compressive stress of -7% can achieve a nearly zero value of \text{Δ}{G}_{\text{H}}, about -0.084 eV, which is close to that of the ideal Pt–SACs for HER. The compressive stress regulates the number of charge transfer between (H)–(B)–(MoS₂), weakening the E_ads(H⁺) on the surface of B@MoS₂. Moreover, B doping and compressive stress reduce the bandgap and move the Fermi energy level to the conduction band, improving the conductivity of the pristine MoS₂. Finally, the SiN/B@MoS₂ heterojunction with a -8.6% compressive stress to B@MoS₂ is found to give a \text{Δ}{G}_{\text{H}} of -0.098 eV. This work not only provides an effective strategy for improving the catalytic activity of MoS₂ for HER, but also inspires more researchers to further explore and optimize the potential of metal-free SACs for HERs.

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