## 2 Methods  
Spin-polarized density functional theory (DFT) calculations using the SIESTA code [26] were performed to investigate the effects of different types of defects on the structure, electronic, and catalytic properties of 2D ZnO supported by graphene. A generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) functional [27] combined with a double-\(\zeta\) polarized (DZP) basis set was employed. Since a 2D ZnO/graphene system can be referred to as a van der Waals (vdW) heterostructure [3], [PERSON]'s D2 empirical correction [28] was included in the calculations to consider the vdW interaction between 2D ZnO and graphene layers. To generate Grimme potential parameters, _fdf2 grimme_ utility was used. The GGA-PBE functional is known to underestimate the ZnO band gap [29], and the Hubbard \(U\) correction (DFT + \(U_{\text{d}}\) + \(U_{\text{p}}\)) [30] was therefore applied to provide a more correct representation of the electronic structure of 2D ZnO. These parameters, \(U_{\text{d}}\) = 10 eV and \(U_{\text{p}}\) = 7 eV, were selected because they effectively reconciled theoretical predictions with the experimental data [31]. For relaxation-type calculations, a \(k\)-grid of 9 \(\times\) 9 \(\times\) 1 Monkhorst-Pack was used to sample the Brillouin zone. For band-structure calculations, partial density of states (PDOSs), spin density, and charge transfer properties, finer sampling with a \(k\)-grid size of 24 \(\times\) 24 \(\times\) 1 \(k\)-points was utilized. For all calculations, the energy shift was set to 200 meV and the force tolerance was set to 0.02 eV A\({}^{-1}\). A convergence tolerance of 0.02 eV/A was selected as it provided a balance between the computational cost and accuracy in capturing the relevant physical properties of interest [32; 33; 34; 35]. However, to ensure the robustness of the findings, additional calculations were performed with a tighter force convergence tolerance of 0.01 eV/A. The results demonstrate that the adhesion energy (Table S1, Supplementary Material) and band structure of all considered systems remain unchanged (Figures S1-S5, Supplementary Material), suggesting that the initial choice of 0.02 eV/A was sufficient to capture the essential physics and achieve accurate results within the scope of the study. Norm-conserving Troullier-Martins pseudopotentials for Zn, O, C, Fe, and H atoms were generated using the ATOM code [36]. The two-dimensional ZnO/graphene heterostructure was constructed by stacking a 3 \(\times\) 3 ZnO monolayer on top of a 4 \(\times\) 4 graphene. The initial choice of unit cell sizes was based on previous studies [13]. According to the available literature data, the lattice mismatch between the 4 \(\times\) 4 graphene and 3 \(\times\) 3 ZnO was about 0.59%. In this study, a full optimization procedure was performed to determine the appropriate unit cell sizes and achieve an accurate representation of the stacking structure. To simulate the formation of oxygen vacancies and Fe-to-Zn substitutional defects, the starting point was a perfect structure of 2D ZnO with one or two oxygen atoms removed from the lattice, as well as with one or two Zn atoms replaced by Fe atoms. In this study, all atomic coordinates, including those of graphene, were allowed to relax during the calculations. This meant that the local structure and composition of ZnO, such as oxygen vacancies and substitutional Fe defects, could potentially influence the local structure of the graphene. In the final stage of the study, the catalytic activity of pristine and defect-engineered 2D ZnO supported by graphene for the hydrogen evolution reaction (HER) was assessed using the slab model and the (DFT+U)-D2 method. HER was modeled by analyzing the Volmer reaction and predicting the Gibbs free energy of hydrogen adsorption (\(\Delta\)G\({}_{\text{H^{+}}}\)) [37].