In our DFT calculations, we selected three different types of point defects with a relatively low formation energy. (1) S monovacancy: Its formation energy is \({E}_{S}^{vac}\)=1.55 eV 22,47 in Morich limit (deficit of Satoms), and \({E}_{S}^{vac}\)=2.81 eV 22,24,47 in Srich limit (deficit of Moatoms). (2) S divacancy: Its formation energy is \({E}_{2S}^{vac}\)=3.2 eV30,48 in Morich limit and\({ E}_{2S}^{vac}\)=5.44 eV in Srich limit22. Since the formation energy of a S divacancy is approximately twice of that of an S monovacancy, this implies that S monovacancies do not have a strong tendency to merge, which is in contrast to graphene, where divacancies are energetically more favorable than monovacancies30. (3) MoS antisite defect (an Moatom substitutes an Satom): Its formation energy is \({E}_{Mo\to S}^{sub}\)=4.2 eV47 in Srich limit, and \({E}_{Mo\to S}^{sub}\)=6.2 eV47 in Morich limit. Due to the relatively higher formation energy, the antisite defects are likely to form at high temperatures26.
Since the formation energy of a Mo monovacancy is \({E}_{Mo}^{vac}\)=7.2 eV 47 and\({E}_{Mo}^{vac}\)=8.2 eV [7] in Morich limit, and \({E}_{Mo}^{vac}\)=4.9 eV30 in Srich limit, and once a Mo monovacancy is formed, its nearby Satoms have a strong tendency to leave behind vacancies since the S monovacancy formation energy around an Mo monovacancy is only ~ 1.1 eV even under Srich condition30, therefore, Mo monovacancies are not observed experimentally alone, but as clusters of vacancies. Yet the clusters of vacancies, which appear via merging of S and Mo monovacancies, such as: V(MoS2) \({E}_{Mo{S}_{2}}^{vac}\)=8.2 eV 30 and V(MoS3) \({E}_{Mo{S}_{3}}^{vac}\)=4.5 eV30 in Srich limit, are rather unstable especially when MoS2 layer is supported by Au substrate22. This is consistent with the experimental observations, where S monovacancies are frequently observed in all samples, but Mo monovacancies were only occasionally found30. We note that when MoS2 monolayer interacts with the Au substrate, the formation energies of these point defects are slightly higher22. For example, the formation energy of an S monovacancy increases by 7% (\({E}_{S}^{vac}\)=2.81 eV > \({E}_{S}^{vac}\)=3.0 eV), and an S monovacancy in the top sublayer is slightly more stable than in the bottom sublayer.
As the first step, we constructed and optimized the Au(111)/MoS2 samples with a vacuum thickness of 20 Å, constraining the lattice constants. The following six Au/MoS2 samples were constructed (see Fig. 1): (1) Defectfree (PF) MoS2, (2) MoS2 with an S monovacancy in the top sublayer (VT), (3) MoS2 with an S monovacancy in the bottom sublayer (VB), (4) MoS2 with an S divacancy (DV), (5) MoS2 with an MoS antisite defect at the top sublayer (AST), and (6) MoS2 with an MoS antisite defect in the bottom sublayer (ASB). The top and side views of the defectfree sample are shown in Fig. 1 (a), while the samples with a VT and AST of MoS2 layer are shown in Fig. 1 (b) and (c), respectively. The defect sites are indicated by red arrows. The samples with an ASB and a DV of MoS2 layer are shown in Figs. 1 (d) and (e), respectively. Planar charge density distribution around the point defects in the Au/MoS2 samples are shown at the bottom panel in Fig. 1.
To study the effect of defect density on the SBH, we varied the lateral size of the computational cell: The supercells of 3x3, 4x4, 5x5 and 6x6 lattice unit cells of MoS2 monolayer accommodated on Au (111) substrate were constructed. The defect density (and vacancy concentration) per unit area in the constructed samples is given in Table 1. For 3x3 MoS2supercell, we built Au (111) substrates containing 4, 5 and 6 Au layers; but found that the difference in the obtained SBH values was rather minor, thus for the remaining samples with 4x4, 5x5 and 6x6 supercells, we constructed Au (111) substrate with 4 layers. The supercells were relaxed, while the positions of atoms in the two bottom layers of the Au (111) substrate were constrained and the positions of the remaining Au atoms were relaxed.
Table 1
Defect density per unit area for the constructed samples.
Au/MoS2 sample size

Defect density per unit area (1/Å2)

Defect concentration

6x6x4

0.003

1%

5x5x4

0.005

2%

4x4x4

0.008

3%

3x3x4

0.014

6%

Periodic boundary conditions were applied along all the directions, while a vacuum layer with the thickness of ~20 Å was added as a padding along the Zdirection (normal to the Au(111) surface, see Fig. 1) to avoid interactions due to periodic boundary conditions. Since there was a small lattice mismatch along the lateral (X, Y) directions between the lattice constants of primitive unit cells of MoS2 monolayer and that of Au(111) surface, the Au(111) substrate was deformed to eliminate the mismatch. This is a common practice43,49, which permits the application of periodic boundary conditions in DFT calculations. The physical basis for this treatment is that a minor tensile deformation of the metallic substrate by a few percent only leads to a minor change in its electronic band structure and work function. The geometry of the constructed samples was optimized by DFT method using conjugategradient optimization.
Two firstprinciplesbased methods, the projection of electronic band structure and the SM rule, have been frequently used to calculate SBH. We use both methods to calculate SBH and compare and assess their reliability and accuracy. Below, we briefly discuss these two methods.
The method based on projection of electronic band structure
In this method, the SBH value is obtained by identifying the position of conduction band minimum (CBM) of the contact MoS2 layer among the bands of the Au(111) /MoS2 heterojunction. The value of SBH is the distance from the Fermi level to the identified CBM18,37. Hence, to calculate SBH, one needs to obtain the electronic band structure of the Au(111)/MoS2 heterojunction, and that of the contact MoS2 monolayer taken from the Au(111)/MoS2 heterojunction. Since when a freestanding MoS2 layer is accommodated on a substrate, its geometry, and therefore its electronic band structure may be changed, therefore, to calculate its electronic band structure, we take the MoS2 layer from the Au(111)/MoS2 heterojunction by freezing its geometry. By using the frozen MoS2, one can obtain its CBM accurately (see the redcolored band structure in Fig. 2 (a)).
Next, the electronic band structure of contact MoS2 layer is projected onto the electronic band structure of Au(111)/MoS2 heterojunction (see Fig. 2 (a)). When the superimposed electronic bands align, one can identify the minimum of the electronic band of the Au(111)/MoS2 heterojunction by looking for a band that overlaps to a greater extent with the CBM band of the frozen MoS2 monolayer. In Fig. 2 (b), the redcolored bottom conduction band matches with one of the bands of the Au/MoS2 heterojunction (see the blue line in Fig. 2 (b)). The distance from the Fermi level to the identified minimum (which is indicated by the red arrow in Fig. 2 (b)) is equal to the SBH of the of Au(111)/MoS2 heterojunction11,18,40
An important restriction to use this approach is that the monolayersubstrate interactions must be weak enough so that the weak interactions should only perturb the band structure of MoS2 to a small extent. In the case of the Au/MoS2 heterojunction, since the interfacial bonding is attributable to van der Waals interactions18,40, this method is applicable to the Au/MoS2 junction. Therefore, we applied this method for both defectfree and defective MoS2 monolayer. It is noted that point defects introduce new occupied defect states below and unoccupied states above the Fermi level in the band gap of MoS2 monolayer. Since the vacancy produces localized states50, we used the CBM position to obtain the SBH.
In addition to the electronic band structure, partial density of states (pDOS) is a convenient way to illustrate the effect of point defects in the MoS2 layer on the electronic properties of Au/MoS2 junction (see Figs. 4–6). The pDOS is calculated separately for the Mo and Satoms of the contact layer as an average over all the atoms and their corresponding orbitals (five 4dorbitals for Moatoms and three 3porbitals for Satoms). The position of CBM cannot be identified from the pDOS plots with high accuracy since the band edge shape in pDOS plot is often fuzzy. The exact position was taken by using the method based on the projected electronic band structure.
The method based on the SM rule
Another commonly used method to calculate the SBH is based on the SM rule15. According to this rule, the value of SBH between a metal/semiconductor junction is proportional to the difference of metal work function, \({W}_{m}\), and the semiconductor electron affinity energy, \(\chi\): \({\Phi }={W}_{m}\chi .\) For a metal, which is in our case Au(111) substrate, the work function is defined as the difference between its vacuum energy level and the Fermi energy. We obtained \({W}_{Au}\)=5.1 eV from our DFT calculations with PBE XCfunctional, and \({W}_{Au}\)=5.27 eV with PBE XCfunctional and DFTD2 van der Waals correction. It is noted that the calculated values are slightly lower than previously reported values of 5.13 eV and 5.3 eV11,18 since we deformed the Au(111) sample to enable the application of periodical boundary conditions.
The electron affinity energy (EAE), denoted as \({\chi }_{{MoS}_{2}}\), is calculated as the difference between the vacuum energy level (obtained as an asymptotic value of planar averaged electrostatic Hartree potential, which is taken sufficiently far off the monolayer, see Fig. 2 (c)) and the energy level of the conduction band minimum, which is identified by using the calculated electronic band structure of the MoS2 layer. In our case, the\({\chi }_{{MoS}_{2}}\) varies within a certain range around\({\chi }_{{MoS}_{2}}=4.2 eV\) for defective monolayer (see Fig. 7(b) and Tables S1S4 in Supplementary Materials).
To account for the interaction between the MoS2 monolayer and the underlying metallic substrate and the corresponding change in the work function of the metal in the presence of MoS2 layer, the SM rule must be modified. When the MoS2 monolayer and Au substrate is integrated into the Au/MoS2 junction, the equalization of the Fermi levels results in the charge transfer from the metal to the MoS2 monolayer (see Fig. 2 (d), where the charge accumulation and depletion zones at the Au (111)/MoS2 junction are exemplified), which alters the SBH. The charge transfer and its redistribution at the Au/MoS2 junction results in the potential step, \(\varDelta V\), given by \(\varDelta V=\frac{{e}^{2}}{A}\iiint z\varDelta n(x,y,z)dxdydz\), where \(A\) is the contact area (measured within the [X,Y] plane), and \(\varDelta n(x,y,z)={n}_{Au/{MoS}_{2}}(x,y,z){n}_{Au}(x,y,z){n}_{{MoS}_{2}}(x,y,z)\) is the difference between the electronic density of Au/MoS2 junction, \({n}_{Au/{MoS}_{2}}\), (which is illustrated in Fig. 2 (e) for the Au/MoS2 junction containing double Svacancies) and the electronic density of Au substrate, \({n}_{Au}\left(x,y,z\right)\) and that of MoS2 monolayer, \({n}_{{MoS}_{2}}\left(x,y,z\right).\)According to the modified SM rule, which includes the effect of the interface potential step, the SBH value is given by: \({{\Phi }}_{Au/{MoS}_{2}}={W}_{m}\)\({\chi }_{{MoS}_{2}}\)\(\varDelta V\)18,40. The interface potential step is attributed to the reduction in the metal work function due to its contact with the MoS2 monolayer. The change in the work function \({W}_{m}\) is a combined effect due to the rehybridization of dorbitals of Auatoms13, polarization of the metal electrons induced by the MoS2 monolayer51, the “pushback” effect (the displacement of surface electron density around the metallic substrate into the metal by the MoS2 monolayer) due to the exchange (Pauli) repulsion at the interface, which is the main contribution to the interface potential step in the weakly interacting regime40,52,53, the presence of interface dipole moment18 and the surface relaxation of metallic substrate37,40.
The potential step at the interface can be calculated either by using the planar average electronic charge density along the zdirection, \(n\)(z), or by using the planeaveraged Hartree potential defined as \(V\left(z\right)=\frac{{e}^{2}}{A}\iint z\varDelta n\left(x,y,z\right)dxdy.\) According to Farmanbar et al.40, the potential step can be obtained by inspecting the asymptotic values of \(V\left(z\right)\) for the Au/MoS2 junction in the vacuum, which are typically attained within a few Å from the metallic surface at the bottom and the MoS2 layer at the top ( see Fig. 2 (f), where the planeaveraged Hartree potential is shown for the Au(111)/MoS2 junction with double Svacancies). Thus, one can calculate the value of \(\varDelta V\) as the difference of \(V\left(z\right)\) taken between two points located at sufficiently large distance deep in the vacuum (at the points where electrostatic potential \(V\left(z\right)\) converges to constant values). Since the periodic boundary conditions are applied in the DFT calculations, one needs to use dipole corrections along the zaxis to obtain the welldefined potential step in\(V\left(z\right).\)
Comparison of the two methods
To compare these two methods, we plot the calculated SBH values obtained from the method based on the band structure projection (see blue circles in Fig. 3 (ad)) and the method based on the SM rule (see red squares in Fig. 3 (ad)). The results from these two methods show a remarkably similar trend between the SBH and defect type as shown in Fig. 3 (ad) (see also Tables S14 and Figure S5 in Supplementary materials). On average, the difference in the SBH values is ~ 3%, while the maximal difference is 7%. We note that the difference in the SBH values obtained by the two methods in this study are smaller than previously reported 0.68 eV and 0.60 eV40.
Details of DFT calculations
All our calculations were carried out by using density functional theory (DFT) with the generalized PerdewBurkeErnzerhof54 and the projectoraugmented wave (PAW) pseudopotential planewave method 55 for the core electrons as implemented in the Vienna ab initio simulation package (VASP) code 56. For the PAW pseudopotentials, we included 5d106s1, 4p6d55s1, and 3s2p4 as valence electrons for Au, Mo, and S, respectively. For DFT calculations, we used 6 × 6 × 1 Monkhorst − Pack57 kpoint grid for the geometry optimizations, and a planewave basis set with an energy cutoff of 520 eV was adopted. Good convergence was obtained with these parameters, and the total energy was converged to 10− 7 eV per atom. The atomic samples were fully relaxed with a residual force of less than 0.02 eV/Å. Spin polarization was considered in this study. The energy minimization was performed using a conjugategradient algorithm to relax the ions into their instantaneous ground state. The DFT calculations were done with van der Waals corrections using Grimme’s DFTD2 approach as realized in the VASP 56. Dipole corrections to the total energy were used along the direction normal to Au(111)/MoS2 interface for all calculations 58. To avoid spurious interactions between replica of the Au(111)/MoS2 interface, a vacuum region of at least 20 Å is included along the same direction normal to the junction in the supercell.
We note that the application of van der Waals corrections not only leads to more accurate results, but it is crucial for Au substrate: In contrast to other more reactive metallic surface like Mo and Ti, where bonds are formed13, the van der Waals nature of the Au − MoS2 interaction is prevalent. Covalent bonds between Au and S atoms cannot be formed since the Au atom with one selectron has fully occupied dorbitals, and hence only weakly interacts with MoS240. In Fig. 3 (eh), we compare the SBH calculated with (green squares) and without (blue circles) van der Waals corrections. It is evident that application of van der Waals corrections systematically lowers the SBH values by ~ 15% (~ 0.1 eV).
Beyond PBE functional: hybrid HSE XCpotential
There is an uncertainty in the calculated SBHs coming from using PBE functional. In our DFT calculations, we used the PBE functional, but it is wellknown that it underestimates the band gap of MoS2 since it does not take into account the many body effect among electrons, only partially accounts for electronic correlation, and neglects longrange exchange and subtle screening effects24,59. We obtained a direct band gap of \({E}_{g}\)=1.7 eV for MoS2 monolayer using PBE with DFTD2 van der Waals corrections, which is in a good agreement with previous GGA calculations11,20, while calculations based on the GWquasiparticle approximation give \({E}_{g}\)=2.8 eV60,61 and application of hybrid HSE XCpotential results in \({E}_{g}\)=2.2 eV26. Even though the electronic band gap for freestanding MoS2 monolayer is not well known, the results obtained with HSE and GWquasiparticle approximations are in excellent agreement with the experimentally measured optical band gap is \({E}_{g}\)=2.9 eV21,62. It must be admitted that the electronic band gap is a fundamentally different from the optical gap, which is generally measured by photoluminescence experiments63. The optical band gap corresponds to the energy required to create an exciton, while the electronic band gap also requires the breaking of the exciton, and is thus higher due to the exciton binding energy. Exciton binding energies between 0.01–0.5 eV have been reported40. A direct comparison of the PBE vs. GWquasiparticle approximation is not truly fair, as the observed difference includes the exciton binding energy, obtained by using the GWquasiparticle approximation.
We estimated the required corrections when the hybrid density XCfunctional potential is applied. Hybrid functionals mix a fraction of the shortrange part of the HartreeFock (HF) exchange interaction with the local functional. There is a range of hybrid functionals, among them, we selected the Heyd, Scuseria, and Ernzerhof (HSE) hybrid functional64. This hybrid density functional is based on a screened Coulomb potential for the exchange interaction which circumvents the bottleneck of calculating the exact (Hartree–Fock) exchange, especially for systems with metallic characteristics. The main reason for the selection is due to its high accuracy combined with its computational advantages for periodic systems64. Moreover, the conduction band in MoS2 consists of dorbitals and PBE functional has significant limitations in proper description of localized delectron states. Therefore, we complement our DFT calculations with hybrid functional calculations for the band structures, Hartree potential and defect states.