The effects of point defect type, location, and density on the Schottky barrier height of Au/MoS2 heterojunction: a first-principles study

Using DFT calculations, we investigate the effects of the type, location, and density of point defects in monolayer MoS2 on electronic structures and Schottky barrier heights (SBH) of Au/MoS2 heterojunction. Three types of point defects in monolayer MoS2, that is, S monovacancy, S divacancy and MoS (Mo substitution at S site) antisite defects, are considered. The following findings are revealed: (1) The SBH for the monolayer MoS2 with these defects is universally higher than that for its defect-free counterpart. (2) S divacancy and MoS antisite defects increase the SBH to a larger extent than S monovacancy. (3) A defect located in the inner sublayer of MoS2, which is adjacent to Au substrate, increases the SBH to a larger extent than that in the outer sublayer of MoS2. (4) An increase in defect density increases the SBH. These findings indicate a large variation of SBH with the defect type, location, and concentration. We also compare our results with previously experimentally measured SBH for Au/MoS2 contact and postulate possible reasons for the large differences among existing experimental measurements and between experimental measurements and theoretical predictions. The findings and insights revealed here may provide practical guidelines for modulation and optimization of SBH in Au/MoS2 and similar heterojunctions via defect engineering.

www.nature.com/scientificreports/ SBH in the Au/MoS 2 contact with the defective MoS 2 monolayers is universally higher than that of the defect-free monolayer. Among the defects considered, Mo S antisite defect and S divacancy significantly increase the SBH, while it is only weakly affected by S monovacancy. Moreover, the defects in the inner sublayer have more influence on SBH than those in the outer sublayer. Finally, an increase in the defect density noticeably increases the SBH.
Our study suggests that the reported variations in the experimentally measured SBH for Au/MoS 2 contact can be accounted (to a certain degree) by the variations in the type, location, and density of point defects in MoS 2 monolayer. However, the predicted SBH values are ubiquitously higher than the experimentally measured values. We suggest that the lower SBH values observed in experiments may be due to the difference in experimental samples. The present study indicates that the value of SBH can be altered via defect engineering in the MoS 2 layer. Our findings provide a guide for tuning the SBH in the Au/MoS 2 heterojunctions.

Computational model
Different types of point (single and double vacancies, antisite defects, etc.) and topological (Stone-Wales) defects can be created in the MoS2 monolayer at the Au/MoS 2 heterojunction (see Table 1). Here, we examined only three types of point defects with low formation energy, which can be experimentally observed with a high probability. The effect of other defects with higher formation energy, such as Mo vacancy, V MoS2 and V MoS3 cluster vacancy and Stone-Wales defects are thus not considered. These three types of point defects are S monovacancy, S divacancy, and Mo S antisite defect (see Table 1). For our DFT calculations, we selected three different types of point defects with a relatively low formation energy: (1) S monovacancy with formation energy of E vac S = 1.55 eV 22,47 in Mo-rich limit (deficit of S-atoms) and E vac S = 2.81 eV 22,24,47 in S-rich limit (deficit of Mo-atoms), (2) S divacancy with E vac 2S = 3.2 eV 30,51 in Mo-rich limit and E vac 2S = 5.44 eV in S-rich limit 22 . Since the formation energy of an S divacancy is roughly twice of S monovacancy, the monovacancies in MoS 2 in contrast to graphene do not have a strong tendency to merge into divacancies 30 . (3) Mo S antisite defect with E sub Mo→S = 4.2 eV 47 in S-rich limit and E sub Mo→S = 6.2 eV 47 in Mo-rich limit. We note that when MoS 2 monolayer interacts with Au substrate, the formation energies of these point defects are slightly higher (for example, the formation energy of S monovacancy increases by ~ 7% 22 ).
Since the formation energy of a Mo monovacancy is E vac Mo = 8.2 eV in Mo-rich limit 47 and E vac Mo = 4.9 eV in S-rich limit 30 , once a Mo monovacancy is formed, there is a strong tendency to form S vacancies from its neighbouring S atoms (since the formation energy of S vacancy around a Mo vacancy is only E vac S = 1.1 eV 30 ). Therefore, Mo monovacancies are not observed experimentally alone, but as clusters of vacancies which appear via merging of S and Mo monovacancies, such as: V MoS2 with E vac MoS 2 = 8.2 eV and V MoS3 with E vac MoS 3 = 7.7 eV in S-rich limit 30 . Yet, the clusters of vacancies are unstable 22 , especially when MoS 2 layer is supported by Au substrate. For this reason, S monovacancies are frequently observed experimentally, but Mo monovacancies are only occasionally found 30 .
As the first step, we constructed and optimized the Au(111)/MoS 2 samples. The following six Au/MoS 2 samples were constructed (see Fig. 1 Fig. 1a, while the samples with a VT and AST of MoS 2 layer are shown in Fig. 1b and c, respectively. The defect sites are indicated by red arrows. The samples with an ASB and a DV of MoS 2 layer are shown in Fig. 1d and e, respectively. Planar charge density distribution around the point defects in the Au/MoS 2 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 3 × 3, 4 × 4, 5 × 5 and 6 × 6 lattice unit cells of MoS 2 monolayer accommodated on Au (111) substrate were constructed. Since defect-free MoS 2 monolayer (PF) and defective MoS 2 monolayers with five different types of defects (VT, VB, DV, AST and ASB) were considered for each supercell, we generated 24 atomic supercells in total. For the defect-free MoS 2 monolayer, we used the 3 × 3, 4 × 4, 5 × 5 and 6 × 6 supercells to verify that the SBH is not affected by sample size. The defect density (and defect concentration) per unit area in the constructed samples is given in Table 2. For 3 × 3 MoS 2 supercell, we built Au (111) substrates containing 4, 5 and 6 Au layers. We found that the difference in the obtained SBH values was rather minor, thus for the remaining samples, we only constructed the Au (111) substrate with 4 layers. All the atom positions were relaxed except for the atoms in the two bottom layers of the Au (111) substrate which kept fixed to mimic an infinite substrate. www.nature.com/scientificreports/ 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 Z-direction (normal to the Au(111) surface, see Fig. 1) to avoid spurious interactions due to periodic boundary conditions. Considering that there is a lattice mismatch (~ 6%) along the lateral (X, Y) directions between the lattice constants of primitive unit cells of MoS2 monolayer and that of Au(111) surface, the metallic substrate was elongated along the lateral directions to eliminate the mismatch. This is a common practice 43,53 , which allows to apply periodic boundary conditions in DFT calculations. The physical basis for this treatment is that small deformation of the metallic substrate leads to a minor change in its electronic band structure and work function. For example, we found that the work function of the Au(111) substrate was reduced only by ~ 2% by the tensile strain. The geometry of constructed samples was optimized by DFT method using conjugate-gradient optimization.
Two DFT-based methods, that is, the method based on the projected electronic band structure and the method based on the SM rule, have been commonly used to calculate the SBH. We used both methods to calculate the SBH and to compare their reliability and accuracy. Below, we briefly discuss these methods.
The method based on the projected electronic band structure. In this method, the SBH is obtained by identifying the position of conduction band minimum (CBM) of the contact MoS 2 layer amongst the bands of the Au(111)/MoS 2 heterojunction. The SBH is the distance from the Fermi level to the identified CBM 18,37 .  www.nature.com/scientificreports/ Hence, to calculate the SBH, one needs to obtain the electronic band structure of the Au(111)/MoS 2 heterojunction, and that of the contact MoS 2 monolayer taken from the Au(111)/MoS 2 heterojunction. In the case when a free-standing MoS 2 layer is accommodated on a substrate, its geometry, and therefore its electronic band structure is altered. Therefore, to calculate its electronic band structure in the Au(111)/MoS 2 heterojunction, we have to take the MoS 2 layer from the heterojunction (while keeping all the atomic positions fixed). By using this MoS 2 monolayer with the fixed contact geometry, one can obtain its CBM accurately (see the red-colored band structure in Fig. 2a). Next, the electronic band structure of contact MoS 2 layer is projected onto the electronic band structure of Au(111)/MoS 2 heterojunction (see Fig. 2a). When the superimposed electronic bands align, one can identify an electronic band of the Au(111)/MoS 2 heterojunction overlapping with the CMB band of the contact MoS 2 monolayer. The two overlapping bands have almost identical shapes. In Fig. 2b, the red-colored bottom conduction band (CBM) of MoS 2 contact monolayer matches one of the electronic bands of the Au(111)/MoS 2 heterojunction (see the blue line in Fig. 2b). The distance from the Fermi level to the minimum of the identified band (which is indicated by the red arrow in Fig. 2b) is equal to the SBH of the of Au(111)/MoS 2 heterojunction 11,18,40 .
The MoS 2 monolayer-substrate interaction places a particular limitation on application of this approach: the interaction must be comparatively weak so that the band structure of MoS 2 is perturbed only to a small extent. This method is applicable to the Au/MoS 2 heterojunction, since the interfacial bonding is attributed to relatively weak van der Waals interaction 18,40 . Therefore, we applied this method for both defect-free and defective MoS 2 monolayer. So far as known, point defects introduce new defect states in the band gap of MoS 2 monolayer: occupied defect states below the Fermi level and unoccupied ones above it. Since the vacancy produces localized states 54 , we used the CBM position (as defined for defect-free monolayer) 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 MoS 2 layer on the electronic properties of Au/MoS 2 junction (see Figs. 4, 5 and 6). The pDOS is calculated separately for the Mo-and S-atoms of the contact layer as an average over all the atoms and their corresponding orbitals (five 4d-orbitals for Mo-atoms and three 3p-orbitals for S-atoms). 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 rule 15 . According to this rule, the value of SBH for a metal/semiconductor junction is proportional to www.nature.com/scientificreports/ the difference of metal work function, W m , and the semiconductor EAE, χ : � = W m − χ. 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 = 5.11 eV from our DFT calculations with PBE XC-functional, and W Au = 5.27 eV with PBE XC-functional and DFT-D2 van der Waals correction. It is noted that the calculated values are slightly lower than previously reported values of W Au = 5.13 eV and W Au = 5.3 eV 11,18 , since we deformed the Au(111) sample to eliminate the lattice mismatch between Au(111) and MoS 2 monolayer to enable the application of periodical boundary conditions. The EAE, denoted as χ MoS 2 , is calculated as the difference between the vacuum energy level (obtained as an asymptotic value of planar averaged electrostatic potential, which is taken sufficiently far off the monolayer, see Fig. 2c) and the energy level of the CBM, which is identified by using the calculated electronic band structure of the MoS 2 layer. In our case, the χ MoS 2 varies within a certain range around χ MoS 2 = 4.21eV for defective monolayer (see Fig. 7b and Tables S1-S4 in Supplementary Materials).
To account for the interaction between the MoS 2 monolayer and the underlying metallic substrate, and for the corresponding change in the work function of the substrate in the presence of MoS 2 monolayer, the original SM rule 15 must be modified. When the MoS 2 monolayer and Au substrate are integrated into the Au/MoS 2 heterojunction, the equalization of the Fermi levels results in the charge transfer from the gold substrate to the MoS 2 monolayer (see Fig. 2d, where the charge accumulation and depletion zones at the Au(111)/MoS 2 heterojunction are exemplified), which alters the SBH. The charge transfer and its redistribution at the Au/MoS 2 heterojunction results in the potential step, V , given by �V = e 2 A z�n(x, y, z)dxdydz , where A is the contact area (measured within the [X,Y] plane), and �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(111)/MoS 2 interface, n Au/MoS 2 , (which is illustrated in Fig. 2e for the Au(111)/MoS 2 heterojunction containing double S-vacancies) and the electronic density of Au substrate, n Au x, y, z and that of MoS 2 monolayer, n MoS 2 x, y, z . According to the modified SM rule, which includes the effect of the interface potential step, the SBH value is given by: 40 . The interface potential step is attributed to the reduction in the metal work function due to its contact with the MoS 2 monolayer. The change in the work function is a combined effect of the rehybridization of d-orbitals of Au-atoms 13 , polarization of the metal electrons induced by the MoS 2 monolayer 55 , the "pushback" effect (the displacement of surface electron density around the metallic substrate into the metal by the MoS 2 monolayer 56 ), the Pauli repulsion at the interface (which is the main contribution to the interface potential step in the weakly interacting regime 40,57,58 ), the presence of interface dipole moment 18 and the surface relaxation of metallic substrate 37,40 .
The potential step at the interface can be calculated either by using the planar average electronic charge density along the z-direction, n(z), or by using the plane-averaged electrostatic (Hartree) potential defined as V (z) = e 2 A z�n x, y, z dxdy. According to Farmanbar et al. 40 , the potential step can be obtained by inspecting the asymptotic values of V (z) for the Au/MoS 2 junction in the vacuum, which are typically attained within a few Å from the metallic surface at the bottom and within a few Å from the MoS 2 layer at the top (see Fig. 2f, where the plane-averaged electrostatic potential is shown for the Au(111)/MoS 2 junction with double S-vacancies). Thus, one can calculate the value of V as the difference of V (z) taken between two points located at sufficiently large distance deep in the vacuum (at the points where electrostatic potential V (z) converges to constant values). Since the periodic boundary conditions are applied in the DFT calculations, one needs to use dipole corrections along the z-axis to obtain the well-defined potential step in V (z).

Comparison of the two methods.
To compare these two methods, we plot the calculated SBH values obtained by using the band structure projection (see blue circles in Fig. 3a-d) and by using the SM rule (see red squares in Fig. 3a-d). The results show a remarkably similar trend between the obtained SBH, and the defect type as shown in Fig. 3a-d (see also Tables S1-S4 and Fig. S5 in Supplementary materials). On average, the difference in the SBH values obtained by these methods 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 40 . Details of DFT calculations. All our calculations were carried out by using density functional theory (DFT) with the generalized Perdew-Burke-Ernzerhof 59 and the projector-augmented wave (PAW) pseudopotential plane-wave method 60 for the core electrons as implemented in the Vienna ab initio simulation package (VASP) code 61 . For the PAW pseudopotentials, we included 5d 10 6s 1 , 4d 5 5s 1 , and 3s 2 3p 4 as valence electrons for Au, Mo, and S, respectively. For DFT calculations, we used 6 × 6 × 1 Monkhorst − Pack 62 k-point grid for the geometry optimizations, and a plane-wave basis set with an energy cut-off 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 conjugate-gradient algorithm to relax the ions into their instantaneous ground state. The DFT calculations were done with van der Waals corrections using Grimme's DFT-D2 approach as realized in the VASP 61 . Dipole corrections to the total energy were used along the direction normal to Au(111)/MoS 2 interface for all the calculations 63 .
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 formed 13 , the van der Waals nature of the Au − MoS 2 interaction is prevalent. Covalent bonds between Au and S atoms cannot be formed since the Au atom with one s-electron has fully occupied d-orbitals, and hence only weakly interacts with MoS 2 40 . In Fig. 3e-h, 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). www.nature.com/scientificreports/ Beyond PBE functional: hybrid HSE XC-potential. 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 well-known that it underestimates the band gap of MoS 2 since it does not take into account the many body effect among electrons, only partially accounts for electronic correlation, and neglects long-range exchange and subtle screening effects 24,64 . We obtained a direct band gap of E g = 1.7 eV for MoS 2 monolayer using PBE with DFT-D2 van der Waals corrections, which is in a good agreement with previous GGA calculations 11,20 , while calculations based on the GW-quasiparticle approximation give E g = 2.8 eV 65,66 and application of hybrid HSE XC-potential results in E g = 2.2 eV 26 . Even though the electronic band gap for free-standing MoS 2 monolayer is not wellknown, the results obtained with HSE and GW-quasiparticle approximations are in excellent agreement with the experimentally measured optical band gap is E g = 2.9 eV 21,67 . It must be admitted that the electronic band gap is fundamentally different from the optical gap, which is generally measured by photoluminescence experiments 68 .
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 and 0.5 eV have been reported 40 . A direct comparison of the PBE vs. GW-quasiparticle approximation is not truly fair, as the observed difference includes the exciton binding energy, obtained by using the GW-quasiparticle approximation. We estimated the required corrections when the hybrid density XC-functional potential is applied. Hybrid functionals mix a fraction of the short-range part of the Hartree-Fock (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 functional 69 . 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 systems 69 . Moreover, the conduction band in MoS 2 consists of d-orbitals and PBE functional has significant limitations in proper description of localized d-electron states. Therefore, we complement our DFT calculations with hybrid functional calculations for the band structures, Hartree potential and defect states.
The effect of spin-orbit coupling. It is well-known that spin-orbit coupling (SOC) is large in the valence band of MoS 2 monolayer 70,71 , thus it may affect the SBH. To verify the effect of SOP on the SBH, we carried out DFT calculation with SOC for the defect-free and defective 4 × 4 × 4 and 5 × 5 × 4 samples in which S monovacancy, VB, and Mo S antisite defect, ASB, located at the bottom sublayer of MoS 2 monolayer were considered. It was found that SOP does not affect the SBH in all the considered cases, which agrees with the results obtained by Szczȩśniak et al. 72 and Chen et al. 73 for defect-free metal/TMD heterojunctions.

Results and discussions
Defect-free sample. First, we examined the effect of different types of point defects on the SBH. To investigate this effect, we used the Au(111)/MoS 2 junction with a defect-free MoS 2 monolayer as a reference, which was compared with the samples containing defects. In Fig. 4a, we plot the pDOS of a defect-free free-standing MoS 2 monolayer, which was calculated as an average over 4d-orbitals of Mo-atoms and 3p orbitals of S-atoms.  Fig. 4b. It should be readily seen that the rearrangement of atomic position in the contact layer due to its interaction with the underlying substrate changes the overall shape of pDOS, but the band gap and the location of the CBM are nearly the same. The pDOS of the Au(111)/MoS 2 heterojunction and the corresponding electronic band structure are shown in Fig. 4c and d, respectively. In addition, we project the electronic band structure of the contact MoS 2 layer onto the Au (111)/MoS 2 band structure (see the red-colored electronic bands in Fig. 4d). The electronic band structure and pDOS are apparently changed due to the interaction of the MoS 2 layer with the underlying Au(111) substrate.
The mid-gap states appear in the band gap of MoS 2 monolayer as shown in Fig. 4c. Direct orbital hybridization occurs between Au-and S-atoms at the Au/MoS 2 interface due to the overlap of their wave functions, while S-atoms mediate indirect orbital hybridization between Au-and Mo-atoms, resulting in formation of mid-gap states 11 . The Fermi level at the interface, which determines the SBH, is now governed by the charge transfer and filling of the mid-gap states. Although the density of mid-gap states is somewhat low, it is sufficient to pin the Fermi level above the middle of MoS 2 band gap, as in an n-type contact 17,22 . Fermi pinning sets the Fermi level close to the MoS 2 CBM, preventing from reaching it.
The position of MoS 2 CBM in the pDOS of the Au(111)/MoS 2 contact is indicated in Fig. 4c. If one compares the pDOS of the free-standing MoS 2 monolayer in Fig. 4a, b with the pDOS of the Au(111)/MoS 2 sample, it is apparent that the position of the CBM edge, which determines the SBH value, is shifted closer to the Fermi level. To accurately pinpoint the CBM edge location, we use the projection method as shown in Fig. 4d, which is consistent with the pDOS estimation. We note that the CBM is located at the Γ point in accordance with previous reports 8, 18,40 . The SBH value for the Au(111)/MoS 2 contact sample with defect-free monolayer obtained with the SM rule based method is reported in Supplementary materials. S monovacancies. Next, VT and VB defects are introduced in the MoS 2 monolayer (see Fig. 4e). An introduction of S monovacancy creates dangling bonds in the neighboring Mo-atoms, which lead to a defect state in the band gap positioned close to the bottom of conduction band (see also the distinct peak in the pDOS of a free-standing MoS 2 monolayer in Fig. 4f). The new defective state is mainly due to the dominant 4d-states of Mo-atoms with only a small mixture of 3p states of S-atoms.
The pDOS for atoms in the MoS 2 layer taken from the Au(111)/MoS 2 sample is fairly similar to that of the free-standing MoS 2 layer (see Fig. 4f for VT defect and Fig. 5b for the VB defect). However, due to the interaction of MoS 2 layer with its underlying substrate, the position of the peak in the band gap corresponding to the defect state shifts closer to the bottom of conduction band.
In the pDOS for Mo-and S-atoms of the Au(111)/MoS 2 sample with the MoS 2 layer containing S monovacancies (see Fig. 4g for the VT defect and Fig. 5c for the VB defect), new states (with small density amplitude) appear in the band gap due to the mixing and hybridization of S-atom orbitals, and to some extent Mo-atom orbitals with the orbitals of surface Au-atoms. The height of the vacancy-related peak somewhat diminishes, while its width broadens. We used the projection method based on the electronic band structure of the Au(111)/MoS 2 www.nature.com/scientificreports/ samples to identify the CBM positions (see Fig. 4h for the VT defect and Fig. 5d for the VB defect, respectively), and found that the SBH increases in the range from ∼ 10 to ∼ 30% due to the S monovacancies. We found that the SBH also depends on the number of S monovacancies per unit area, which will be discussed later.
S divacancies. Next, we calculated the pDOS for S-and Mo-atoms of a free-standing MoS 2 monolayer with S divacancies, which were created by removing S-atoms from both the top and bottom sublayers of MoS 2 layer as shown in Fig. 1e. The pDOS results are shown in Fig. 5e. It is seen that S divacancies result in the three distinct peaks in the band gap located above the Fermi level, with one of them being near the bottom of the conduction band.
The pDOS for S-and Mo-atoms of the MoS 2 monolayer with S divacancies taken from the Au(111)/MoS 2 sample is shown on Fig. 5f.The rearrangement in the atomic positions of the defective MoS 2 (due to its interaction with Au(111) substrate) modifies the pDOS, particularly the shape of peak in the proximity to the bottom of conduction band.
In Fig. 5g, we plot the pDOS for S-and Mo-atoms of the MoS 2 layer with S divacancies accommodated on Au(111) surface. The interaction of the defective MoS 2 layer with the Au substrate changes its pDOS substantially. As can be seen in Fig. 5g, new states, with a low-density amplitude, appear in the band gap around the two divacancy-related peaks. The peaks merge to some extent, forming a double hump shape, while the third peak merges with the bottom of the conduction band.
The applications of the method based on projection of the electronic band structure (see Fig. 5h) and the method based on the SM rule show that in the presence of S divacancies, the SBH increases by ∼ 20-40% as compared to that of the Au (111)/MoS 2 contact sample with a defect-free monolayer. The effect of S divacancy is almost twice as larger as that of S monovacancy, and thus can be approximately considered as a linear superposition of two monovacancies.

Antisite defects. The effect of antisite defects introduced in the top (AST) or bottom (ASB) sublayer of
MoS 2 layer (see Fig. 1c, d) on the pDOS, band structure and SBH of the Au(111)/MoS 2 sample are illustrated in Fig. 6. It is seen that antisite defects markedly change the pDOS of a free-standing MoS 2 monolayer (see Fig. 6e). Five localized defect states occur within the band gap of the free-standing MoS 2 : Two states below the Fermi level and three above it, with one being in the vicinity of the bottom of conduction band (see Fig. 6a).
When the MoS 2 layer is placed on an Au(111) substrate, its electronic structure changes significantly as the result of its interaction with the underlying gold surface. In Fig. 6b, we plot the pDOS for atoms of MoS 2 layer with an AST defect taken from the Au (111)/MoS 2 heterojunction (see also Fig. 6f for MoS 2 with an ASB defect). The geometry and the corresponding pDOS of the contact MoS 2 layer are markedly modified by the underlying substrate: the two peaks of a free-standing layer located above the Fermi level now merge into a single peak for the MoS 2 with an AST defect (see also the corresponding band structure in Fig. 6h). In contrast, three peaks located near the Fermi level now merge into one for the MoS 2 with an ASB defect (see Fig. 6f and the corresponding band structure in Fig. 6h). Contrary to the monovacancy defects, the difference in the pDOS between the AST and ASB defect in the MoS 2 monolayer is considerably larger. www.nature.com/scientificreports/ Even more revealing is the change in the pDOS for S-and Mo-atoms of the MoS 2 monolayer with the ASB defects. As can be seen in Fig. 6c, in the case of MoS 2 layer containing AST defects, the two defect states located above the Fermi level merge into one, while many additional states appear around it within the band gap. However, the shape of pDOS resembles that of the contact monolayer (or the free-standing monolayer). In the case of MoS 2 layer containing the ASB defects, the changes in the pDOS are substantial as compared with the contact (or free-standing) MoS 2 layer (see Fig. 6g). The different defect-related peaks merge with the new states within the band gap and form a broad continuum. This indicates that the interaction of the MoS 2 layer with the ASB defects is stronger than that with the AST defects. We calculated and compared the relative changes in the binding energy and the interfacial distance between the defective MoS 2 monolayer and its defect-free counterpart and found that the effect of ASB defects is noticeably stronger than that of AST defects (see Fig. S7 in Supplementary materials).
The methods based on the projection of electronic band structure (see Fig. 6d and h) and on the SM rule were applied to calculate SBH for MoS 2 layer with antisite defects. It was found that the presence of AST defects increases SBH in the range from ∼ 10 to 25% (the SBH increases in direct proportion to antisite density). However, when ASB defects are present, the effect on the SBH is much more profound: the increase in SBH is in the range of ∼ 40 to ∼ 60% according to the number of the ASB defects per unit area. Fig. 7a, we summarize the obtained SBH results for Au(111)/ MoS 2 heterojunctions with a defect-free MoS 2 monolayer, as well as MoS 2 monolayer with VT, VB, DV, AST, and ASB defects. The impact of ASB and DV is the strongest, and that of VT and VB defects is in the middle, while that of AST defects is the weakest (see also Table S5 in Supplementary Materials).

Comparison of different point defects. In
The SBH is calculated according to SBH = W Au − χ MoS 2 − �V , where W Au is the work function of Au (111), χ MoS 2 is the EAE of MoS 2 monolayer, and V is the step in the plane averaged electrostatic potential, which represents the reduction of the work function of Au substrate in contact with MoS 2 layer. Since W Au of the substrate is fixed, the SBH increases when the value of χ MoS 2 decreases. As can be seen in Fig. 7b, the introduced point defects (especially the antisite defects and double vacancies) reduce noticeably the χ MoS 2 . Besides that, the SBH increases when the V decreases. As can be seen in Fig. 7c, the hosted point defects (except for the AST defects) reduce the ∆V. However, even for the AST defects, the effect of the reduction in the value of χ MoS 2 is stronger than that of an increase in the value of V , thus the overall result is a minor increase in the SBH for these defects.
An interesting question is why the defects reduce χ MoS 2 ? It is known that the EAE is the energy required to transfer an electron from the bottom of the conduction band to the vacuum level. The χ MoS 2 is measured as the energy difference between the CBM and vacuum level, and since the introduced point defects move the CBM position further away from Fermi level, the energy difference (and the corresponding χ MoS 2 ) decreases. As can be seen in Fig. S6c, d (see Supplementary Materials), the introduction of point defects changes the electrostatic Hartree potential profile (especially in their vicinity). Since the minimum value of the Hartree potential rises, the corresponding χ MoS 2 , which is required to transfer an electron from the bottom of the conduction band to the vacuum level, is reduced. Thus, in the presence of point defects, the χ MoS 2 , which is considered as an average over all possible sites of MoS 2 layer, including the defect sites, decreases. The magnitude of the effect depends on both the type of point defects and their density per unit area.
As can be seen from Fig. 7c, all the considered point defects, except for the AST defects, reduce the value of V in the Au (111)/MoS 2 heterojunctions. The degree of reduction in V depends on the type of point defects,  www.nature.com/scientificreports/ which affects the interfacial dipole moment (see Fig. 2d). The smaller is the interfacial dipole, the smaller is the V . We found that both the amount of charge transferred from the Au(111) substrate to the defect-free MoS 2 and the resulting interface dipole moment are rather small, in agreement with previous studies 17,22 . The introduction of point defects further reduces the magnitude of interfacial dipole (see Fig. S6b in Supplementary Materials), and hence that of ∆V, ultimately leading to the larger SBH.
The effect of defect density on SBH. We next investigated how the SBH depends on the defect density.
To illustrate the effect of defect density on the electronic structure of MoS 2 monolayer at Au(111)/MoS 2 heterojunction, we plot the pDOS for Mo-and S-atoms of a MoS 2 monolayer with VT defects (see Fig. 8a) and AST defects (see Fig. 8b) at the different defect densities per unit area. As can be seen in Fig. 8a, the overall shape of pDOS for MoS 2 monolayer with VT defects varies insignificantly with the defect density. The height of the peak within the band gap (and to some extent its width) enhances with an increase in the vacancy density. The distance between the Fermi level and CBM, which is the measure of SBH, increases proportionally with the defect density.
The pDOS for Au(111) /MoS 2 sample with the MoS 2 monolayer containing AST defects is shown in Fig. 8b at the different defect densities. Once more, one can see that at the different defect densities, the pDOS shape is similar: there are well-defined peaks located above and below the Fermi level in the band gap. It is evident that the height of the peaks grows, and their width broadens with the increase in defect density. The higher is the defect density, the more distorted is the pDOS. The similar changes in the pDOS are found for the DV, VB, and AST defects (see Figs. S1-S3 in Supplementary Materials). www.nature.com/scientificreports/ The main effect of defect density on the SBH for these different point defects is summarized in Fig. 7d: the SBH monotonically increases with the defect density: The increase in the SBH is nearly linear for VT and AST defects, while it is strongly non-linear for ASB and DV defects, especially at high defect densities. In general, the higher the defect density, the stronger its impact on the electronic structure of the Au(111)/MoS 2 heterojunction.
The defect densities per unit area for the 3 × 3, 4 × 4, 5 × 5, and 6 × 6 supercells are 0.014, 0.008, 0.005, and 0.003 (1/Å 2 ). To further investigate the effect of defect density, DFT calculations on the SBH for a considerably larger 12 × 12 supercell with the defect density 0.001 (1/Å 2 ) have also been carried out. As expected, the SBH for the VB, DV and ASB has been reduced to 0.58 eV, 0.59 eV, 0.61 eV, respectively. Hence, the calculated SBH values agree with the trends derived from the results using smaller supercells. These results also suggest that the higher the defect density, the higher the SBH.

Discussions
Our study shows that the results obtained by the two methods predict remarkably similar trends between the SBH and defect type, and between the SBH and defect density. On average, the difference in the SBH values is ∼ 3%, while the maximal difference is only ∼ 7%. Interestingly, the difference in the SBH values obtained by these two methods in this study is smaller than previously reported 40 . This may be ascribed to two factors: first, we calculated the metal work function for the deformed Au(111) sample, and second, we used the contact MoS 2 monolayer instead of the free-standing one to calculate the corresponding EAE. Clearly, these two improvements make the method based on the SM rule more accurate.
We note that the experimentally measured values of SBH for Au/MoS 2 heterojunction fall in a broad range between 0.06 and 0.92 eV 1,8,38,39 . Our present study shows that the values of SBH can vary from 0.57 to 0.92 eV, depending on the type, density and location of point defects studied here. Hence, the present study can partially explain the large dispersion observed in experiments. In particular, the defect type and defect density play an important role. For example, the defect-free Au/MoS 2 heterojunction has the SBH of 0.57 eV and while that with ASB defects at a high density can lead to the SBH of 0.92 eV.
It is understood that the MoS 2 samples used in experiments can be quite inhomogeneous, and the type, density, and location of defects in the samples can vary to a great extent, which can result in the large scattering of the SBH values. The previous study 1 has shown that the method used to fabricate the electrode to create the metal/ MoS 2 heterojunction can have a profound effect on the SBH. When a deposition method is used, the deposited ''high energy'' metal atoms can damage the lattice structure of MoS 2 , which can lead to the substantial chemical disorders, namely formation of numerous S and Mo vacancies, and even metallic-like defects (metallic impurities) at the interface 3,15,42 . These chemical disorders can have a profound effect on the SBH. In particular, these metallic-like defects can lead to local Ohmic contacts, and thus can significantly reduce the overall SBH at the metal/MoS 2 junction, which might explain the very low values of SBH observed in some of the experiments. In contrast, when atomically flat metal thin films are transferred onto MoS 2 monolayer (without direct chemical bonding) by using the damage-free electrode transfer technique 1 , the observed interface is effectively free from chemical disorder, and this leads to much higher values of the measured SBHs. Hence, defect engineering, for example, by controlling the type, location, and defect density per unit area can play an effective role in modulating the SBH. www.nature.com/scientificreports/

Conclusions
We performed first-principles calculations to investigate the effects of the type, location, and density of point defects in MoS 2 layer on the SBH of the Au (111)/MoS 2 junction. The values of SBH were calculated by applying two different methods: the method based on the projection of the electronic band structure and the method based on the SM rule. We found that these methods predict the same trend. With a couple of corrections, the two methods can lead to comparable values of SBH. Three types of point defects were studied: S monovacancy, S divacancy, and Mo S antisite defects. For S monovacancy and antisite defects, their presence in the top sublayer and bottom sublayer is differentiated. Overall, the SBH is sensitive to the type, density, and location of point defects in the MoS 2 layer. In general, the SBH for the defective MoS 2 layer is universally higher than its defectfree counterpart, which will lead to a higher contact resistance and a lower electron injection efficiency. Among these defects, we found that the ASB and DV defects significantly increase the SBH, while the effect of VT, VB and AST defects is relatively weaker. Furthermore, the SBH monotonically increases with the defect density initially but gradually slows down. The effect of defect density for VT, VB and AST defects is smaller than that for ASB and DV defects. The present work suggests that the reported dispersion of the experimentally measured SBH for Au/MoS 2 junction can be at least partially accounted by the existence of point defects in MoS 2 monolayer. The present study also suggests that the SBH can be modulated via defect engineering of MoS 2 layer, for example, by controlling the type, location, or density of defects. Hence, our findings can serve as a guide for the control and optimization of the SBH in Au/MoS 2 heterojunctions.

Data availability
Most of the data generated or analyzed during this study are included in this published article and its Supplementary Materials file. The remaining data used and analyzed during the current study is readily available from the corresponding author on reasonable request.