Exploring atomic defects in molybdenum disulphide monolayers

Defects usually play an important role in tailoring various properties of two-dimensional materials. Defects in two-dimensional monolayer molybdenum disulphide may be responsible for large variation of electric and optical properties. Here we present a comprehensive joint experiment–theory investigation of point defects in monolayer molybdenum disulphide prepared by mechanical exfoliation, physical and chemical vapour deposition. Defect species are systematically identified and their concentrations determined by aberration-corrected scanning transmission electron microscopy, and also studied by ab-initio calculation. Defect density up to 3.5 × 1013 cm−2 is found and the dominant category of defects changes from sulphur vacancy in mechanical exfoliation and chemical vapour deposition samples to molybdenum antisite in physical vapour deposition samples. Influence of defects on electronic structure and charge-carrier mobility are predicted by calculation and observed by electric transport measurement. In light of these results, the growth of ultra-high-quality monolayer molybdenum disulphide appears a primary task for the community pursuing high-performance electronic devices.

T he success of graphene 1,2 offers a paradigm for the exploration of novel low-dimensional physical phenomena 3 and physical properties in two-dimensional (2D) crystal systems 4,5 . However, its intrinsic shortcoming lies in its zero bandgap, which strongly hinders its application in logical electronic devices. Among the post-graphene development, 2D semiconducting molybdenum disulphide and other transition metal dichalcogenides 6,7 have recently appeared on the horizon of materials science and condensed matter physics. Monolayer MoS 2 is a direct-gap semiconductor that exhibits a substantially improved efficiency in its photoluminescence 8,9 . Valley polarization occurs due to significant spin-orbit coupling and leads to optical circular dichroism [10][11][12] in the monolayer system. The electronic transport of MoS 2 -based field effect transistors (FETs) shows steep sub-threshold swing of 70 mV dec À 1 (refs 13-15) and a high on/off ratio up to 10 8 (ref. 16). A metalinsulator transition happens when carrier densities reaches 10 13 cm À 2 , which also increases the effective mobility [17][18][19] . Owing to its unique optical and electric properties, MoS 2 is believed to be a promising candidate as a building block for future applications in nanoelectronics and optoelectronics 6 .
Wafer-scale production of atomically thin layers is paramount for MoS 2 to be used as a candidate channel material for electronic and optoelectronic devices [13][14][15][16][17][18][19] . Among the currently available preparation methods, mechanical exfoliation (ME) is deemed less efficient for these large-scale applications, even though it produces the highest-quality samples exhibiting the best electric performance. Physical and chemical vapour deposition 17,[20][21][22][23][24][25] methods are more compatible for the scalable growth of high-quality samples. However, the experimentally attainable mobility is still one or two order-of-magnitude lower than the theoretical value of 410 cm 2 V À 1 s À 1 (refs 13,18,26-29). For back-gated m-MoS 2 FET devices, the highest mobility reported so far reaches 81 cm 2 V À 1 s À 1 for ME sample 28 , 45 cm 2 V À 1 s À 1 for chemical vapour deposition (CVD) 17 ando1 cm 2 V À 1 s À 1 for physical vapour deposition (PVD) 25 . The major scattering mechanism for the mobility deterioration has been recently suggested as due to the presence of plentiful localized band tail states 30 caused by short-range disordered structural defects (such as vacancies 28,31 and grain boundaries 32 ), and Coulomb traps 33,34 . However, the roles played by various defects in electric and optoelectronic properties are yet to be explicitly understood. There have been few investigations reported on point defects, mostly vacancies, and grain boundaries in m-MoS 2 (refs 28,30-32,35,36). These studies are, however, often performed with samples made by preparatory methods, for example, ME or CVD, which essentially limits the scope of those studies.
Here we present a systematic investigation of the point defects in distinctly prepared m-MoS 2 by combining atomically resolved annular dark-field scanning transmission electron microscopy (ADF-STEM) imaging, density functional theory (DFT) calculation and electric transport measurements. We observe, for the first time, that antisite defects with molybdenum replacing sulphur are dominant point defects in PVD-grown MoS 2 , while the sulphur vacancies are predominant in ME and CVD specimens. These experimental observations are further supported qualitatively by the growth mechanism and quantitatively by the defects' formation energies calculations. The DFT calculations, in addition, predict the electronic structures and magnetic properties of m-MoS 2 with antisite defects. We also discuss the influence of defects on the phonon-limited carrier mobility theoretically, and further examine them by electric transport in defective m-MoS 2 -based FETs. Our systematic investigation of point defects, especially antisites, will further deepen our understanding of this novel 2D atomically thin semiconductor and pave the way for the scalable electronic application of the family of atomically thin transition metal chalcogenides.

Results
Statistics of point defects. For the purpose of the analysis of defects and their concentration, we have chosen about ten samples prepared under the optimized fabrication condition (see the Methods section and Supplementary Note 1 for the details of sample synthesis) from each method (ME, PVD and CVD) and then transferred each sample onto at least two TEM grids independently for ADF-STEM characterizations. The crystalline quality and the choice of samples for statistical analysis are presented in Supplementary Figs 1-9. Figure 1 summarized the most significant results obtained from our analysis on these MoS 2 samples. For the PVD specimen, antisite defects with one Mo atom replacing one or two S atoms (Mo S or Mo S2 ) are frequently observed, marked with red dashed circles shown in Fig. 1a, while the dominant defects for the ME and CVD samples are S vacancies with one (V S ) or two (V S2 ) S atoms absent, as marked by green dashed circles in Fig. 1b. As the STEM's Z-contrast mechanism 37 , that is, IBZ 1.6-2.0 (I and Z are the image contrast and atomic number, respectively), predicts, Mo and S atoms can be unambiguously discriminated, with Mo (Z ¼ 44) showing bright contrast and two superposed S atoms showing dim contrast in the lattice of m-MoS 2 . Following a similar argument and quantitative image analysis, various defects, for example, Mo S or V S where the lattice image presents abnormal intensity variation, can be clearly identified individually through direct imaging and their atomic structures further verified by ab-initio calculations (please refer to Supplementary Table 1 for details of each atomic defect).
We show the relative importance of each type of point defects in Fig. 1c,d. Figure 1c presents the total counts of different point defects based on over 70 atomically resolved ADF-STEM images for each type, that is, ME, PVD or CVD MoS 2 samples. It is found that the dominant type of point defects in each sample highly depends on the specific sample preparation method. The V S vacancy is the predominant point defects in ME and CVD samples, with its concentration of about (1.2 ± 0.4) Â 10 13 cm À 2 ( Supplementary Fig. 6), close to the results reported previously 30 . Atomic defects V Mo (one Mo atom missing) and S Mo (one S atom replacing Mo site) were also found, but with much lower concentrations as shown in Fig. 1c. In contrast, the histogram also shows that antisite defects Mo S2 and Mo S are dominant in PVD samples, with their concentrations higher than that of V S . The density of Mo S2 and Mo S reaches (2.8 ± 0.3) Â 10 13 and 7.0 Â 10 12 cm À 2 , corresponding to an atomic percent of 0.8% and 0.21%, respectively (counted on the total number of all Mo and S atoms). Such a defect concentration is surprisingly high if the defects were regarded as impurity doping, which is usually only achieved in degenerate semiconductor 38 (for instance 10 À 2 B10 À 4 ). It is, therefore, of vital importance to understand how they modify the electronic properties of m-MoS 2 as elucidated below.
Structural characterization of point defects. So far, there have been few reports concerning the structures of sulphur vacancies 31,39 and their impacts on the electronic transport properties 30,40 of m-MoS 2 ; by contrast, there is still a lack of detailed knowledge on the antisite defects, which is at such an unexpected high doping level in PVD samples. Hence, we focus more on antisite defects. In Fig. 2a-e, we highlight all the images of the experimentally observed antisite defects in m-MoS 2 (see also Supplementary Fig. 10), which can be grouped into two categories. One is the antisite defects with Mo atom(s) substituting S atom(s), including Mo S , Mo S2 and Mo2 S2 (Fig. 2a-c). The other category is the antisite defects with S atom occupying the site of Mo, namely S Mo and S2 Mo (Fig. 2d,e). The experimental identification of these antisite defects can be further unambiguously supported by the quantitative image simulation based on DFT-predicted atomic structures of all antisite defects. The fully relaxed DFT-predicted atomic structures of antisite defects were shown in Fig. 2k-t, where the atomic displacement and structure deformation are explicitly observable, especially for antisites Mo S2 and S2 Mo . Associated side views of these relaxed structures are available in Fig. 2p-t. The ADF image simulations ( Fig. 2f-j) based on the calculated structures fit quite well with the experimental images shown in Fig. 2a-e, respectively, especially for the off-centre feature observable in antisites Mo S2 and S2 Mo , which can be tentatively attributed to Jahn-Teller distortions.
Energetics of predominant point defects in different m-MoS 2 . The distribution of different atomic defects in m-MoS 2 certainly depends on their preparation process. A full exploration of the growth dynamics requires a comprehensive experiment-theory joint investigation, which is beyond the scope of the present work, although it is of fundamental interest. Here we provide a qualitative explanation, based on our DFT calculations, to reveal the microscopic physical mechanism of the preparation-processdependent defect formation. The formation energy (DE Form ) for all the point defects are calculated and summarized in Table 1, as consistent with a previous report 35 . Here, chemical potentials of elements Mo and S were employed to calculate the formation enthalpy (DH Form ) of defects. To account for a range of different possible reservoirs (for example, bulk element or bulk MoS 2 ), each enthalpy is given with a range, as listed in Table 1. Two widely used DFT codes (CASTEP 41 and VASP (Vienna Ab-initio Simulation Package) 42 ) are adopted to exploit the full range of functionality available and demonstrate the consistency of the calculated results in Table 1.
An ME sample is exfoliated from MoS 2 natural mineral. After the MoS 2 mineral was formed and/or extracted, either element S or Mo of MoS 2 is prone to reach a solid-gas phase equilibrium. Owing to a higher saturated vapour pressure of S, the mineral-form MoS 2 has to release more S than Mo atoms into the gas phase and thus S is prone to be deficient in MoS 2 . Reflecting this fact, vacancies V S and V S2 have the lowest DE Form of 2.12 eV and 4.14 eV, respectively, among all the defects. The formation energies of all the antisite defects are higher than 5 eV, indicating that the S-deficient mineral-form MoS 2 favours the formation of S vacancies, leading to the observation of the most common defect of V S , followed by V S2 , and almost no antisite defect in ME samples.
In a typical PVD process, MoS 2 precursor is sublimated into the gas phase with clusters and atoms, carried by Ar gas (mixed with H 2 ), and then condensed into a solid-phase MoS 2 . Sulphur has a larger saturated vapour pressure so that more S atoms in the gas phase will leave the preparation chamber, thus establishing a S-deficient and Mo-rich condition. These clusters and atoms are highly mobile and are thus prone to form an ordered structure of MoS 2 in the lowest total energy. Considering n þ 1 Mo and 2n-1 S atoms for example, they have two options, namely, forming (i) n MoS 2 units with one Mo S antisite or (ii) n þ 1 MoS 2 units with three V S vacancies. The exact value of n does not affect the energetic difference between these two types of defects. We thus arbitrarily instantiate n as 107, namely 108 Mo and 213 S atoms in total. The total energy of the antisite option is À 1,774.83 eV, while that for the vacancy case is À 1,774.26 eV which is 0.57 eV less stable than the former. A similar relation also applies to antisite defect Mo S2 with an energy gain of 0.92 eV. A sample with one antisite Mo S or Mo S2 shares the same number of Mo and S atoms with another sample that has three or four S vacancies, respectively. The formation energies of antisites Mo S and Mo S2 were, therefore, divided by three and four, respectively, to make  The CVD process is distinctly different from ME or PVD. Extra S vapour is supplied to replace O in the MoO 3 precursor under an S-rich condition. We suspect that there are small amount of residual O atoms taking the position of S atoms in the resulting MoS 2 sheets, due to the competition between Mo-O and Mo-S bonding in the reaction chamber. Our ab-initio calculation, not shown here, suggests that these O atoms are 1.99 eV less stable than corresponding 2S and usually tend to desorb into the gas phase leaving vacancies at the S sites, that is, S vacancies. It is argued that Mo atoms may jump into the S vacancies and form Mo antisites. Despite of the S-rich condition, even if Mo is rich in a certain local environment, Mo atoms may be firmly bonded with oxygen in the precursor, which strongly limits the diffusion of Mo, making the formation of Mo antisite from mobile Mo atom and S vacancy much less likely.
Electronic structures. The electronic structure of point defects plays a crucial role in determining the electric properties of these defective m-MoS 2 . Vacancy V S and its effect on electronic structures have been recently reported 30,35 ; we thus focus on the less-studied antisite defects (please refer to Supplementary Fig. 11 for our ADF imaging and DFT calculation of V S ). Figure 3 shows the theoretically predicted band structures and projected densityof-states of two primary antisites Mo S and Mo S2 . Our results give a bandgap of 1.73 eV for a defect-free m-MoS 2 ( Supplementary  Fig. 12), close to the experimentally observed optical bandgap of 1.8 eV (ref. 8). Defect states with nearly flat band dispersion for Mo S2 and Mo S reside inside the band gap of a perfect m-MoS 2 (Fig. 3a,d). These states mostly comprises the d orbitals of four Mo atoms around the defect. In addition, the orbital hybridization of Mo and S atoms results in extended wavefunctions involving the surrounding atoms, forming a 'superatom' with a radius of roughly 6 Å, as shown in Fig. 3b,c,f.
Magnetic properties of m-MoS 2 have not been reported yet, as it is believed to be a non-magnetic material. Nevertheless, we did  find a local magnetic moment of 2 m B in antisite Mo S , while the values for other defects, for example, V S , V S2 and Mo S2 , are smaller than 0.1 m B , and hence are negligible. The magnetic moment of 2 m B for a Mo S antisite is not localized only on the central Mo atom, with the surrounding atoms contributing roughly 20% of the total moment probably due to the strong hybridization among these atoms in a 'superatom', as shown in the visualized total spin density (Fig. 3e). Detailed distribution of magnetic moment is available in Supplementary Fig. 13. The spin-resolved real-space distribution of a defect-induced state (state 3) marked in Fig. 3d was plotted in Fig. 3f  Carrier mobility in defective samples. As there have been plentiful reports on the transport of ME MoS 2 -based FETs with electron mobility 1B81 cm 2 V À 1 s À 1 (refs 15,16,18,26,28,43,44), we focus on the transport properties of CVD and PVD monolayers. Figure 4a-d presents the output and transfer characteristics of fabricated FETs based on PVD and CVD MoS 2 , respectively. Our transport measurements (in Fig. 4a-d) of defective MoS 2 -based FETs reveal that the PVD and CVD MoS 2 has electron mobility 0.5 and 11 cm 2 V À 1 s À 1 , respectively. All these results are well comparable with the reported mobilities of m-MoS 2 of 1B81 cm 2 V À 1 s À 1 for ME 15,16,18,26,28,43,44 , 5B45 cm 2 V À 1 s À 1 for CVD 17,29 and the reported values of o1 cm 2 V À 1 s À 1 for PVD m-MoS 2 (ref. 25) respectively. Theoretically, we focus on the effect of the defects on the phonon-limited carrier mobilities [45][46][47] . Table 2 lists the calculated effective masses, deformation potentials and estimated mobilities derived based on the predicted electron mobility of 410 cm 2 V À 1 s À 1 in a perfect m-MoS 2 (ref. 48). MoS 2 samples are usually n-type, we thus primarily focus on the electron mobility. It is found that the phonon-limited mobility of electrons flowing in the intrinsic conduction band is, exceptionally, nearly unaffected by the presence of vacancies (V S or V S2 ), but reduced by three times in the samples with antisite defects, whereas the phonon-limited mobility of holes carried by the intrinsic valence band is more sensitive to these defects and reduces roughly three times for vacancy and more than four times for antisite. Both vacancy and antisite are strong electron-scattering centres that the mobility derived from the defect states (d-e and d-h) for either electron or hole is fairly small, mostly smaller than 1 and 10 cm 2 V À 1 s À 1 , respectively. The defect states strongly affect, but not overwhelmingly dominate, the overall carrier mobility of the samples, owing to the relative low density of defects and the strongly localized defect states. On the other hand, in a real FET device the measured mobility can be affected by the contact resistance 45 or the carrier density 19 . Furthermore, the trapped charges would act as a scattering centre 15,40 . A hopping transport caused by localized disorder is also observed 30,33 . Both can effectively reduce the mobility of the device. Nevertheless, our theory shows that the measured mobility is, most likely, correlated with the primary type of defects in a sample.

Discussion
We have to address the possible influence of electron beam irradiation on the formation of atomic defects and to distinguish the native from irradiation-induced defects. The observed antisite defects are believed to be native, not caused by electron beam irradiation. We envisage that two steps are involved in the formation of a Mo S defect from a well-prepared sample, namely the formation of a V S vacancy and Mo adatom, followed by the capture of the Mo adatom by the S vacancy. Although sulphur vacancy could be created by electron beam sputtering 31,39 , the formation of a Mo adatom and adjacent Mo vacancy need substantially high energy transfer from electron irradiation, which is less likely. The in-situ experiments show that the Mo adatoms are very mobile, but rarely jump into S vacancies to form antisite defects (Supplementary Note 3 and Supplementary Figs 15 and 16). On considering these experimental and simulation results, the observed Mo S and Mo S2 antisites should be confidently regarded as intrinsic defects. In terms of S vacancies, as suggested by early studies 30,31 , the concentration of sulphur vacancies may be slightly overestimated due to beam damage even if the microscope works at low accelerating voltage ( Supplementary Fig. 17).
MoS 2 sheets are extensively adopted in electronic devices. These point defects, as localized disorders, are significant scattering centres of carriers, which may reduce the mobility of charge carriers through the intrinsic conduction or valence band, especially for samples with antisite defects. Therefore, growth of ultra-high-quality m-MoS 2 is of crucial importance to fabricate high-performance electronic devices. On the other side, the presence of defects may provide us novel routes to tailor the properties of m-MoS 2 . We predicted theoretically for the first time that antisite Mo S shows a magnetic moment of 2 m B . Our prediction of local magnetic moments may promote further investigations on the magnetic properties of defective MoS 2 monolayers. Given the strong optoelectronic response of MoS 2 layers with Mo S defects, it is a probable material that is capable for optical manipulation of local magnetic moment. If the defect density goes sufficiently high, it may expect an appreciably large magnetic exchange interaction between defects, and thus become a promising model system for the studies of dilute-magnetic semiconductors and 2D magnetism.
Based on these findings, we propose an application-oriented strategy for fabricating atomically thin MoS 2 . In terms of electric   Carrier types 'e' and 'h' denote 'electron' and 'hole' for the original conduction and valence bands in perfect MoS 2 monolayer, and 'd-e' and 'd-h' represent those for the defect states, respectively. m* is carrier effective mass and E is the deformation potential. All mobilities are estimated based on the value for the electron mobility in the perfect monolayer of 410 cm 2 V À 1 s À 1 (ref. 48). The change of elastic moduli of perfect and defective samples is so small, which can be safely neglected. Each defective monolayer consists of only one type of defects (either vacancy or antisite) in a 6 Â 6 supercell, close to an equivalent defect density revealed in the statistics.
applications, extra S should be introduced into the growth process or post-growth treatment, to restrain the formation of antisite defects for PVD specimen or to heal the abundant S vacancies for CVD and ME specimens, whereas in respect to magnetism, varied pressure of S in the PVD growth could produce different densities of Mo Sx antisites that remain to be explored for magnetic applications.
In summary, our systematic investigation of geometric and electronic structures of antisites and vacancies of m-MoS 2 by ADF-STEM imaging and DFT calculation has led to a considerable progress in our understanding of the variation of the electric and magnetic properties induced by these point defects. We have demonstrated that minimizing point defects, especially antisites, is paramount for electric transport applications, while controllably introduced antisites may produce atomic size local magnetic moments. All these results considerably improve the understanding of point defect in atomically thin transition metal dichalcogenides and should benefit their potential applications in optoelectronic and nanoelectronic devices.

Methods
Sample preparations and transfer. ME m-MoS 2 was prepared by micro-cleavage 8 of natural bulk crystal (SPI Supplies) using scotch tapes. The monolayer was identified from the optical contrast of thin flakes under an optical microscope (Zeiss A2m) and then transferred onto copper TEM grids covered with holey carbon films.
CVD monolayers were synthesized through the reduction of precursor MoO 3 by sulphur vapour flow at ambient pressures following the previously reported method 21,32 . PVD MoS 2 monolayers used in this study were synthesized by thermal evaporation of MoS 2 powders (Sigma-Aldrich, 99%) at a temperature of 950°C. Ar (2 s.c.c.m.) and H 2 (0.5 s.c.c.m.) were used as the carriers gases, following the reported method in ref. 25. The pressure of the growth chamber was about 8 Pa and the growth time was usually 10 min. In terms of sample synthesis, the advantages and disadvantages of these methods and liquid-phase exfoliation are compared in Supplementary Table 2. PVD and CVD MoS 2 monolayers were transferred onto the TEM grid as follows: first, the SiO 2 substrates with monolayer samples were covered with polymethyl methacrylate (PMMA) film after spin coating and then dried in air at 120°C for 5 min. The substrates were immersed into the boiling sodium hydroxide solution (1 mol l À 1 ), which was heated up to 200°C to etch away the underneath SiO 2 layers. The floating PMMA film was picked up with a clean glass slide and then transferred into the distilled water for several cycles to wash away surface residues. In the next step, the PMMA film was lifted out by a TEM grid covered with lacey carbon film and then dried naturally in ambient. This TEM grid was heated at 120°C for 5 min in air before immersion into hot acetone for about 24 h, to remove the PMMA. Before the ADF-STEM characterization, all the monolayer specimens on TEM grids were annealed at 200°C in air for 10 min to reduce surface residues and/or contaminations.
STEM characterization and image simulation. Most of the structural characterizations of m-MoS 2 were carried out with a probe-corrected Titan ChemiS-TEM (FEI, USA). We operated this microscope at an acceleration voltage of 80 kV to alleviate specimen damage induced by beam radiation. A low probe current was selected (o70 pA) and the convergence angle was set to be 22 mrad. Under such a condition, the probe size was estimated to be close to 1.5 Å. To enhance the contrast of the sulphur sublattices, the so-called medium-range ADF mode rather than the high-angle ADF mode was used by adjusting the camera length properly. Some experiments (such as Figs 1a and 2a-c) were done with an ARM 200CF (JEOL, Japan), equipped with a cold field-emission gun. The advantage of higher energy resolution (0.3 eV) and smaller probe size (o1.2 Å) provides higher resolution, thus giving rise to sharper contrast of the atomic images. All the experimental images shown in the main text and Supplementary Information were filtered through the standard Wiener deconvolution to partially remove the background noise for a better display (Supplementary Fig. 10).
It should be noted that intrinsic adatom defects were seldom observed experimentally and, therefore, they are not considered here. All the image statistics were done on the clean regions of the examined samples by ADF-STEM. On considering the good homogeneity of these samples prepared under the optimized conditions (see Supplementary Information), it shall not lead any large variations in the analysed defect population.
STEM-ADF image simulations of relaxed antisite defects were done by software QSTEM 49 . The input parameters were set according to the experimental conditions. Probe size, convergence angle and acceptance angle of the ADF detector are critical and accounted for in the image simulation.
DFT calculations. The defect formation enthalpy for the first column of Table 1 was calculated using the total energy method with the plane-wave pseudopotential DFT code CASTEP 41 . The basic methodology is well known and has been widely used before for defect calculations. In this study, the Perdew-Burke-Ernzerhofgeneralized gradient approximation 50 with ultrasoft potentials is used, as supplied in the CASTEP library. In addition, the dispersion interactions were added using the semi-empirical scheme of Grimme 51 . Structural optimizations of both ionic positions and cell vectors are performed using a modified Broyden-Fletcher-Goldfarb-Shanno-like scheme. The calculations were performed in a slab geometry of a 6 Â 6 supercell of m-MoS 2 with a 15 Å vacuum space in the c axis direction perpendicular to the monolayer. A comparative study of the defect formation enthalpy and energy, together with the electronic and magnetic properties were also done by VASP simulation code 42 using the same slab model. The projector augmented-wave method 52 combined with a plane wave basis is adopted in the calculations. The energy cutoff for plane wave is 400 eV in structural relaxation and increases to 500 eV while calculating the energy and electronic properties. The optB86b exchange functional 53 together with the vdW correlation 54,55 was adopted for exchange-correlation functional. The Brillouin zone of the supercell is sampled by a 3 Â 3 Â 1 k-mesh. All these structures are fully relaxed until the residual force for each atom is less than 0.02 eV Å À 1 .
Estimation of formation energy and enthalpy. The formation energy was defined as,  Table 1, the former one was computed by choosing m Mo and m S equal to m* Mo and m* S , respectively, indicating that the removed (added) atoms come from (go to) the pure bulk form of Mo and S. For the latter value, we set m Mo ¼ m Ã Mo þ 0:5ÂDH MoS2 and m s ¼ m Ã s þ 0:25ÂDH MoS2 . In this case, the source and drain of defect atoms are pure MoS 2 ML. For antisite defects Mo S2 and S2 Mo , both schemes give the same result.
Estimation of 'phonon-limited' carrier mobility. In 2D, the carrier mobility is given by the expression [45][46][47] where m Ã e is the effective mass in the transport direction and m d is the average effective mass determined by m d ¼ ffiffiffiffiffiffiffiffiffiffiffiffi m Ã x m Ã y p . The term E 1 represents the deformation potential constant of the valence-band maximum for holes or conduction-band minimum for electrons along the transport direction, defined by Here DV i is the energy change of the i th band under proper cell compression and dilatation, l 0 is the lattice constant in the transport direction and Dl is the deformation of l 0 .
FET fabrication and transport. The monolayer MoS 2 -based FET devices are fabricated through the following process. First, source (S) and drain (D) electrodes of the devices were defined via e-beam lithography and a 5/45 nm Ti/Au film was then evaporated followed by a standard lift-off process. In addition, the back-gated MoS 2 FETs were then finished. Second, the top-gated devices were begun with forming gate insulator. Gate oxide layer (30 nm HfO 2 film) was grown under 90°C through Atomic Layer Deposition (Cambridge NanoTech Inc.). Lastly, the gate electrode window was also defined by e-beam lithography, followed by evaporation of 5 nm Ti and 45 nm Au thin film, and the top-gated MoS 2 FETs are finished after a lift-off process. The as-fabricated devices were measured through Keithley 4200 semiconductor analyser on a probe station at room temperature and in air. An example of the device architecture is shown in Supplementary Fig. 18.