Tunable Electronic Properties of Lateral Monolayer Transition Metal Dichalcogenide Superlattice Nanoribbons

The structural stability and structural and electronic properties of lateral monolayer transition metal chalcogenide superlattice zigzag and armchair nanoribbons have been studied by employing a first-principles method based on the density functional theory. The main focus is to study the effects of varying the width and periodicity of nanoribbon, varying cationic and anionic elements of superlattice parent compounds, biaxial strain, and nanoribbon edge passivation with different elements. The band gap opens up when the (MoS2)3/(WS2)3 and (MoS2)3/(MoTe2)3 armchair nanoribbons are passivated by H, S and O atoms. The H and O co-passivated (MoS2)3/(WS2)3 armchair nanoribbon exhibits higher energy band gap. The band gap with the edge S vacancy connecting to the W atom is much smaller than the S vacancy connecting to the Mo atom. Small band gaps are obtained for both edge and inside Mo vacancies. There is a clear difference in the band gap states between inside and edge Mo vacancies for symmetric nanoribbon structure, while there is only a slight difference for asymmetric structure. The electronic orbitals of atoms around Mo vacancy play an important role in determining the valence band maximum, conduction band minimum, and impurity level in the band gap.


Introduction
Properties of two-dimensional (2D) materials have been the focus of a great deal of experimental and theoretical research over the past few decades due to their unique electronic structure. 2D materials such as graphene, silicene, boron nitride, phosphorene, transition metal chalcogenides (MoS 2 , WS 2 , MoTe 2 , WTe 2 ), etc., show extraordinary mechanical, electronic and optical properties which make them suitable candidates for future optoelectronic and thermoelectric applications different from bulk materials [1][2][3][4][5][6][7][8][9]. Apart from the unique properties of these materials, there are also important areas of emphasis, and aspects that have attracted extensive attention in the academic field, especially transition metal dichalcogenides (TMDCs) materials, which may be important for transport measurements and applications [6,[10][11][12].
Although TMDCs themselves exhibit many unique characteristics, making them powerful candidates for future electronics and sensors, heterostructures or superlattice composed of TMDCs can further achieve the novel electronic properties that the individual materials do not exhibit alone. Given that experimental techniques are available to synthesize (a) TMDCs lateral superlattice [20] and (b) TMDCs nanoribbons [23], it is expected that TMDCs superlattice nanoribbons can also be synthesized. Such a system should offer a richer variety of electronic structure due to the presence of either two cations (such as in MoS 2 /WS 2 ) or two anions (such as in MoS 2 /MoTe 2 ) at ribbon edges.
In this paper, we apply a first-principles method to investigate the structural and electronic properties of lateral monolayer transition metal chalcogenide superlattice nanoribbons. We consider the geometry, edge atom modification and vacancy effect of symmetric and asymmetric armchair (MoS 2 ) 3 /(WS 2 ) 3 and (MoS 2 ) 3 /(MoTe 2 ) 3 nanoribbons. The H-, H-S-and H-O-saturated nanoribbons were studied. The results show that the band gap opens up when these armchair nanoribbons (ANR) are passivated by H, S and O atoms. Especially, the (MoS 2 ) 3 /(WS 2 ) 3 ANR-H-O exhibits remarkable large band gap. The effects of increasing the ribbon width and superlattice period and of biaxial strain on the band gap for the armchair nanoribbons have also been studied, which can provide insight and information for applications in electronic devices.

Method
In this work we employ the density functional theory (DFT) based plane-wave pseudopotential first-principles method as implemented in the Quantum-Espresso (QE) computer package [48,49]. The ion-electron interaction was modeled by using norm-conserving pseudopotentials, except for the W atom, for which an ultrasoft pseudopotential was used [50]. The electron exchange-correlation energy is calculated using the generalized gradient approximation as outlined by Perdew, Burke, and Ernzerhof [51]. The cutoff energy for the plane-wave basis was 60 Ry. The Brillouin zone summation was carried out using the 8 × 8 × 1 k-points grid mesh for the MoS 2 monolayer. A 12 × 1 × 1 k-points grid mesh is used for zigzag nanoribbons and a 8 × 1 × 1 k-points grid mesh is used for armchair nanoribbons. Within the periodic geometry construction, the vacuum space was considered to be more than 12 Å in order to avoid the interaction between neighboring nanoribbons. Atomic geometry optimization was carried out using the BFGS (Broyden-Fletcher-Goldfarb-Shanno) algorithm as implemented in the Quantum Espresso package.
It is well known that DFT, whether within the local density approximation (LDA) or a generalized gradient approximation (GGA), underestimates the electronic band gap of semiconductors. Use of hybrid functionals and/or a quasi-particle theory is required to obtain accurate band gap results. While these considerations are beyond the scope of the present study, it is important to point out that our consistent use of PBE-GGA will provide a consistent set of band gap results across all the structures considered in this work.

Results and Discussion
We will present band structure results with the energy zero set at the Fermi level (E F ) for metallic systems and at the top of the valence band (E V ) for semiconducting systems. We modeled a lateral superlattice (SL) as an artificially periodic system, with a unit cell normal to the monolayer containing a vacuum region of 12 Å. We first optimized the MoS 2 and WS 2 monolayer systems, obtaining the equilibrium lattice constant of 3.186 Å for both. Our calculated direct bandgap of 1.71 eV for MoS 2 and 1.82 eV for WS 2 are in good agreement with reported experimental and theoretical results [12,15,26,32,34,52,53]. The similarity of lattice parameters and different band gaps is suitable to build superlattices. We focus on (MoS 2 ) 3 /(WS 2 ) 3 lateral superlattice (SL), as shown in Figure 1a. The band structure, shown in Figure 1b, indicates that this superlattice is also semiconducting and has direct band gap close to that of monolayer MoS 2 : the band gap is 1.71 eV for the (MoS 2 ) 3 /(WS 2 ) 3 lateral superlattice, with both the valence band maximum (VBM) and conduction band minimum (CBM) located at the high symmetry point K of the monolayer system. The energy bands close to VBM and CBM are relatively flat along ΓY and dispersive along ΓX. The states near the band gap at the high symmetry point K are mainly derived from the d orbital of Mo atom and the p orbital of S atom. nanoribbons. Within the periodic geometry construction, the vacuum space was considered to be more than 12 Å in order to avoid the interaction between neighboring nanoribbons. Atomic geometry optimization was carried out using the BFGS (Broyden-Fletcher-Goldfarb-Shanno) algorithm as implemented in the Quantum Espresso package. It is well known that DFT, whether within the local density approximation (LDA) or a generalized gradient approximation (GGA), underestimates the electronic band gap of semiconductors. Use of hybrid functionals and/or a quasi-particle theory is required to obtain accurate band gap results. While these considerations are beyond the scope of the present study, it is important to point out that our consistent use of PBE-GGA will provide a consistent set of band gap results across all the structures considered in this work.

Results and Discussion
We will present band structure results with the energy zero set at the Fermi level (EF) for metallic systems and at the top of the valence band (EV) for semiconducting systems.

MoS2/WS2 System
We modeled a lateral superlattice (SL) as an artificially periodic system, with a unit cell normal to the monolayer containing a vacuum region of 12 Å. We first optimized the MoS2 and WS2 monolayer systems, obtaining the equilibrium lattice constant of 3.186 Å for both. Our calculated direct bandgap of 1.71 eV for MoS2 and 1.82 eV for WS2 are in good agreement with reported experimental and theoretical results [12,15,26,32,34,52,53]. The similarity of lattice parameters and different band gaps is suitable to build superlattices. We focus on (MoS2)3/(WS2)3 lateral superlattice (SL), as shown in Figure 1a. The band structure, shown in Figure 1b, indicates that this superlattice is also semiconducting and has direct band gap close to that of monolayer MoS2: the band gap is 1.71 eV for the (MoS2)3/(WS2)3 lateral superlattice, with both the valence band maximum (VBM) and conduction band minimum (CBM) located at the high symmetry point K of the monolayer system. The energy bands close to VBM and CBM are relatively flat along ΓY and dispersive along ΓX. The states near the band gap at the high symmetry point K are mainly derived from the d orbital of Mo atom and the p orbital of S atom.

Zigzag (MoS 2 ) 3 /(WS 2 ) 3 Lateral Nanoribbons
Nanoribbons were constructed by cutting the lateral (MoS 2 ) 3 /(WS 2 ) 3 monolayer superlattice. There are zigzag (zz-NR) and armchair (ac-NR) nanoribbons with different chiralities and edge structures. As shown in Figure 2a, for zz-NR we used the rectangular unit as repeat unit cell of length a along x direction and (3 √ 3a plus a vacuum region of 12 Å) along y direction, where a is the lattice constant. There are 3 Mo, 3 W, 6 S atoms in the repeat unit.  Figure 3 shows the structure of armchair MoS2-WS2 lateral superlattice nanoribbon. The vertical red arrow line marks the width of armchair nanoribbon. We use the notation Na-ac-(MoS2)m/(WS2)n-NR to describe a (MoS2)m/(WS2)n armchair nanoribbon. Here the width of the armchair nanoribbon is characterized by the number of atomic layers Na in the width direction. The numbers m and n denote the numbers of MoS2 and WS2 layers in the superlattice period. The x-axis is the superlattice period direction and the y-axis is the width direction. The high-symmetry points Γ, X and Y in the Brillouin zone are marked. The structural stability of a bare nanoribbon was examined by eValuating the edge energy per interface unit cell using the formula Here E ribbon is the total energy of the nanoribbon unit cell, E unit (MoS 2 ) is the total energy of bulk MoS 2 monolayer, E unit (WS 2 ) is the total energy of bulk WS 2 monolayer, E(Mo) is is the total energy of bcc bulk Mo, E(W) is the total energy of bcc bulk W, we have considered E(S) as half of the total energy of a S 2 molecule, n 1 is the number of bulk MoS 2 monolayers, n 2 is the number of bulk WS 2 , n Mo is the number of edge Mo atoms, n W is the number of edge W atoms, and n S is the number of edge S atoms, and L is the number of edge unit cells. For the zz-NR in Figure 2a the calculated edge energy is 0.89 eV. This result indicates that such a nanoribbon, terminated on one side by the metallic atom Mo and on the other side by the non-metallic atom S, is not a stable structure. Other zz-NR geometries can be constructed, as explained in [32], but we have not studied them all here. The study in [32] asserts that only S-terminated MoS 2 zz-NR are stable geometries.
We also estimated the ''interface energy per interface unit cell" between the constituent materials of the superlattice nanoribbon (e.g., between MoS 2 and WS 2 ) by using the formula where n MoS2 and n WS2 are the numbers of MoS 2 and WS 2 units in the supercell used in any of the zigzag (zz-NR) and armchair (ac-NR) nanoribbons. For each of these structures, our computed interface energy is −2.27 eV, indicating that these interface formations are energetically favorable. The structural stability of a passivated nanoribbon was examined by using the energy formula where E passivated ribbon is the total energy of the passivated ribbon, E bare ribbon is the energy of the unpassivated (bare) ribbon, E(H) is taken as the half of the total energy of a H 2 molecule, E(X) is half of the total energy of the X 2 molecule, n H is the number of H atoms, and n X is the number of the X passivating atom. The passivation energy for the zz-NR structure in Figure 2b is −0.59 eV. This clearly suggests that this bare zz-NR in Figure 2a will become stable upon hydrogen passivation.
The high-symmetry points Γ, X and Y in Brillouin zone are marked. Due to lack of real periodicity along the y axis, the energy bands are flat along ΓY and we find dispersive bands along ΓX. As shown in Figure 2, we find that the zigzag (MoS 2 ) 3 /(WS 2 ) 3 nanoribbon is a metal, similar to the results obtained in [32] for a MoS 2 nanoribbon. Clearly, the H passivation of the zigzag (MoS 2 ) 3 /(WS 2 ) 3 nanoribbon does not make the system semiconducting.
3.1.3. Armchair (MoS 2 ) m /(WS 2 ) n Lateral Nanoribbons Figure 3 shows the structure of armchair MoS 2 -WS 2 lateral superlattice nanoribbon. The vertical red arrow line marks the width of armchair nanoribbon. We use the notation N a -ac-(MoS 2 ) m /(WS 2 ) n -NR to describe a (MoS 2 ) m /(WS 2 ) n armchair nanoribbon. Here the width of the armchair nanoribbon is characterized by the number of atomic layers N a in the width direction. The numbers m and n denote the numbers of MoS 2 and WS 2 layers in the superlattice period. The x-axis is the superlattice period direction and the y-axis is the width direction. The high-symmetry points Γ, X and Y in the Brillouin zone are marked.  Both the symmetric and asymmetric armchair nanoribbons may lower their energy via formation of corrugation [33] or the bond reconstruction mechanism, as is widely realized for semiconductor surfaces, particularly the (001) surface of diamond structure materials and the (110) surface of zincblende materials [54]. However, in this work we have not considered any possible reconstructions of geometries discussed in the text.
In order to examine the effect of edge modification on the electronic properties we Both the symmetric and asymmetric armchair nanoribbons may lower their energy via formation of corrugation [33] or the bond reconstruction mechanism, as is widely realized for semiconductor surfaces, particularly the (001) surface of diamond structure materials and the (110) surface of zincblende materials [54]. However, in this work we have not considered any possible reconstructions of geometries discussed in the text.
In order to examine the effect of edge modification on the electronic properties we first focus on the m = n = 3 superlattice structure (MoS 2 ) 3 /(WS 2 ) 3 . Symmetric (7-ac-(MoS 2 ) 3 /(WS 2 ) 3 -NR-sym) and asymmetric (8-  Both the symmetric and asymmetric armchair nanoribbons may lower their energy via formation of corrugation [33] or the bond reconstruction mechanism, as is widely realized for semiconductor surfaces, particularly the (001) surface of diamond structure materials and the (110) surface of zincblende materials [54]. However, in this work we have not considered any possible reconstructions of geometries discussed in the text.
In order to examine the effect of edge modification on the electronic properties we first focus on the m = n = 3 superlattice structure (MoS2)3/(WS2)3. Symmetric (7-ac-(MoS2)3/(WS2)3-NR-sym) and asymmetric (8-  After atomic relaxation, the structure of zigzag nanoribbon does not change much. But the structure of armchair nanoribbons, both symmetric and asymmetric, has changed a lot. Compared to the S atoms, the Mo and W atoms on the edge shrink to the inside of the armchair nanoribbons, and the hexagonal ring structure is twisted and deformed, as shown in Figure 5a,b. The stability of a nanoribbon was examined by evaluating the edge After atomic relaxation, the structure of zigzag nanoribbon does not change much. But the structure of armchair nanoribbons, both symmetric and asymmetric, has changed a lot. Compared to the S atoms, the Mo and W atoms on the edge shrink to the inside of the armchair nanoribbons, and the hexagonal ring structure is twisted and deformed, as shown in Figure 5a,b. The stability of a nanoribbon was examined by eValuating the edge energy formula in Equation (1). With E edge (asym) = +0.03 eV the asymmetric structure in Figure 4b is unstable. In contrast, with E edge (sym) = −0.26 eV the symmetric structure in Figure 4a is clearly stable. Our estimates of edge energy for these MoS 2 /WS 2 superlattice nanoribbons are similar in trend, but different in estimate, compared to the finding in [32] for MoS 2 armchair nanoribbons of different period and width than considered in this work. Using Equation (3) we find that H passivation stabilizes the symmetric as well the asymmetric nanoribbon structures. The stabilizing energy upon H passivation is −8.84 eV and −9.15 eV for the symmetric and asymmetric structures, respectively. That is, the asymmetric structure becomes more stable than the symmetric structure by 0.31 eV. For H-S and H-O co-passivation the relative stability energies of the asymmetric structure over the symmetric structure are, respectively, 0.31 eV and 0.36 eV. Our estimates of the results for the thin MoS2/WS2 nanoribbon are somewhat higher than those for the asymmetric MoS2 nanoribbon of different period and width considered in [34]. Energy gain results in passivating the armchair nanoribbon by H, H and S, and H and O are listed in Table 1.  In contrast to zigzag nanoribbons, armchair (MoS 2 ) 3 /(WS 2 ) 3 nanoribbons exhibit semiconducting property. Semiconducting nature of armchair MoS 2 nanoribbons were previously established in [32,34]. below E V ) are flat for the asym structure. There are also flat bands for the sym structure, but these are further below E V . These differences are due to W and Mo being different chemical species.
Using Equation (3) we find that H passivation stabilizes the symmetric as well the asymmetric nanoribbon structures. The stabilizing energy upon H passivation is −8.84 eV and −9.15 eV for the symmetric and asymmetric structures, respectively. That is, the asymmetric structure becomes more stable than the symmetric structure by 0.31 eV. For H-S and H-O co-passivation the relative stability energies of the asymmetric structure over the symmetric structure are, respectively, 0.31 eV and 0.36 eV. Our estimates of the results for the thin MoS 2 /WS 2 nanoribbon are somewhat higher than those for the asymmetric MoS 2 nanoribbon of different period and width considered in [34]. Energy gain results in passivating the armchair nanoribbon by H, H and S, and H and O are listed in Table 1.

System
Edge Energy (eV)  [34]. We find that there is some difference in unoccupied states for the H-, H-S-and H-O-passivated 7-ac-(MoS 2 ) 3 /(WS 2 ) 3 -NR-sym armchair nanoribbon structures. In particular, the states around 0.75 eV for the H-passivated system have been removed for the H-S-and H-O-systems. The peak above E V around 0.75 eV has a much bigger contribution from the d orbital of edge Mo and W atoms of H-passivated 7-ac-(MoS 2 ) 3 /(WS 2 ) 3 -NR-sym (in Supplementary Materials Figure S2), but there are no contributions from the d orbital of edge Mo and W atoms for the H-S-and H-O-systems in the same energy range above E V (in Supplementary Materials Figures S3 and S4).  To investigate such features, we first studied the band structure of the lateral superlattice (SL) as (MoS 2 ) 3 /(MoTe 2 ) 3 and the alloy superlattice [MoS 2(x) Te 2(1−x) ] 3×3 (x = 0.5) using the same unit cell. The normal SL structure has the two bond lengths arranged periodically, whereas the alloy superlattice has the two bond lengths distributed randomly throughout the structure (we chose S and Te positions in a random manner). The total energy results show that the superlattice geometry (MoS 2 ) 3 /(MoTe 2 ) 3 is energetically favorable and more stable than the separated constituents alloy geometry [MoS 2(x) Te 2(1−x) ] 3×3 (x = 0.5). The band structure and dos results are shown in Figure 6. The (MoS 2 ) 3 /(MoTe 2 ) 3 superlattice is semiconductor with a band gap of 1.14 eV, close to that of monolayer MoTe 2 (1.16 eV). The band gap of [MoS 2(x) Te 2(1−x) ] 3×3 (x = 0.5) is 1.26 eV, larger than that of the (MoS 2 ) 3 /(MoTe 2 ) 3 superlattice. After alloying, the band edge of CBM is slightly increased, so that the band gap increases about 0.12 eV. The energy band near VBM is relatively flat along ΓY and dispersive along ΓX. Due to different bond length distributions there are some differences in the band structure for the two systems. The alloy superlattice has a slightly larger band gap and its energy bands show splitting at symmetry points, notably at the zone center Γ. so that the band gap increases about 0.12 eV. The energy band near VBM is relatively flat along ΓY and dispersive along ΓX. Due to different bond length distributions there are some differences in the band structure for the two systems. The alloy superlattice has a slightly larger band gap and its energy bands show splitting at symmetry points, notably at the zone center Γ.

Zigzag and Armchair (MoS2)3/(MoTe2)3 Lateral Nanoribbons
Zigzag and armchair (MoS2)3/(MoTe2)3 lateral nanoribbons are investigated, with results shown in Figure 7 and Figure 8. After atomic relaxation, the structure of zigzag nanoribbon does not change much. But the structure of armchair nanoribbons has greatly changed. At the edge, compared to the anionic S atoms, the cationic Mo atoms move towards the ribbon and the anionic Te atoms move away from the ribbon, and the hexagonal ring structure is twisted and deformed. The Te atoms relax more than the S atoms.
The armchair (MoS2)3/(MoTe2)3 ribbon is stable in both symmetric and asymmetric structures, with edge energies, respectively, of −1.89 and −2.70 eV per edge atom. As shown in Table 1, the symmetric structure becomes more stable upon passivation, with energies, for H-, H-S and H-O passivations, of −8.48, −1.72 and −3.24 eV. The corresponding values for the asymmetric structure are −8.26, −1.50 and −3.10 eV.
The interface energy between the MoS2 and MoTe2 parts, using equation (2), is −5.89 eV, −3.48 eV and −4.08 eV for the zigzag (Figure 7a), symmetric armchair ( Figure 8a) and asymmetric armchair (Figure 8b) structures, respectively. It is interesting that for the MoS2/MoTe2 nanoribbon system the interface energy shows large structural dependence, while we did not find any noticeable difference for different structures of the MoS2/WS2 nanoribbons. These differences are due to different amounts of the atomic relaxation at the interface and edges for the three types of nanoribbon structures considered here (see Table 2).
The armchair (MoS 2 ) 3 /(MoTe 2 ) 3 ribbon is stable in both symmetric and asymmetric structures, with edge energies, respectively, of −1.89 and −2.70 eV per edge atom. As shown in Table 1 The interface energy between the MoS 2 and MoTe 2 parts, using equation (2), is −5.89 eV, −3.48 eV and −4.08 eV for the zigzag (Figure 7a), symmetric armchair ( Figure 8a) and asymmetric armchair (Figure 8b) structures, respectively. It is interesting that for the MoS 2 /MoTe 2 nanoribbon system the interface energy shows large structural dependence, while we did not find any noticeable difference for different structures of the MoS 2 /WS 2 nanoribbons. These differences are due to different amounts of the atomic relaxation at the interface and edges for the three types of nanoribbon structures considered here (see Table 2).   Table 2. The vertical displacements δ W , δ S , δ Te and the tilt angles ω 1 (for Mo-S bond), ω 2 (for W-S bond), ω 3 (for Mo-Te bond) for symmetric, asymmetric (MoS 2 ) 3 /(WS 2 ) 3 and (MoS 2 ) 3 /(MoTe 2 ) 3 armchair nanoribbons. δ Mo−S , δ W−S and δ Mo−Te are defined as |δ Mo -δ S |, |δ W -δ S | and |δ Mo -δ Te |, respectively (see Figures 5 and 9).   For the armchair (MoS 2 ) 3 /(MoTe 2 ) 3 ribbon structure the edge Te atoms relax much more than the S atoms for the armchair (MoS 2 ) 3 /(WS 2 ) 3 ribbon structure, as shown in Figures 5 and 9. Following the practice adopted for semiconductor surface relaxation [54], we show in Table 2

Symmetric and Asymmetric MoS 2(x) Te 2(1−x) Alloy (x = 0.5) Armchair Nanoribbons
The alloy has the two bond lengths Mo-S and Mo-Te distributed randomly throughout the structure (we chose S and Te positions in a random manner). Compared to bare symmetric and asymmetric (MoS 2 ) 3 /(MoTe 2 ) 3 armchair nanoribbons, the asymmetric MoS 2(x) Te 2(1−x) alloy (x = 0.5) armchair nanoribbon and the symmetric MoS 2(x) Te 2(1−x) alloy (x = 0.5) armchair nanoribbon are still metallic. The results are shown in Figure 12. After alloying, the band edge of CBM of symmetric MoS 2(x) Te 2(1−x) alloy (x = 0.5) armchair nanoribbon (in Figure 12a) is slightly increased than that of symmetric (MoS 2 ) 3 /(MoTe 2 ) 3 armchair nanoribbon (in Figure 8c). Due to different bond length distributions, there are some differences in the band structure for the two systems. ilar to the (MoS2)3/(WS2)3 system. As for the 8-ac-(MoS2)3/(MoTe2)3-NR-asym structure, the states around 0.6 eV for the H-passivated system have been removed for the H-S-and H-O-systems. The peak above EV around 0.6 eV has a much bigger contribution from the d orbital of edge Mo atoms of H-passivated 8-ac-(MoS2)3/(MoTe2)3-NR-asym, but there are no contributions from the d orbital of edge Mo atoms for the H-S-and H-O-systems above EV around 0.6 eV.   Figure 12a) is slightly increased than that of symmetric (MoS2)3/(MoTe2)3 armchair nanoribbon (in Figure 8c). Due to different bond length distributions, there are some differences in the band structure for the two systems.

Effects of Atomic Vacancies in (MoS2)3/(WS2)3 Armchair Nanoribbon
Point defects in a material play an important role in its physical properties. For twodimensional materials, defects, such as impurities and vacancies, can cause local deformation and stress and have impact on their electronic, magnetic and transportation properties [27,[55][56][57][58]. We consider two cases: (1) a vacancy inside the ribbon and (2) a vacancy at the ribbon edge. There is also significant difference between sym and asym structures. Figure 13 shows the bands and dos of symmetric (7-ac-(MoS2)3/(WS2)3-NR-sym) and asymmetric (8-ac-(MoS2)3/(WS2)3-NR-asym) armchair nanoribbons with an interior and an edge S vacancy. Among the edge defects, one is the defect of S connected to Mo atom (S1), and the other is the defect of S connected to W atom (S2). The edge S2 vacancy both in the symmetric and asymmetric nanoribbon produces a much-reduced band gap. The band gap is 0.48 eV and 0.46 eV for inside S vacancy and edge S1 vacancy in the symmetric

Effects of Atomic Vacancies in (MoS 2 ) 3 /(WS 2 ) 3 Armchair Nanoribbon
Point defects in a material play an important role in its physical properties. For two-dimensional materials, defects, such as impurities and vacancies, can cause local deformation and stress and have impact on their electronic, magnetic and transportation properties [27,[55][56][57][58]. We consider two cases: (1) a vacancy inside the ribbon and (2) a vacancy at the ribbon edge. There is also significant difference between sym and asym structures. Figure 13 shows the bands and dos of symmetric (7-ac-(MoS 2 ) 3 /(WS 2 ) 3 -NR-sym) and asymmetric (8-ac-(MoS 2 ) 3 /(WS 2 ) 3 -NR-asym) armchair nanoribbons with an interior and an edge S vacancy. Among the edge defects, one is the defect of S connected to Mo atom (S1), and the other is the defect of S connected to W atom (S2). The edge S2 vacancy both in the symmetric and asymmetric nanoribbon produces a much-reduced band gap. The band gap is 0.48 eV and 0.46 eV for inside S vacancy and edge S1 vacancy in the symmetric (MoS 2 ) 3 /(WS 2 ) 3 armchair nanoribbon, which is similar to bare symmetric (MoS 2 ) 3 /(WS 2 ) 3 armchair nanoribbon (0.47 eV). We can find similar bands and dos characters around the CBM and VBM with bare symmetric (MoS 2 ) 3 /(WS 2 ) 3 armchair nanoribbon. The edge S2 vacancy in the symmetric (MoS 2 ) 3 /(WS 2 ) 3 armchair nanoribbon reduces the band gap from 0.47 eV to 0.20 eV. For both inside and edge S1 vacancies in the asymmetric (MoS 2 ) 3 /(WS 2 ) 3 armchair nanoribbon, the band gap only changes a little, being 0.51 eV and 0.45 eV, respectively. The edge S2 vacancy in the asymmetric (MoS 2 ) 3 /(WS 2 ) 3 armchair nanoribbon reduces the band gap from 0.53 eV to 0.34 eV. The bands and dos change a lot. There are new energy levels in the band gap region. The peak around E V comes from the contribution of the d orbitals Mo and W atoms at the edges around the defect.  Figure 13. The relaxed geometry, band structure and dos of symmetric and asymmetric (MoS2)3/(WS2)3 armchair nanoribbons with a S vacancy: (a) symmetric structure with inside S vacancy, (b) symmetric structure with edge S1 vacancy and (c) symmetric structure with edge S2 vacancy, (d) asymmetric structure with inside S vacancy, (e) asymmetric structure with edge S1 vacancy and (f) asymmetric structure with edge S2 vacancy.  Figure 13. The relaxed geometry, band structure and dos of symmetric and asymmetric (MoS 2 ) 3 /(WS 2 ) 3 armchair nanoribbons with a S vacancy: (a) symmetric structure with inside S vacancy, (b) symmetric structure with edge S1 vacancy and (c) symmetric structure with edge S2 vacancy, (d) asymmetric structure with inside S vacancy, (e) asymmetric structure with edge S1 vacancy and (f) asymmetric structure with edge S2 vacancy. Figure 14 shows the bands and dos of symmetric (7-ac-(MoS 2 ) 3 /(WS 2 ) 3 -NR-sym) and asymmetric (8-ac-(MoS 2 ) 3 /(WS 2 ) 3 -NR-asym) armchair nanoribbons with inside and edge Mo vacancy. There is a clear difference in the band gap states between inside and edge Mo vacancy for symmetric structure, while there is only a little difference for asymmetric structure, inside and edge Mo vacancy both reduce the band gap. The band gap for inside Mo vacancy in symmetric (MoS 2 ) 3 /(WS 2 ) 3 armchair nanoribbons is 0.40 eV. The p orbital of S atom and d orbital of Mo atoms around Mo vacancy have more contribution below the E V . The edge Mo vacancy in symmetric (MoS 2 ) 3 /(WS 2 ) 3 armchair nanoribbons changes the band gap from 0.47 eV to 0.10 eV. The p orbital of S atom around Mo vacancy has more contribution below the E V than that of bare symmetric (MoS 2 ) 3 /(WS 2 ) 3 armchair nanoribbon. Inside and edge Mo vacancies in asymmetric (MoS 2 ) 3 /(WS 2 ) 3 armchair nanoribbons both change the band gap: the value decreases to 0.27 eV and 0.26 eV, respectively. There are new energy levels around 0.28 eV and 0.35 eV for inside Mo vacancies in asymmetric (MoS 2 ) 3 /(WS 2 ) 3 armchair nanoribbon. The peak comes from the contribution of the p orbitals of S atoms around the Mo defect. The new energy level around E V mainly comes from the contribution of the d orbitals of edge W atom, and there is less contribution the p orbitals of the S atoms around the Mo defect. There is strong vacancy-vacancy interaction in the thin-period nanoribbons considered above. In order to get a realistic understanding of the role of a single vacancy for band gap changes, we made bulk vacancy calculations using a large unit cell for MoS 2 including 216 atoms. We created a Mo vacancy (or S vacancy) somewhere in the middle of the unit cell, as shown in Figure 15a,b. Vacancies in neighboring unit cells will be 'reasonably far' to interact and we get vacancy related flat electronic bands inside the bulk band gap, and pdos will allow us to find out the chemical and orbital signature of that band. For Mo vacancy, impurity energy levels around 0.00 eV, 0.35 eV, 0.73 eV are found to be located near the valence band maximum (VBM) and exhibit strong local characteristics, shown in Supplementary Materials Figure  There is strong vacancy-vacancy interaction in the thin-period nanoribbons considered above. In order to get a realistic understanding of the role of a single vacancy for band gap changes, we made bulk vacancy calculations using a large unit cell for MoS2 including 216 atoms. We created a Mo vacancy (or S vacancy) somewhere in the middle of the unit cell, as shown in Figure 15a,b. Vacancies in neighboring unit cells will be 'reasonably far' to interact and we get vacancy related flat electronic bands inside the bulk band gap, and pdos will allow us to find out the chemical and orbital signature of that band. For Mo vacancy, impurity energy levels around 0.00 eV, 0.35 eV, 0.73 eV are found to be located near the valence band maximum (VBM) and exhibit strong local characteristics, shown in Supplementary Materials Figure

Band Gap Variation with Ribbon Width and Period
We now investigate how the band gap changes as we fix the period size and change the width for the armchair (MoS2)m(WS2)n lateral nanoribbon. We change the width Na from 7 to 23 and obtain the relation between band gap and width as shown in Figure 16a. The band gap is oscillating as we increase the width and fix the period as (m, n) = (3, 3), and finally converges to a value about 0.59 eV, which are in agreement with the previous calculations for the armchair MoS2 and WS2 nanoribbons in [26][27][28]30,59]. Similarly, we investigate how the gap changes as we fix the width and change the period. With change in the superlattice period (m, n), the relationship between the band gap and superlattice period is shown in Figure 16b. It is not the same as increasing the width, as the band gap decreases with increasing superlattice period. This is due to reduction in the confinement effect with increase in period size from (m, n) = (3, 3) until (m, n) = (9,9).

Band Gap Variation with Ribbon Width and Period
We now investigate how the band gap changes as we fix the period size and change the width for the armchair (MoS 2 ) m (WS 2 ) n lateral nanoribbon. We change the width N a from 7 to 23 and obtain the relation between band gap and width as shown in Figure 16a. The band gap is oscillating as we increase the width and fix the period as (m, n) = (3, 3), and finally converges to a value about 0.59 eV, which are in agreement with the previous calculations for the armchair MoS 2 and WS 2 nanoribbons in [26][27][28]30,59]. Similarly, we investigate how the gap changes as we fix the width and change the period. With change in the superlattice period (m, n), the relationship between the band gap and superlattice period is shown in Figure 16b. It is not the same as increasing the width, as the band gap decreases with increasing superlattice period. This is due to reduction in the confinement effect with increase in period size from (m, n) = (3, 3) until (m, n) = (9,9).

Bang Gap Variation with Biaxial Strain
Application of strain can tune electronic properties of materials. We examine the effect of biaxial strain on the band gap of the nanoribbons. The electronic band structures of 7-ac-(MoS2)3/(WS2)3-NR-sym, 7-ac-(MoS2)3/(MoTe2)3-NR-sym armchair nanoribbons under tensile and compressive biaxial strain have been studied. We applied different values of biaxial strain by changing lattice constant. The band gap under strain is shown in Figure  17. The energy gap of 7-ac-(MoS2)3/(WS2)3-NR-sym increases under compressive strain and decreases under tensile strain, which is similar to the results obtained previously for monolayer MoS2 [42]. The energy gap of 7-ac-(MoS2)3/(MoTe2)3-NR-sym increases under both the compressive and tensile biaxial strains, which is different from the 7-ac-(MoS2)3/(MoTe2)3-NR-sym system. When the system is stretched or compressed, the relative position changes between the atoms, which affects the bonding properties and the coupling between different orbitals, so the band structure changes. The edge states play an important role in nanoribbon. The Te atom has a higher atomic number and lower electronegativity than the S atom. After atomic relaxation, the structures of 7-ac-(MoS2)3/(WS2)3-NR-sym and 7-ac-(MoS2)3/(MoTe2)3-NR-sym are very different, especially at the edge of the nanoribbon as discussed earlier, resulting in a big difference in the band structure with strain.

Bang Gap Variation with Biaxial Strain
Application of strain can tune electronic properties of materials. We examine the effect of biaxial strain on the band gap of the nanoribbons. The electronic band structures of 7-ac-(MoS 2 ) 3 /(WS 2 ) 3 -NR-sym, 7-ac-(MoS 2 ) 3 /(MoTe 2 ) 3 -NR-sym armchair nanoribbons under tensile and compressive biaxial strain have been studied. We applied different values of biaxial strain by changing lattice constant. The band gap under strain is shown in Figure 17. The energy gap of 7-ac-(MoS 2 ) 3 /(WS 2 ) 3 -NR-sym increases under compressive strain and decreases under tensile strain, which is similar to the results obtained previously for monolayer MoS 2 [42]. The energy gap of 7-ac-(MoS 2 ) 3 /(MoTe 2 ) 3 -NR-sym increases under both the compressive and tensile biaxial strains, which is different from the 7-ac-(MoS 2 ) 3 /(MoTe 2 ) 3 -NR-sym system. When the system is stretched or compressed, the relative position changes between the atoms, which affects the bonding properties and the coupling between different orbitals, so the band structure changes. The edge states play an important role in nanoribbon. The Te atom has a higher atomic number and lower electronegativity than the S atom. After atomic relaxation, the structures of 7-ac-(MoS 2 ) 3 /(WS 2 ) 3 -NR-sym and 7-ac-(MoS 2 ) 3 /(MoTe 2 ) 3 -NR-sym are very different, especially at the edge of the nanoribbon as discussed earlier, resulting in a big difference in the band structure with strain.

Bang Gap Variation with Biaxial Strain
Application of strain can tune electronic properties of materials. We examine the effect of biaxial strain on the band gap of the nanoribbons. The electronic band structures of 7-ac-(MoS2)3/(WS2)3-NR-sym, 7-ac-(MoS2)3/(MoTe2)3-NR-sym armchair nanoribbons under tensile and compressive biaxial strain have been studied. We applied different values of biaxial strain by changing lattice constant. The band gap under strain is shown in Figure  17. The energy gap of 7-ac-(MoS2)3/(WS2)3-NR-sym increases under compressive strain and decreases under tensile strain, which is similar to the results obtained previously for monolayer MoS2 [42]. The energy gap of 7-ac-(MoS2)3/(MoTe2)3-NR-sym increases under both the compressive and tensile biaxial strains, which is different from the 7-ac-(MoS2)3/(MoTe2)3-NR-sym system. When the system is stretched or compressed, the relative position changes between the atoms, which affects the bonding properties and the coupling between different orbitals, so the band structure changes. The edge states play an important role in nanoribbon. The Te atom has a higher atomic number and lower electronegativity than the S atom. After atomic relaxation, the structures of 7-ac-(MoS2)3/(WS2)3-NR-sym and 7-ac-(MoS2)3/(MoTe2)3-NR-sym are very different, especially at the edge of the nanoribbon as discussed earlier, resulting in a big difference in the band structure with strain.

Conclusions
(MoS 2 ) 3 /(WS 2 ) 3 and (MoS 2 ) 3 /(MoTe 2 ) 3 lateral monolayer superlattices, and their zigzag and armchair nanoribbons were studied using DFT calculations. The structural, edge atom modification, vacancy effect and biaxial strain effect of (MoS 2 ) 3 /(WS 2 ) 3 and (MoS 2 ) 3 /(MoTe 2 ) 3 nanoribbons were investigated. The bare zig-zag and asymmetric (MoS 2 ) 3 /(WS 2 ) 3 nanoribbons are unstable, but can be stabilized by H passivation. The bare symmetric (asymmetric) structure is found to be stable (unstable). Both the symmetric and asymmetric structures are stable when passivated by H atoms, or co-passivated by H and S, or by H and O atoms; however, the asymmetric structure is relatively more stable. In contrast to the (MoS 2 ) 3 /(WS 2 ) 3 nanoribbons, zig-gag and armchair (MoS 2 ) 3 /(MoTe 2 ) 3 nanoribbons are stable with or without passivation. Because of large lattice mismatch the band structure of (MoS 2 ) 3 /(MoTe 2 ) 3 superlattice is quite different from that of (MoS 2 ) 3 /(WS 2 ) 3  atom; (f) Mo44 atom; (g) Mo47 atom; (h) S106 atom; (i) S118 atom; (j) S180 atom of MoS 2 SL with a Mo vacancy in the middle, Figure S11: The band structure and dos of MoS 2 SL with a S vacancy in the middle, Figure S12: (a) The geometry of MoS 2 SL with a S vacancy in the middle and the atoms around S vacancy are labeled. The partial density of states (pdos) for atoms around S vacancy: (b) Mo25 atom; (c) Mo27 atom; (d) Mo28 atom; (e) S180 atom of MoS 2 SL with a S vacancy in the middle.
Funding: Jinhua Wang is supported by China Scholarship Council (CSC).

Data Availability Statement:
No data sets were generated or analyzed during the current study.