Tunable band gaps and optical absorption properties of bent MoS2 nanoribbons

The large tunability of band gaps and optical absorptions of armchair MoS2 nanoribbons of different widths under bending is studied using density functional theory and many-body perturbation GW and Bethe–Salpeter equation approaches. We find that there are three critical bending curvatures, and the non-edge and edge band gaps generally show a non-monotonic trend with bending. The non-degenerate edge gap splits show an oscillating feature with ribbon width n, with a period \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta n=3$$\end{document}Δn=3, due to quantum confinement effects. The complex strain patterns on the bent nanoribbons control the varying features of band structures and band gaps that result in varying exciton formations and optical properties. The binding energy and the spin singlet–triplet split of the exciton forming the lowest absorption peak generally decrease with bending curvatures. The large tunability of optical properties of bent MoS2 nanoribbons is promising and will find applications in tunable optoelectronic nanodevices.

strains, and spontaneous emission enhancement 31 , which is explored for ultracompact single-photon quantum light emitters.
Mechanical bending can provide effective LNS on nanoribbons, as shown by Yu et al. 32 and Nepal et al. 33 It was found that the bending induced shifting of edge bands and the charge localization of top valence bands can mitigate or remove the Fermi-level pinning and change the conductivity both along and perpendicular to the width direction of doped nanoribbons. Conduction along the width direction could be realized by attaching electrodes to the edges. Since edge band positions in band structures and edge band gaps are important for the optical absorption of nanoribbons, it is appealing to show how the edge bands will evolve with bending for varied widths of nanoribbons and how this will modify the optical properties. In this work, with density functional theory (DFT) [34][35][36][37][38][39] and many-body perturbation G 0 W 0 computations 40,41 , we systematically investigate the band structures and band gaps of semiconducting armchair 1L MoS 2 nanoribbons with widths from 1.3 to 3.6 nm under different bending curvatures. It is found that the evolution of the edge gap has more features and shows a nonmonotonic trend with bending curvatures. The phenomenon is correlated with the complex strain patterns experienced in the bent nanoribbons. Furthermore, we use the GW + BSE approach to calculate the optical absorption spectra and reveal a large tunability of optical absorption by bending nanoribbons.

Results and discussion
Band gap tunability. The nanoribbon (denoted as AnMoS 2 ) is formed by cutting from monolayer hexagonal MoS 2 and its two armchair edges are hydrogen passivated (Fig. 1a-c). n represents the number of MoS 2 units in one repeating unit along axis c. We study AnMoS 2 nanoribbons with n from 9 to 24 and widths from 1.3 to 3.6 nm. The band structures of nanoribbons show a large tunability with bending. As shown for A13MoS 2 (Fig. 1d), with an increase in bending curvature κ ( κ = 1/R , where R is an average curvature radius, Supporting Fig. S1), the two nearly degenerate conduction bands C1 and C2 slowly approach to the Fermi energy, while the two valence bands V1 and V2 approach to the valence band continuum (VBC) and eventually merge into it at R = 10 Å . C1, C2, V1 and V2 are mostly developed from the d orbitals of the edge Mo atoms 42 . Both EG (edge band gap, see Fig. 1e and caption) and NEG (non-edge band gap, Fig. 1f) can change up to or over 50% with bending curvatures. There are three critical curvatures, namely κ 0 , κ c1 and κ c2 , dividing curvatures into four Regions I, II, III, and IV, shown in Fig. 1e,f. For A13MoS 2 , κ 0 = 0.04/Å ( R = 25Å ), κ c1 = 0.0625/Å ( R = 16 Å ) and κ c2 = 0.100/Å ( R = 10 Å ). With curvatures from zero to κ 0 (Region I), EG is almost unchanged, and NEG slightly increases. From κ 0 to κ c1 (Region II), EG slightly decreases, while NEG increases further. From κ c1 to κ c2 (RegionIII) , EG remains nearly constant. Region IV is for κ > κ c2 . Both in Regions III and IV, NEG decreases. At κ c1 , the border line between Regions II and III, NEG shows a hump (maximum). At κ c2 , V1 and V2 merge into VBC and EG turns to a quick decrease. All nanoribbons AnMoS 2 with n from 9 to 23 show similar features (Supporting Figs. S2, S3). However, with increasing width (larger n), all κ 0 , κ c1 and κ c2 become smaller and Regions I-III gradually merge, and the NEG hump feature is reducing. Eventually, for A24MoS 2 , EG shows nearly a constant behavior followed by a decrease with curvatures, while NEG shows a monotonic decrease. As shown in Fig. 1e,f, the trends of gap with curvatures from PBE 34 , SCAN 35 , TASK 36 , mTASK 37 , HSE06 38 and G 0 W 0 40,41 are nearly the same, although the gaps themselves are not. PBE underestimates the band gaps, since it only has the ingredients of the local density and its gradient, without explicit inclusion of the nonlocal exchange effect, a factor important for an accurate description of band gaps. At meta-GGA level, SCAN can include some nonlocal exchange effects through the orbital dependent ingredient, and slightly improves the gaps. TASK with more nonlocality in the exchange part than SCAN, further improves the results, especially for EG. mTASK further increases the nonlocality in the exchange over TASK by lifting the tight upper bound for one-or twoelectron systems and lowering the limit of the interpolation function f x (α) , resulting in a better description for low dimensional materials. For d-orbital MoS 2 nanoribbons, mTASK underestimates the gaps more than HSE06 does.

Complex local strains in nanoribbons.
The strains in bending nanoribbons are highly non-uniform, complex, and closely related to the observed tunability of band gaps. For A13MoS 2 , the strain in the xy-plane for the middle Mo atom layer (SXYM, Fig. 2a) steadily decreases and gets more negative (compression) with R from ∞ to 16 Å , while the strain along the z direction for this middle Mo atom layer (SZM, Fig. 2c) increases and gets more positive (tensile strains) with R from ∞ to 16 Å . After the critical κ c1 = 0.0625/Å ( R = 16 Å ), SXYM mostly increases (Fig. 2b) and SZM mostly decreases (Fig. 2d). The length of vector c of the supercell (LC) (Supporting Fig. S4) of the nanoribbon also reaches the maximum at κ c1 , consistent with the turning of SZM at κ c1 . Since the relaxed structure of A13MoS 2 nanoribbon is nearly flat for R > 16 Å or κ < κ c1 = 0.0625/Å (Supporting Fig. S5), and SXYM is mostly negative for R > 16 Å , as in Fig. 2a, the nanoribbons experience a compression along the width direction, leading to a tensile expansion in the ribbon periodic direction, as expected for materials with a positive Poisson's ratio. So, LC increases up to κ c1 = 0.0625/Å ( R = 16 Å ). Furthermore, for R < 16 Å or κ > κ c1 , the nanoribbon begins to substantially bend outwards. The compression along the width starts to release and the nanoribbon is gradually getting tensile strains in the width direction, as in Fig. 2b (the gradually positive strains of SXYM). At the same time, the expansion along the cell vector c starts to reduce and LC gradually becomes less and less.
The strain in the xy-plane for the outer S atom layer (SXYOS) and the strain along the z direction for the outer S atom layer (SZOS) (Supporting Fig. S6) have the similar change trends with SXYM and SZM, respectively. The strain in the xy-plane for the inner S atom layer (SXYIS) is mostly compressive and shows mostly a decrease trend with κ , while the strain along the z direction for the inner S atom layer (SZIS) shares an approximately similar change trend with SZM, with more complex compressive and tensile patterns. For AnMoS 2 with even n, i.e., A12MoS 2 (Supporting Fig. S7 www.nature.com/scientificreports/ trends to those of A13MoS 2 , while the strains along the z direction (SZM, SZOS, and SZIS) have more complex patterns, due to the asymmetrical structures of the two edges. The narrow A11MoS 2 shows an additional feature. In Region III ( κ from 0.095 to 0.133/Å , Supporting Figs. S3, S8), EG shows a sudden increase within the range of κ from 0.1 to 0.125/Å . This relates to the special arrangement of edge atoms and strain patterns in the nanoribbon. For example, at κ = 0.111/Å ( R = 9 Å ), the four Mo atoms on the two edges move along the z direction dramatically (Supporting Fig. S9) and the strains associated with the four atoms are significantly large.
Non-degenerate splits of edge bands. The PBE value of NEG of flat nanoribbon shows a monotonic decrease with widths and almost reaches 1.7 eV (the PBE value of monolayer MoS 2 ) at A24MoS 2 , as shown in Fig. 3a, while EG shows a vibrating feature with widths with a period n = 3 (Fig. 3b). EG as a function of ribbon width under different curvatures is shown in Fig. 3c,d. As can be seen for both PBE and SCAN, under low curvatures ( κ ≤ 0.04/Å) , EG still basically follows the same periodicity. However, for a large curvature, EG shows no obvious periodicity, although having a less regular oscillating feature. The bending space in nanoribbons will drastically alter the distribution and symmetry of wavefunctions, as well as the electronic band structures. For flat nanoribbons, the same period n = 3 is shown for the non-degenerate splits of edge bands (Fig. 3e,f), namely, E C = E C2 − E C1 , the energy difference between C2 and C1 at the Ŵ point, and E V = E V1 − E V2 , the energy difference between V1 and V2 at the Ŵ point. For flat nanoribbons AnMoS 2 with n = 9, 12, 15, 18, 21, and The results from PBE, SCAN, TASK, mTASK, HSE06 and G 0 W 0 are shown, and have nearly the same trend. EG is defined as the energy difference between C1 and V1 at the Ŵ point, and NEG as that between C3 and V3. When V1 and V2 are merged into the valence band continuum, NEG is the difference between C3 and V1. www.nature.com/scientificreports/ 24 (i.e., n = 3p , where p is an integer), both E C and E V are minimal, showing nearly degenerate edge bands. The flat nanoribbons with other n values show non-degenerate edge bands around the Ŵ point, with larger E C and E V for narrower nanoribbons (smaller n). For the same nanoribbon, E V is approximately three times E C . Note that the flat nanoribbons with widths n = 3p have larger EG values than those of neighboring n's. This differs from the previous study 43 of n = 3p − 1 without hydrogen passivated nanoribbons. Hydrogen (H) passivation does not change the oscillating feature of EG with widths n , but shifts the maximum n 's by + 1. As shown in Fig. 3g for the flat A13MoS 2 nanoribbon, the distance u, which is the horizontal distance between the outmost Mo atom and the next outmost Mo atom, is approximately equal to the distance v, the horizontal distance between the outmost Mo and H atoms. For other flat AnMoS 2 nanoribbons, the situation is also similar. For non-passivated nanoribbons, the distance u is usually slightly less, since the outmost Mo atom will relax more towards the ribbon center. So, adding H atoms for a flat nanoribbon of width n will make its effective width increased (Supporting Fig. S10) and approximately equal to the width m of the nanoribbon without H passivation, where m = n + 2 . The H passivated nanoribbon of width n = 3p has a high EG value, so does the unpassivated nanoribbon of width m. Since m = n + 2 = 3p + 2 = 3 p + 1 − 1 = 3p ′ − 1 , where p ′ is also an integer. This basically explains the "− 1" difference in the maximum n's for nanoribbons with and without H passivation.  NEG approximately shows a monotonic decrease trend with n, while EG shows an oscillating feature with a period n = 3 . E C = E C2 − E C1 , defined as the energy difference between C2 and C1 at the Ŵ point (see Fig. 1d), and E V = E V1 − E V2 , the energy difference between V1 and V2 at the Ŵ point. In (g), the distance u, which is the horizontal (ribbon width direction, or cell-vector a direction) distance between the outmost Mo atom and the next outmost Mo atom, is approximately equal to the distance v, the horizontal distance between the outmost Mo and H atoms. www.nature.com/scientificreports/ Figure 4 shows the nondegenerate splitting of the lower ( E V ) and the upper ( E C ) edge band gap as a function of bending curvature for different nanoribbon width from SCAN. As can be seen, the splits E C and E V are robustly kept when bending is applied. For example, for A13MoS 2 , the split E V is kept from κ = 0 to 0.091/Å , until V1 and V2 merge into VBC at κ = 0.10/Å . Bending can even enhance the split E V ( E C ), as can be seen for A11MoS 2 (A9MoS 2 ) from κ = 0 to 0.091/Å ( 0 to 0.10/Å ), and induce the split E V , as can be seen for A12MoS 2 from κ = 0 to 0.10/Å . The splits E C and E V on flat nanoribbons may be due to the quantum confinement effect along the width direction. Bending can couple with the quantum confinement effect and enhances the splits, especially for narrower nanoribbons.
In Fig. 4, for about κ ≤ 0.02/Å , the ordering of the curves for different width n is basically the same as that of the flat case, implying that E V and E C still have the same periodicity and oscillating behavior as that of the flat nanoribbons. When increasing curvatures, the ordering of the E V curves for wider nanoribbons (n from 18 to 24) is also approximately unchanged, before the lower edge bands merge into the valence continuum at about κ = 0.07/Å . However, the orderings of the E V and E C curves for narrower nanoribbons (n from 9 to 17) all show a more complex feature with larger curvatures. This is echoed with the complex changing feature of EG vs. large curvatures (supporting Fig. S2). In those large curvature regions, E V and E C , especially for narrow nanoribbons, will not follow the same periodicity and oscillating behavior as that of the flat ones.  (Table 1), more and more bright exciton states appear within the energy range below the fundamental gap, and the bending nanoribbon shows a broad absorption within this energy range.
At R = ∞(flat), peak A ′ consists of two degenerate (energy difference < 3 meV) exciton states at energy 0.57 eV, and each of them is due to a combination of four transitions, namely, V1 to C1, V1 to C2, V2 to C1, and V2 to C2, mainly around the Ŵ point. The wavefunction of the exciton is shown in Fig. 6. The binding energy of these excitons is 1.41 eV, which is bigger than that (0.96 eV 20 ) of the lowest exciton in 1L MoS 2 , due to the further reduced screening and enhanced electron-electron interaction in the one-dimensional nanoribbon. Peak B ′ mainly consists of two degenerate exciton states at 0.94 eV. One is mainly due to the transition around Ŵ from V3 to C1, and the other from V3 to C2. There are two degenerate and relatively weak exciton states at 1.01 eV, merged in the right bottom of peak B ′ . They are the excited states of the excitons forming peak A ′ . Peak C ′ consists of two sets of exciton sates. The first set, with an energy at 1.11 eV, is two degenerate exciton states mainly corresponding to the transition V4 to C1 (or V4 to C2) around Ŵ , while the second set, with a slightly higher energy at 1.17 eV, is two degenerate exciton states mainly due to V3 to C1 (or V3 to C2) around k points slightly away from, but near Ŵ . The wavefunction of this second set exciton has a nodal feature, indicating that it is the excited state of the exciton forming peak B ′ at 0.94 eV. Peaks D ′ and E ′ are relatively weak and mainly due to transitions from lower valence bands (V5 or V6) to upper edge bands (C1 or C2), and their wavefunctions in k space have more complex nodal features. Peak F ′ is mainly due to transition V1 → C3 around Ŵ . This exciton has an energy very close to the fundamental gap and hence a small binding energy of 0.09 eV.
At R = 13 Å , the excitonic composition of peak A ′′ is similar to that of peak A ′ . However, due to bending, the two originally degenerate exciton states red shift to 0.5 and 0.55 eV, respectively. This makes the overall peak A ′′ broadened and located at a lower energy than peak A ′ . Peak B ′′ becomes complex. Overall, it is still similar to peak B ′ , and consists of exciton states corresponding to transitions around Ŵ for V3 → C1 (or V3 → C2), and the excited states of excitons forming peak A ′′ . The overall peak position of B ′′ red shifts, compared to peak B ′ . Similarly, peaks C ′′ to G ′′ are complex and involve mixtures of transitions involving lower valence bands (V3 to V6) and upper edge bands (C1 and C2).
At R = 9 Å , peaks A ′′′ and B ′′′ are, respectively, due to two excitons, each of which is mainly from the mixed transitions involving V2 → C1, V3 → C1, V2 → C2, and V3 → C2 around Ŵ . The exciton corresponding to peak A ′′′ is at 0.5 eV, while the one for B ′′′ is at 0.57 eV, and the energy difference may be mainly due to bending effects. Peaks A ′′′ and B ′′′ are related to, but different from peaks A ′ and A ′′ . At R = 9 Å , the lower edge bands, named as V1 and V2 before merging into VBC and named as V2 and V3 after the merging, are already merged into VBC. As can be seen (Fig. 5k), V1, V2 and V3 are very close to each other around Ŵ , and the transition V1 → C1/C2 has very small contributions (< 4%) to peaks A ′′′ and B ′′′ . In fact, the exciton states corresponding to transitions V1 → C1/C2 are moved to 0.65 and 0.7 eV, and are relatively weak and embedded in the valley between B ′′′ and C ′′′ . Yu et al. 32 showed that the charge density of the continuum valence band maximum (V3 at Ŵ before the merging) changes from a nearly uniform and symmetric distribution along the nanoribbon width for flat case to a nonuniform distribution concentrating in the middle region of the nanoribbon for large bending curvatures, resulting in an asymmetric wavefunction over the inner and outer sulfur layers. This may also change the parity of the wavefunction. On the other hand, both the upper and lower edge bands are located mainly near the two edges, thus less affected by the bending. The symmetry and parity of wavefunctions involved is important to the formation of optically active bright excitons 44 . So, the band inversion of V1/V2 with V3 around Ŵ at R = 9 Å may change the symmetry and parity of wavefunction of the continuum valence band maximum, in a manner less favoring a large oscillator strength for the optical transition, leading to a weak absorption contribution from www.nature.com/scientificreports/ the transition V1 → C1/C2. Peaks C ′′′ , D ′′′ and E ′′′ form a broad composite peak and consist of many exciton states involving mixed transitions from V1, V2 and V3 to C1 and C2. Some of these excitons bear features of the excited states of the excitons forming peaks A ′′′ and B ′′′ . Peaks F ′′′ and G ′′′ are weak and involve transitions from lower valence bands (V4, V5, and V6) to C1 and C2. Peak H ′′′ , similar to peak F ′ , is mainly due to exciton states involving transitions from V1, V2, V3 to C3, since V1, V2, and V3 are very close around Ŵ . The binding energy of the exciton states mainly contributing to peak H ′′′ is 0.24 eV, larger than that for peak F ′ . At R = 6 Å , the main contribution to peak A ′′′′ is two degenerate excitons at 0.33 eV and they are mainly due to transitions V1/V2 → C1/C2 around Ŵ . This is mainly due to the closeness of V1 and V2 around Ŵ , and V3 is lower than V1 and V2 there (Fig. 5l). Note that the height of peak A ′′′′ is apparently much lower than other main peaks. Peak B ′′′′ is mainly due to excitons of transitions V3 → C1/C2 and V4 → C1/C2. Peaks from C ′′′′ to I ′′′′ become more complex. The bending lowers EG, and the conduction band continuum (CBC) also shifts downwards. This makes the upper edge bands C1 and C2 closer to CBC, and more transitions relating lower valence bands (V3, V4, V5, etc.) and higher conduction bands (C3, C4, etc.) occur. For example, peak G ′′′′ has contributions of excitons involving mixed transitions from V3-V5 to C1-C4. Figure 5e-h show the exciton energy spectra under different curvatures, with no feature of the 2D hydrogenic model 45 , due to the varying dielectric screening effect in the confined and layered nanoribbon structures. The A12MoS 2 nanoribbon also has similar tunable properties (Supporting Fig. S21). With increasing curvature, the main absorption peaks shift to lower frequencies and become broader. Bending activates more exciton states and many of them contribute to the optical absorptions. The lowest energy exciton's binding energy generally decreases with bending (Table 1), basically consistently with an increase in the static dielectric constant (Supporting Figs. S22 and S23) with bending, while the wavefunction distortions induced by the bending surface (or space) in bent nanoribbons may also influence the binding. The spin singlet-triplet splitting of the lowest energy exciton is about 0.22 eV for flat ribbons and shows a tunability (decrease) with bending ( Table 1). The bending space reduces the overlap between electron and hole wave functions and thus results in a decrease in e-h exchange interactions. We think that the edge states in the nanoribbons have similar properties to those of the defect states 46 in monolayer TMDs and the relatively large and tunable singlet-triplet splitting can make the nanoribbon system suitable for quantum information applications 47 . Figure 7 shows the energy levels of several low energy singlet and triplet excitons for the A13MoS 2 nanoribbon at three bending curvatures. Since the typical lifetime of an exciton in monolayer MoS 2 can be about 10 nanoseconds 48 , it is feasible to optically excite the system from one triplet state to another in the bent nanoribbon. Also, the intersystem crossing in MoS 2 system is realizable, evidenced by the phosphorescence application of MoS 2 quantum dots 49 , and it becomes more realizable in bent nanoribbons since bending usually increases the spin-orbit coupling, which assists in changing the spin during the intersystem crossing process. Besides, by appropriate doping or introducing defects, one may realize combination or hybridization between localized states and edge states in the nanoribbons, creating possible novel quantum states.
The band structures of the A12MoS 2 and A13MoS 2 nanoribbons with SOC are also calculated (Supporting Fig.  set SA). It shows that there is a negligible difference in the band structures between those with SOC and without SOC, before the lower edge bands merge into the valence continuum. Otherwise, there are small SOC induced splittings (~ 50 meV) in the top valence bands at k points away from Ŵ . Since the SOC induced splitting is much smaller than the quasiparticle fundamental gaps (~ > 1.5 eV) and not located at Ŵ , this will have very limited influence on the calculated optical absorption. However, even the small SOC effects may have an important consequence to the dynamic process involving excitons, such as the intersystem crossing.
The large tunability of electron energy loss spectrum (EELS) and absorption coefficient α with bending is shown in Fig. 8. As can be seen, both EELS and α are very low at energy around 1.5 eV for low bending curvatures, while they are large at large curvatures. Since the electron energy loss spectrum and the optical absorption coefficient are the quantities conveniently accessible by experiments, this can be utilized as a means to detect or control the bending for nanoribbon devices. Also, the data presented in this work can serve as a guide for the future experiments. As shown here, the mechanical bending is an effective means to control and fine tune the optical properties of nanoribbons.

Conclusion
In conclusions, from first-principles calculations DFTs and GW + BSE, we assessed the large tunability of band gaps and optical absorptions of armchair MoS 2 nanoribbons of different widths under different bending curvatures. We find that there are three critical bending curvatures κ 0 , κ c1 and κ c2 with κ 0 < κ c1 < κ c2 . Below κ 0 , the edge gap is almost unchanged, while from κ 0 to κ c1 , it slightly decreases. The non-edge gap slowly increases from zero curvature to κ c1 . From κ c1 to κ c2 , the edge gap nearly keeps constant, and beyond κ c2 , it decreases. From κ c2 and on, the lower edge bands merge into the valence band continuum. The non-edge gap decreases from κ c1 and on, and has a maximum around κ c1 , consistent with the maximum strain along the ribbon length direction around κ c1 . For wider nanoribbons (width = 3.6 nm), Regions I, II and III merge, and the non-edge gap shows a monotonic decrease with bending, while the edge gap shows a constant behavior followed by a decrease under increasing curvatures. The edge gaps and the non-degenerate edge gap splits show an oscillating feature with ribbon width n, with a period n = 3 , due to quantum confinement effects. The non-degenerate edge gap splits generally persist with bending. Bending generally induces more exciton states and they contribute to controllable optical absorptions. The induced excitons are related to the bands near the gap and the subtle changes of these bands with bending. The binding energy and the spin singlet-triplet split of the exciton forming the lowest absorption peak generally decreases with bending curvatures. This latter phenomenon opens opportunities for bent nanoribbons to utilize excitons in quantum information science. Since MoS 2 nanotubes have already been synthesized 50 www.nature.com/scientificreports/ nano-troughs and etching or eroding out some parts of the nanotubes. The large tunability of optical properties of bending MoS 2 nanoribbons is appealing and will find applications in tunable optoelectronic nanodevices.

Methods
Computational details for band gaps and optical absorption. Density functional theory (DFT) calculations were conducted in the Vienna Ab initio Software Package (VASP) 39 with projector augmented-wave pseudopotentials 52,53 . PBE 34 , SCAN 35 , TASK 36 , mTASK 37 , HSE06 38 approximations were used to calculate the band structures of nanoribbons. The vacuum layer of more than 12 Å is added along the direction of nanoribbon width and inserted along the direction perpendicular to the 2D surface of the nanoribbon, to avoid the interactions between the nanoribbon and its periodic images. The energy cutoff is 500 eV. The k-point mesh of 1 × 1 × 8 was used for all nanoribbons. All nanoribbons were fully structurally relaxed with PBE with all forces less than 0.008 eV/Å . During the relaxation, the x and y coordinates of the two outer most metal atoms on the two edge sides were fixed, while their coordinates along the ribbon axis direction, which is the z direction, and all the coordinates of other atoms were allowed to relax. The G 0 W 0 40,41 and G 0 W 0 + BSE 21   www.nature.com/scientificreports/ (for spin triplet), the binding energy of this exciton E b (for spin singlet), and the spin singlet-triplet split S−T for A12MoS 2 and A13MoS 2 nanoribbons under different bending curvature radii. E g is from GW calculations.