Variation of the fundamental band gap nature in curved two-dimensional WS2

In this Supplemental Material we (i) provide details about the finite-element method employed to calculate the strain tensor over spherically-deformed WS2 membranes or domes, (ii) evaluate the biaxial strain acting on the domes’ summit according to a continuum mechanical model, (iii) show micro-Raman measurements over the dome surface to account for the strain gradient, (iv) report the effect of methylpentane capping on the temperature dependence of the dome volume, (v) explain how the funneling effect was taken into account to determine the -K cross-over in the valence band, (vi) show results about the direct-to-indirect band gap transition in MoS2 and WSe2.

shows an optical image of the same flake, excited by a defocused 532 nm laser. The image was acquired at room temperature by filtering the laser out, thus letting the red luminescence (at 690 nm) generated by the domes be revealed.
Peculiarly, the brightly emitting region is restricted to an outer ring-like area independently of the dome footprint. This excludes interference (which would be largely dependent on the dome size) to be at the origin of the observed luminescence pattern, differently from what recently reported in WS 2 bubbles obtained after annealing of chemical-vapor-deposition-grown MLs 13 . In that work, interference effects are likely enhanced by the SiO 2 substrate located right beneath the ML bubbles and strongly modulate the emission. On the contrary, in the present case, the peculiar ring-like emitting area stems from the strain field acting over the domes, as detailed in the following.
To model the spatial evolution of the strain tensor and the height profile of the domes we performed FEM calculations within the framework of the nonlinear membrane theory 14,15,16 . The AFM-derived radius and height of the domes and the elastic properties of the material were used as input parameters. Fig. 2(a) (left axis) successfully compares the experimental (circles) and calculated (solid line) height profile along a radius (0  r  R, where r is the position with respect to the center) of the WS 2 dome, whose AFM image is shown in the inset. The right axis of Fig.   2(a) displays the calculated r dependence of the principal components of the strain tensornamely, along the circumferential ( t ) and radial ( r ) in-plane directions 14 and along the perpendicular ( z ) out-of-plane direction. At the dome's summit, the (tensile) strain is isotropic biaxial ( t = r =2.09%) in agreement with Hencky's model 4,14,16,17 , whereas at the dome edges -where  t =0-strain is uniaxial. The negative value of  z all over the surface is caused by the membrane thinning following the in-plane tensile strain. The strain field across the dome is expected to induce remarkable changes in the electronic properties of the curved TMD membrane 5,18,1920,21,22,23,24,25 , giving rise to the peculiar phenomenology displayed in Fig. 1

(c).
Spatially-resolved micro-PL/Raman measurements were then performed on the dome shown in Fig. 2(a), which was chosen since its size (R=2.85 m) is much larger than the probing laser spot (0.23 m) 12 , thus minimizing diffraction effects. However, it is important to note that the dome aspect ratio and, consequently, the strain distribution remain unchanged with the dome size 8 ensuring the general significance of the following results. To begin with, the micro-Raman measurements described in the Supplemental Material 16 show a progressive softening of the in-plane and out-of-plane vibrational modes while moving from the edge towards the center of the dome, in agreement with the expected tensile-strain increase 13,26,27 . However, the full extent of the effects of strain on the optoelectronic properties of TMD MLs can only be appreciated by looking at the dome's PL emission. Fig. 2(b) depicts a room temperature micro-PL scan taken along a diameter of the dome displayed in the inset of panel (a). The vertical axis indicates the energy of the emitted photons, while the base-10 logarithm of the PL intensity is shown in a false color scale. On moving from the edge toward 8 the summit of the dome, the marked red-shift of the emission wavelength is accompanied by an equally striking decrease (about a factor of 20) of the PL intensity. Figure 2(c) describes in more detail the dramatic changes of the emission spectra from the dome's edge (r = -2.85 m) to the dome's center (r = -0.06 m). Each spectrum is labelled also with the pertinent values of the radial and circumferential strain components [see panel (a)]. The micro-PL spectra recorded close to the edge are dominated by the direct (K CB -K VB ) band gap exciton (A), whose energy (equal to 2.00 eV in a strain-free reference WS 2 ML 8 ) is red-shifted by the tensile strain exerted on the dome. As the excitation laser moves toward the center, the direct exciton keeps redshifting and concomitantly a new, less intense band, labeled I, takes over and eventually dominates the spectrum. We ascribe this band to the K CB - VB indirect band gap exciton. In fact, as predicted by numerous theoretical works 5,18,19,21,22,23,24,25 the presence of strain in TMD MLs should result in a significant reordering of the energies of the critical points of the band structure.
In particular, for tensile biaxial strains ≳1% in WS 2 18,19,21,22,23,25 , the valence band maximum should change from the K to the Γ point of the reciprocal space. Even though this change (i) is expected to occur for values of  that are well within reach of current strain modulation techniques 28,29,30,31 , and (ii) should result in rather dramatic variations of the optical properties of the material, the currently available experimental evidence of this direct-to-indirect transition is either not particularly apparent 32,33 or absent 13,29,30,31 . This is possibly due to a less-than-perfect adhesion between the sample and the strain-inducing devices employed in some of the previous studies, resulting in an incomplete transfer of the applied stress to the TMD ML. In the present work, however, this is not an issue, as large biaxial strains-in the range between 1 -3% -are induced by the pressure exerted on the TMD ML by the H 2 gas trapped and perfectly sealed within the dome. To confirm the previous attributions, we investigated the temporal decay of the micro-PL signal of WS 2 domes. These were cooled down at 50 K to minimize the contribution of nonradiative decay channels 34 . Interestingly, the reduction of the dome's volume at cryogenic temperatures 8 -due to the contraction of the H 2 gas trapped inside the dome-is nearly brought to a halt by the deposition of a thin methylpentane layer on the sample surface (see Supplemental Material), thus making it possible to spatially resolve the PL signal from different zones of the dome. Fig. 3(a) shows the micro-PL spectra of a WS 2 dome recorded at the edge (where the A exciton dominates) and center (where the I exciton can be observed along with the red-shifted A exciton recombination); see pictures in the insets. Fig. 3(b) shows the micro-PL decay curve relative to the different transitions displayed in (a). Most notably, the A and I excitons exhibit largely different temporal behaviors: The decay time of the A exciton is instrument-limited (<250 ps) consistent with other reports 34, 35 . Instead, the I exciton shows a much longer temporal decay that can be fitted by a double exponential function with two decay times equal to (0.40 ± 0.06) ns and (2.9 ± 0.7) ns, which clearly points to an indirect optical transition 35 .
We now establish the strain conditions that induce the K- crossover in the VB. This is an especially important aspect with regard to the optoelectronic properties of TMD MLs and to the enormous potential that mechanical stress holds to engineer those properties. For instance, application of a seamless gradient of strain in these materials could be exploited as an efficient broad-band concentrator of photogenerated carriers in flexible solar cells 5 where Δ A,I is the shift rate with strain of the A (I) exciton. To correctly interpret the data shown in Fig. 4 Such a funnel effect, combined with the finite exciting/collecting area of the objective, alters the correspondence between the coordinate r (and thus p ) and the exciton energy derived from the emission spectra 6 . The solid curves displayed in Fig. 4 This analysis permits to set the direct-to-indirect band gap crossover point at p = (2.70.3)% highlighted by vertical dashed lines in Fig. 4(b). Finally, the top-right inset in Fig. 4(b) provides the A and I exciton energy as a function of the dome radial coordinate. The displayed fits yield  16 . This gives to our results a particular relevance regarding the general electronic properties of TMD 2D crystals.
In conclusion, we investigated the intertwined strain and electronic properties of sphericallydeformed TMD monolayers. We observed that sufficiently high tensile in-plane strains ( p~2 .7% in WS 2 -ML) turn a direct band gap material into an indirect-gap one. This general behavior, common to other TMDs-like MoS 2 and WSe 2 -must be considered when 2D crystal are to be employed in flexible optoelectronic devices, or possibly exploited for the observation of quantum many-body effects involving long-lived k-space indirect excitons 36     The results obtained by means of this mechanical model for the strain tensor are in good agreement with Hencky's model (described in the next paragraph). In Table 1 we report a comparison between Hencky's model and our model, for the 3 domes of Fig. S1.
Besides the height and strain profile, the mechanical model here described also allows us to estimate the internal pressure of the gas contained within the domes. The estimation of the pressure dependence on the dome footprint radius is displayed in Fig. S2 for WS 2 domes. Figure S2. Internal pressure estimation. Predicted dependence of the H 2 pressure as a function of the dome footprint radius in WS 2 domes. The data are evaluated by using the same mechanical model employed for simulating the dome profile and determining the strain tensor components.

Evaluation of biaxial strain at the dome summit (Hencky's model)
The expression for biaxial strain at the top of the dome is accounted for in Refs. 8 and 9 and is given by the Hencky's formula: where h m and R are the maximum height and the footprint radius of the dome, respectively. 0 and are two quantities that depend only on the Poisson's ratio . The values of for all the materials illustrated in Fig. S1 are listed in the following table 10 .  Table 3. Aspect ratios and strain at the summit. First row: Average values of the aspect ratios (h m /R) for the three compounds. Second row: Average values for the strain at the summit, calculated for the three compounds according to Hencky's formula and using the aspect ratios in the first row and the values of f(v) reported in Table 2.

Micro-Raman measurements
The following figure shows a micro-Raman mapping performed along the diameter of the same WS 2 dome, whose micro-PL mapping is shown and discussed in Fig. 2 of the main text. Details about micro-Raman measurements are reported in the caption of following Figure S3. as the laser spot is scanned from the dome's right edge (bottom) to its apex (top). Some selected spectra are labeled with the position of the laser spot and with the values of the radial ( r ) and circumferential ( t ) components of the strain tensor (see Fig. 2a of the main text and Supplementary  Fig. S1). The dotted line marks the position of the E 1 2g mode in bulk WS 2 . Right: Same as on the left, but for the A 1g mode. The black arrow follows the mode shift as the laser is scanned across the dome. (D) Right: Comparison between the normalized micro-Raman spectrum (T = 297 K) in monolayer WS 2 (bottom) and at the edge (middle) and in the center (top) of the dome, in the spectral region of the E 1 2g mode. The dotted line marks the position of the E 1 2g mode in bulk WS 2 ; the arrows follow the Raman shifts of the E 1 2g and 2LA(M) modes. The mode at 330 cm -1 in the ML spectrum was also observed in Ref. 14. We tentatively ascribe this mode to a LA replica. Right: Same as on the left, but for the A 1g mode. (E) Bottom: Dependence of the integrated micro-PL intensity on the position of the laser spot. In order to obtain the displayed intensity profile, the micro-PL spectra displayed in panels b-c of Fig. 3a in the main text were integrated between 1.7 and 1.95 eV. Middle: Evolution of the intensity of the A 1g Raman mode as the laser is scanned across the dome. The reported intensity values were obtained by fitting each Raman spectrum (see panel A and the right-hand side of panel C) with the function I tot = I dome +I bulk +I bkg . Here, I tot is the total spectrum, whereas I bkg is a flat background. I dome and I bulk are Gaussian functions, respectively taking into account the Raman peaks associated with lattice vibrations in the dome layer and in the underlying bulk WS 2 . In the panel we report I dome +I bkg , thereby excluding the contribution of bulk WS 2 to the spectrum. Top: Normalized reflectance at 532.2 nm as a function of the position of the laser spot. The displayed profile was obtained by collecting the light reflected by the sample as the laser was scanned across the dome, under the same experimental conditions used for micro-Raman and micro-PL measurements (laser wavelength  = 532.2 nm, T = 300 K, confocal configuration). Note the much different dynamic range spanned by the three intensity profiles with a ratio between the maximal and minimal value equal to 13 and 33 for the micro-PL and micro-Raman profiles, respectively. For micro-PL it is straightforward to ascribe this intensity reduction to the direct-toindirect band gap transition taking place as one moves from the edge to the center of the dome (see main text, Fig. 2). An analogous behavior to that of the dome discussed in Fig. 2b,c is found also for other WS 2 domes despite of their different dimensions, as also attested by the same ring-like pattern of the laser-excited red luminescence observed for all the domes in Fig. 1c, which suggests a minor role to be played by interference. It seems reasonable to assume a similar origin for the observed reduction of the micro-Raman signal at the dome's center. In this case, some interferential effects can be noticed in between the edge and the center of the dome, where an analogous modulation to that of the reflectance profile can be seen (see, in particular, the minimum for r ~ ±2 µm). We can therefore conclude that the modulation of the micro-PL and micro-Raman profiles are chiefly due to the strain-induced variations of the electronic properties across the surface of our domes, and interference does not play a primary role.

Effects of methylpentane deposition on the T dependence of the dome's size
As reported in Ref. 15, the contraction of the H 2 gas trapped inside a dome leads to a progressive reduction of the dome's volume at cryogenic temperatures, which culminates in the dome's disappearance upon reaching the vapor-to-liquid transition temperature at about 32 K. Even though this phenomenon is fully reversible (each dome reappears in its original position when the temperature is increased), the dome's shrinkage at cryogenic temperatures makes it increasingly difficult to spatially resolve the micro-PL signal from different zones of the dome. As noted in the main text, this is potentially highly problematic for time-resolved micro-PL measurements, which must be performed at low temperature to fully appreciate the existing differences in the temporal behavior of the direct and indirect exciton. At room temperature, indeed, non-radiative decay channels dominate the exciton dynamics, as also reported, e.g., in Ref. 16. As illustrated in Fig. S4, however, this issue can be conveniently overcome by covering the sample surface with a thin layer of methylpentane. Indeed, the adhesion of the latter to the dome's walls is enough to nearly stop the dome's contraction with decreasing T, without sizably altering the dome's emission properties (see main text).

Figure S4. Slowing down the dome's contraction at low T by methylpentane deposition
(a) Optical microscope images of the WS 2 dome on which the time-resolved micro-PL measurements discussed in the main text (see Fig. 3) were acquired. The image on the left was acquired at room temperature (T~290 K), whereas the image on the right was taken at T=50 K (i.e., the temperature at which the time-resolved micro-PL measurements were performed). Even though the same 100× objective (NA=0.75) was used for both images, the quality of the one acquired at T=50 K is affected by the aberrations due to the presence of the optical cryostat window in between the objective and the sample. Nevertheless, the effects of methylpentane deposition are clearly visible, i.e., the dome's size becomes nearly insensitive to temperature changes. This is in sharp contrast with the situation depicted in panel (b), which displays the T dependence of the dome's size in a "bare" (i.e., not covered with methylpentane) sample. The dome's shrinkage with decreasing T is clearly visible in the displayed optical images (acquired with a 50× objective with NA=0.5).
the photogenerated carriers drift towards the minimum energy available within their diffusion length (funnel effect) 18,19 , i.e., towards the dome's center. Combined with the finite exciting/collecting area of the objective, this alters profoundly the correspondence between the coordinate r (and thus p ) and the exciton energy resulting from the emission spectra. The curves superimposed on the experimental data in Fig. 4   Shift rate values  of the direct (A) and indirect (I) exciton energy. E A,I (0) indicates the direct and indirect exciton energies at zero strain. In order to make meaningful a comparison with theoretical papers (where the absolute value of the band gap is underestimated), we report the difference between these quantities.

Direct-to-indirect band gap transition in MoS 2 and WSe 2 monolayers
The findings reported for WS 2 are general and were observed also in other TMD compounds. In the following figure S5, we show the micro-PL spectra recorded in different points of a single MoS 2 (left) and WSe 2 (right) dome, showing the dramatic changes of the emission spectrum on going from the dome's edge to its center. (a) micro-PL spectra acquired in the positions highlighted by the colored dots in the MoS 2 dome (with footprint radius R = (2.46±0.06) µm) shown as inset. The spectra are peak-normalized for ease of comparison. At the edges of the dome (red dot in the inset) the direct A exciton (corresponding to the K C -K V transition) dominates the spectrum (red spectrum) and the B exciton involving the conduction band minimum at K C and the lower split band at K V in valence band can also be observed. According to our mechanical model, at the edges of MoS 2 domes the strain is strongly anisotropic with a radial component equal to ~ 2 % and an almost null circumferential component (see Supplementary Figure S1). The A peak appears to be broad and composite due to the complex strain distribution coupled to the finite resolution of our optical system (as discussed in the main text) and to funneling phenomena. These factors are also likely responsible for irregular behaviors close to the edges, such as the unexpected blueshift (of ~ 30 meV) observed while moving towards the dome center (green dot in the inset and corresponding green spectrum). Notice that deviations from the expected behavior towards higher energies were similarly observed also for uniaxially-strained MoS 2 monolayers in Ref. 27, for strains ≳ 1.4 %. The A band then redshifts while further going towards the center, where strain features a more regular biaxial distribution with  r ≈  t . In between the edges and the center (see green dot in the inset and corresponding green spectrum) a new band (I) appears, which we attribute to the indirect K C -Γ V transition. This band starts dominating the spectrum at the dome summit (purple dot in the inset and corresponding purple spectrum). Notice that at the center the I and A band are redshifted by about 20 meV and 10 meV, respectively, with respect to the green dot, accordingly to the increase of biaxial strain and the theoretically predicted higher shift rate for the K C -Γ V transition with respect to the K C -K V transition 22,23 . The dashed orange line highlights the redshift of the I exciton. Concomitantly, a reduction in the PL signal is observed, the PL signal at the edge being ~ 10 times more intense than at the center. Based on our µ-PL measurements on several MoS 2 domes, the bandgap crossover is found to occur for this compound for an in-plane strain ~ 3-4 %. (b) Same as in panel (a) for the WSe 2 dome (with footprint radius R = (1.42±0.06) µm) shown as inset. The direct A exciton (corresponding to the K C -K V transition) dominates the spectrum at the edges (red dot in the inset and corresponding red spectrum). The indirect I band appears while moving towards the center (green dot and corresponding green spectrum), redshifts while approaching the dome summit (as highlighted by the dashed orange line) and finally dominates the spectrum (purple dot and spectrum). For this compound, no significant quenching of the PL signal is observed for increasing tensile strain, the signal often increasing while going from the edges towards the domes' summit. An increase in the PL emission was also observed in Ref. 28 for WSe 2 monolayer under uniaxial strain. In our domes, however, the signal is observed not to decrease even in presence of the directto-indirect bandgap transition (for the dome here shown the signal at the center is found to be almost 10 times higher than at the edges). This enhancement in the PL efficiency is likely aided by funneling effects combined with the small dimensions of the WSe 2 domes we can create: The larger domes have dimensions comparable to that of the excitation laser spot, resulting in funneling of excitons at the dome summit and thus favoring the indirect transition. For WSe 2 , the bandgap crossover is found to occur for an in-plane strain ~ 2-3 %.