Biaxial strain tuning of interlayer excitons in bilayer MoS2

We show how the excitonic features of biaxial MoS2 flakes are very sensitive to biaxial strain. We find a lower bound for the gauge factors of the A exciton and B exciton of (-41 +- 2) meV/% and (-45 +- 2) meV/% respectively, which are larger than those found for single-layer MoS2. Interestingly, the interlayer exciton feature also shifts upon biaxial strain but with a gauge factor that is systematically larger than that found for the A exciton, (-48 +- 4) meV/%. We attribute this larger gauge factor for the interlayer exciton to the strain tunable van der Waals interaction due to the Poisson effect (the interlayer distance changes upon biaxial strain).

The isolation of atomically thin MoS2 by mechanical exfoliation in 2010 opened the door to study the intriguing optical properties of this 2D semiconductor material. 1 demonstrated that uniaxial strain could be used to tune the energy of the interlayer exciton. 16 In this work we employ biaxial strain to modify the band structure, and thus the excitonic resonances, in bilayer MoS2 flakes. We observe that both the A and B excitons, as well as the interlayer exciton, substantially redshift upon biaxial tension. Interestingly, unlike to what has been reported for uniaxial strain, we found that the interlayer exciton is more effectively tuned upon straining than the A and B excitons. We attribute this effect to a modification of the interlayer interaction as an in-plane biaxial expansion of the bilayer MoS2 is expected to come hand-by-hand of an out-of-plane compression due to the MoS2 Poisson's ratio.   Figure 1(b) shows differential reflectance spectra acquired on a mono-, bi-and tri-layer MoS2 flake with a home-built micro-reflectance microscope. We address the reader to Ref. 22 for technical details about the experimental setup. To obtain the differential reflectance spectra we first collect the light reflected from the substrate (Rs) by means of a fiber-coupled compact CCD spectrometer (see Materials and Methods).
Then we collect the light reflected by the desired MoS2 flake (Rf) and we calculate the differential reflectance as: ΔR/R = 1 -Rf/Rs. 19,23 All the spectra displayed in Figure 1 show two strong transitions in all of them assigned to the A and B excitons (~1.9 eV and ~2.05 eV respectively) originated from direct band gap transitions at the K point of the Brillouin zone. 1,2 Interestingly, in bilayer MoS2 one can see another prominent peak between the A and B excitons. That peak can be also observed in trilayer and even multilayer MoS2 but it cannot be as easily resolved as in the case of bilayer MoS2. This feature in the reflectance spectra have been recently demonstrated (through temperature dependent optical spectroscopy studies, magneto-optical measurements and density functional theory calculations) to be originated by the generation of interlayer (IL) excitons. 14,15,24 These excitons are, similarly to the A and B excitons, due to direct transitions at the K point but unlike them the electron and hole are spatially separated in the different MoS2 layers (see the cartoon in Figure 1(b)).
In order to biaxially strain the MoS2 bilayers we exploit the large thermal expansion mismatch between the PP substrate (~130×10 -6 K -1 ) and MoS2 (1.9×10 -6 K -1 ) 25 . PP has also a relatively high Young's modulus (1.5-2 GPa) for a polymer, which is essential to guarantee an optimal strain transfer from substrate to flake. One can then biaxially stretch (or compress) the flakes by warming up (or cooling down) the substrate. [26][27][28] We used a Peltier element to control the temperature of the substrate around room temperature (27-28ºC) that allows us to cool down to 17ºC (-0.13%) and to warm up to 95ºC (+0.87%).
The substrate temperature can be translated to biaxial expansion/compression through the thermal expansion coefficient of PP (see the Supporting Information).  For the interlayer exciton we find a gauge factor of (-48 ± 4) meV/% which is substantially larger than that found for the A exciton. We address the reader to the Supporting Information for datasets acquired on other bilayer MoS2 flakes (with gauge factor up to -55 meV/%) and one trilayer flake that also have larger gauge factor for the interlayer exciton. This contrasts with what has been recently reported for uniaxially strained bilayer MoS2 flakes by Niehues et al. where the gauge factor of the interlayer exciton was slightly lower than that of the A exciton. We attribute the larger gauge factor observed in our experiment to a reduction (or increase) of the bilayer interlayer spacing upon biaxial tension (or compression) as expected from the Poisson effect: as the out-of-plane Poisson's ratio of MoS2 is νo ~ 0.2 a biaxial tension of 1% would yield a reduction of 0.2% in the interlayer distance. 33 A similar tunability of the interlayer van der Waals interaction upon biaxial strain has been recently reported in black phosphorus by Huang and co-workers. 34 The strain tunable interlayer distance could explain the large gauge factor observed for bilayer MoS2 upon biaxial strain as Deilmann and Thygesen demonstrated though density functional theory calculations that the interlayer exciton position strongly depends on the interlayer distance. 24 In Figure 3 we test the reproducibility of the biaxial strain tuning exploiting the thermal expansion of the substrate. We modulated the temperature of the substrate between ~30ºC and ~40ºC (see the registered temperature vs. time in the top panel of Figure 3). The color map in the bottom panel shows the time evolution of the differential reflectance spectra and the extracted position of the A, IL and B excitons, extracted from fits similarly to  . Time evolution of the differential reflectance spectra of bilayer MoS2 (bottom axis) while the temperature of the substrate is cycled between 30ºC and 40ºC (top axis). The intensity of the differential reflectance spectra is displayed in the color axis of the colormap. The A, B and IL exciton position is also displayed through the black solid lines.

CONCLUSIONS
In summary, we have exploited the large thermal expansion of polypropylene substrates to subject biaxial MoS2 flakes to biaxial strain. We find that the excitons redshift upon biaxial tension with gauge factors that are larger than those reported for monolayer MoS2.
Interestingly, the interlayer exciton gauge factor is systematically larger than that of the A and B excitons (contrasting the results reported for uniaxially strained bilayer MoS2).
We attribute this larger gauge factor of the interlayer exciton to the strain tuning of the van der Waals interaction upon biaxial in-plane straining due to the Poisson effect.

MATERIALS AND METHODS
Optical microscopy images have been acquired with an AM Scope BA MET310-T upright metallurgical microscope equipped with an AM Scope MU1803 camera with 18 megapixels. The trinocular of the microscope has been modified to connect it to a fiber-coupled Thorlabs spectrometer (part number: CCS200/M) to perform the differential reflection spectroscopy measurements. 19 Supporting Information: Biaxial strain tuning of interlayer excitons in bilayer MoS 2 Section S1 -Polypropylene substrate thermal expansion calibration Section S2 -Fitting of the differential reflectance spectra Section S3 -Additional samples Section S2 -Fitting of the differential reflectance spectra

Section S4 -Disentangling temperature and strain effects
In our experiments the biaxial strain is applied by changing the temperature of the PP substrate therefore we need a way to disentangle the effect arising simply by the temperature change from the effects originated by the biaxial strain. Therefore, we performed a set of measurements on SiO2/Si, a substrate with a negligible thermal expansion ~1·10 -6 K -1 (as compared with the 128·10 -6 K -1 of the PP substrate). We selected substrates with 50 nm SiO2 capping layer because they allow to directly resolve the exciton position from differential reflectance measurements. For other SiO2 thicknesses the substrate Fresnel interference could be so strong that hampers the exciton observation.