Spatial Tuning of Light–Matter Interaction via Strain-Gradient-Induced Polarization in Freestanding Wrinkled 2D Materials

To date, controlled deformation of two-dimensional (2D) materials has been extensively demonstrated with substrate-supported structures. However, interfacial effects arising from these supporting materials may suppress or alter the unique behavior of the deformed 2D materials. To address interfacial effects, we report, for the first time, the formation of a micrometer-scale freestanding wrinkled structure of 2D material without any encapsulation layers where we observed the enhanced light–matter interactions with a spatial modulation. Freestanding wrinkled monolayer WSe2 exhibited about a 330% enhancement relative to supported wrinkled WSe2 quantified through photoinduced force microscopy. Spatial modulation and enhancement of light interaction in the freestanding wrinkled structures are attributed to the enhanced strain-gradient effect (i.e., out-of-plane polarization) enabled by removing the constraining support and proximate dielectrics. Our findings offer an additional degree of freedom to modulate the out-of-plane polarization and enhance the out-of-plane light–matter interaction in 2D materials.


Supplementary Note 1: Working mechanism of photo-induced force measurement
Photo-induced force microscopy (PiFM) measures the near-field optical interaction based on sample polarizability, primarily the induced dipole-dipole interaction (attraction force) between the optically driven dipole and the dipole in metal-coated atomic force microscopy (AFM) tip when illuminated with a coherent light source, through the deflection of the AFM cantilever tip, allowing for a true near-field technique 1,2 .
Here, the metal-coated AFM tip (in our case gold-coated tip) enables large values of tip polarization and thus offers the measurement of electric dipole-dipole interaction forces.The local polarization of the sample by the near-field excitation results in highly localized forces exerting between the tip and sample.Such near-field detection minimizes far-field backgrounds contribution of scattered photons surrounding the tip apex which have been one of the general challenges for tip-enhanced measurements 1 .When the light illuminates at the tip-sample junction, it induces dipoles in the tip and the sample mutually interacting with each other generating an attractive Coulombic force.The time-averaged localized photo-induced force () can be described with the dipole approximation of point dipoles (i.e., polarizable spheres) assuming a nonvarying spatial phase of the electric fields: where   and   are the polarizabilities of sample and the tip, respectively, z is the distance from the center of the dipole to surface, and   is out-of-plane components of the incident field.As shown in Eq. 1, both   and 1  4 dependence result in the highly localized optical force detection with the high spatial resolution 1 .
In our experiments, the scan speed was 0.5 lines per second at 256 x 256 pixels.We note that we used a low scan rate of 0.5 lines per second to ensure an identical tip-sample distance in both trace and retrace scans and adequate integration time for the lock-in amplifier to minimize any undesirable tip-sample perturbations.We used an excitation light source having a fixed wavelength at 658 nm with a time constant of 10 ms and a 50% laser duty cycle.
the same incident power (Fig. S1b,e,h).In the particular case of freestanding MoS 2 /WSe 2 heterostructure (Fig. S1h), the enhanced PL signal enables to clearly show the PL peak of momentum-indirect (Γ − Κ) interlayer excitonic resonance at 1.58 eV, where the measured PL spectra show substantially quenched A excitons of MoS 2 (blue spectrum in the inset of Fig. S1h) and WSe 2 (violet) with the dominant interlayer exciton peak (green).
Next, we compared Raman spectra between the supported and freestanding monolayers and heterostructures.Raman spectroscopy characterizes lattice vibration modes that can provide strain, doping, and the number of layer configurations.Similar to PL spectra comparison, we also observed that freestanding 2D materials can enhance both the intensity and resolution of their Raman signals.Raman intensities of the dominant first-order Raman modes of in-plane E 2g and out-of-plane A 1g in freestanding monolayer WSe 2 (Fig. S1c) and MoS 2 (Fig. S1f) were stronger compared to those in their supported counterparts on PDMS substrate.In the particular case of freestanding WSe 2 in Figure S1c, we observed the enhanced sensitivity towards phonon modes ranging from 300 to 400 cm -1 that are associated with second-order processes involving two phonons within the interior of the Brillouin zone (i.e., second-order overtone Raman modes), which are relatively weak to be defined in the supported WSe 2 monolayer.In cases of freestanding MoS 2 and MoS 2 /WSe 2 heterostructure, we observed that both in-plane (E 2g ) and outof-plane (A 1g ) modes of MoS 2 were softened compared to the supported counterparts.Both red-shifted E 2g (~ -0.7 cm -1 ) and A 1g (~ -0.2 cm -1 ) Raman peaks indicate that the freestanding area is slightly under tension and possesses the reduced out-of-plane interaction with the underlying substrate.
We attribute the enhanced sensitivities and intensities of PL and Raman to the changes in the effect of the dielectric environment induced by the underlying substrate (i.e., the substrate decoupling effect) on the local electromagnetic field, where the reduced substrate interaction may enhance the local electromagnetic field and the resultant optical signals.Our results agree well with previous studies where suspended graphene and MoS 2 have shown enhanced Raman and PL signals, which are attributed to the diminishment of detrimental interactions between the materials and substrates, such as nonradiative recombination, charge transfer, and excitonic transition 9,10 .Also, it has been reported that PL efficiencies can be increased via substrate engineering from supported to suspended monolayers of MoS 2 and WSe 2 11 .
Another study showed that the dielectric surroundings around the 2D materials can optically regulate the incoupling and outcoupling of light, yielding a dramatic variation of the emission intensity either enhanced or suppressed depending on the thickness of the underlying substrates 12 .
Supplementary Note 3: In-plain strain effects on the photo-induced force in the freestanding wrinkled WSe 2 .
We first spatially resolved PL over the wrinkle to probe in-plane strain exerting in the freestanding wrinkled WSe 2 .Figure S4 shows comparison plots between the measured PL line scan, the measured wrinkle topography and photo-induced force, and the topography-driven in-plane strain.PL results (Fig. S4a) suggest that the valleys of wrinkles are under tension where they show the maximum PL peak wavelength shift of +24.3 nm, which corresponds to ~51.8 meV shift in resonance PL energy.The apex of the wrinkle is under negligible strain where it shows the PL peak at 750.7 nm (~1.652 eV).According to the previously reported deformation potential of WSe 2 (55 meV/%), the valleys of wrinkles are under tension ranging from 0.44% to 0.96% in-plane strain.We also observed a substantial PL intensity increase over the freestanding wrinkle (Fig. S4b), which is similar to what we observed in the flat freestanding monolayers.When we compare the measured topography of wrinkles and the measured PL intensity, we found that the PL intensities started to increase near the valley of the wrinkle suggesting the in-plane strain probed via PL is due to the edge pinning effects from the periphery of the cavity where there is an abrupt transition from supported to freestanding.We did not find any strong correlation between topographydriven in-plane strain and the measured photo-induced force.
Additionally, we performed photo-induced force microscopy over two different monolayer WSe 2 freestanding samples to explore in-plane strain effect without substantial wrinkle formation.As shown in Figure S5, we did not observe any significant spatial tuning or changes in the measured photo-induced force between the samples even though one flat freestanding WSe 2 was slightly under tension, which was indicated by the shifted A exciton peak wavelength at about 762 nm (Fig. S5d) compared to the unstrained resonance A exciton peak wavelength of 754 nm (Fig. S5b).We did not find any strong correlation between spatial modulation of photo-induced force and in-plane strain.
When there is an increased in-plane tension exerted in semiconducting 2D materials, we expect to see two dominant effects − electronic bandgap modulation and an increase in the piezoelectric potential.If the in-plane strain-induced bandgap modulation plays a critical role in spatial tuning of photoinduced we should have observed an increased photo-induced force with increased tension, because 1) the electronic polarizability of 2D materials is inversely proportional to their optical bandgap energy 13 and thus with the increased tension, which results in lowering the optical bandgap energy, we would expect to see increased electronic polarizability enhancing the photo-induced force.2) Both experimental 14 and first-principle calculations using density functional theory 15 suggested that a tensile strain increases the optical absorption of WSe 2 , which will then lead to an increase in the photo-induced force.Our observation of the reduced photo-induced force with the increased in-plane tension shows a clearly opposite behavior to these effects, ruling out the possible underlying mechanisms of both strain-induced optical bandgap modulation and strain-induced absorption effects.Next, when a static strain is introduced in a strong piezoelectric semiconductor including monolayer WSe 2 , the presence of the localized polarization charges can effectively modulate its optical properties through the corresponding strain-induced electrostatic potentials.
It has been reported that the strain-induced piezoelectric effects induce various destructive effects on the optical light-matter interactions such as a reduction in exciton recombination and exciton binding energy owing to the piezoelectric polarization field reducing the electron-hole wavefunction overlap 16 , and local electronic band tilting 17 .Furthermore, theoretical studies by using transport model including drift and diffusion of electrons and holes on the effect of spontaneous and piezoelectric polarization on the optical characteristics suggested that the influence of piezoelectric polarization on optical properties is more severe than the spontaneous polarization reducing internal quantum efficiency where the best optical performance was predicted without the built-in piezoelectric electrostatic fields 18 .Thus, we hypothesize that the built-in electrostatic potentials generated by the increased in-plane strain resulting in the increased piezoelectric effects may have a destructive effect on the photo-induced force by disturbing optically-generated dipole in freestanding 2D materials and the incident electromagnetic fields.However, since we observed in-plane strain only around the cavity periphery owing to the effect where also may be influenced by substrate dielectric due to the proximity, it is inconclusive that in-plane strain would have destructive effects on the photo-induced force.
We studied correlations between the topography-driven in-plane strain, and the measured photoinduced force profile.Similar to the curvature studies with the 2D model discussed in the main text, inplane strain can be expressed as a function of the square of the first derivative of wrinkle geometry and a bending-induced in-plane strain term consisting of the 2 nd derivative of the wrinkle geometry 2 ), where  is the out-of-plane deflection, and t is the thickness of 2D materials.Our topographydriven in-plane strain corresponds well to the PL line scan result (Fig. S4e) where it shows a lower bandgap (i.e., a longer wavelength) at the valleys with the increased estimated tension.Again, the in-plane strain did not show any strong correlations with the measured photo-induced force (Fig. S4f) where the maximum photo-induced force was observed at the apex with negligible in-plane strain.
Lastly, we performed photoluminescence spectroscopy with peak fittings using Lorentzian function over freestanding monolayer WSe 2 for both unstrained (A-exciton resonance peak at 754 nm in Fig. S6a) and strained (A-excitonic resonance peak shifted to 762 nm in Fig. S6b) samples.As shown in Figure S6, we did not observe any significant peak broadening near the A-exciton resonance peak of monolayer WSe 2 , which is an indication of negligible trion formation and negligible contribution of any trions to the generated photo-induced force.

2 Figure S1 .
Figure S1.Far-field optical characterization of the fabricated freestanding 2D materials via

Figure S2 . 4 Figure
Figure S2.Characterizations of the fabricated freestanding wrinkled structure of monolayer WSe 2 .

Figure S4 .
Figure S4.Strain correlation plots between spatially resolved PL spectroscopy, photo-induced force

Figure S5 .
Figure S5.In-plane strain analysis of freestanding, flat monolayer WSe 2 under different levels of in-

Figure S6 .
Figure S6.Photoluminescence spectra measurement with peak fitting using Lorentzian function