Modulation of the Superconducting Phase Transition in Multilayer 2H-NbSe2 Induced by Uniform Biaxial Compressive Strain

Strain is a powerful tool for tuning the properties of two-dimensional materials. Here, we investigated the effects of large, uniform biaxial compressive strain on the superconducting phase transition of multilayered 2H-NbSe2 flakes. We observed a consistent decrease in the critical temperature of NbSe2 flakes induced by the large thermal compression of a polymeric substrate (>1.2%) at cryogenic temperatures. For thin flakes (∼10 nm thick), a strong modulation of the critical temperature up to 1.5 K is observed, which monotonically decreases with increasing flake thickness. The effects of biaxial compressive strain remain significant even for relatively thick samples up to 80 nm thick, indicating efficient transfer of strain not only from the substrate to the flakes but also across several van der Waals layers. This work demonstrates that compressive strain induced from substrate thermal deformation can effectively tune phase transitions at low temperatures in 2D materials.

Figures S1a (S1b) and S2a (S2b) present the optical image in reflection (transmission) mode of the 10 nm thick NbSe2 flakes presented in Figure 1 of the main text, which were eventually deposited onto a Si/SiO2 and a polycarbonate (PC) substrate, respectively.
To obtain the relative transmittance spectrum of the NbSe2 flakes, the samples were illuminated with a white light source positioned beneath them.The Motic BA310 microscope was modified to perform spectroscopy measurements following Ref.[s1].Light from a spot measuring a few micrometers (~3 μm) in diameter at the center of the sample was collected and guided through an optical fiber (105 μm in diameter) to a CCS200/M compact spectrometer (Thorlabs).The acquired spectra from the sample and from the substrate are used to calculate the relative transmittance (RT) as RT = (Isample)/Isubstrate, where Isample and Isubstrate are the transmitted light intensities at the sample and substrate, respectively.
The relative transmittance spectrum can be simulated using the transfer matrix method, which applies Fresnel equations for light propagation in optical multilayers to model the intensity of the light beam transmitted with normal incidence across the sample [s2].This approach yields an estimation of the layers thickness if their refractive index is known.In this model, we utilized the available refractive index data for bulk NbSe2 from Ref. [s3] and considered each flake sandwiched between two semi-infinite media: air and Gel-Film, with real refractive indices of nair = 1 and nPDMS = 1.43, respectively.These values are considered to be constant for the range of experimental wavelengths.It's important to note that the refractive index value for the Gel-Film substrate has been chosen to match that of polydimethylsiloxane (PDMS) [s4].
Figures S1c and S2c present the relative transmittance spectrum at the center of the two 10 nm thick NbSe2 flakes, which are presented in the Figure 1 of the main text after being deposited onto a Si/SiO2 and a polycarbonate (PC) substrate, respectively.

S2. Parameters from fitting two-terminal resistance data to a broadened step function
Table S1.Fitting parameters of critical temperature (Tc), broadening (∆), height (R0) and residual resistance (Rcontact) of the superconducting phase transition to a broadened step function.Error is estimated from 95% confidence bounds of the fit.

S3. Two-terminal resistance analysis for all devices
A. PC-based devices

Figure S1 :
Figure S1: Optical image in (a) reflection and (b) transmission mode of the 10 nm thick NbSe2 on Gel-film substrate.This same flake is presented in Figure 1c of the main text after being deposited on Si/SiO2.(c) Relative transmittance spectrum obtained from the spectra acquired at the flake and gel-film substrate spots marked (red circles) in panel b.The solid lines correspond to the simulated results from the transfer-matrix model.

Figure S2 :
Figure S2: Optical image in (a) reflection and (b) transmission mode of the 10 nm thick NbSe2 flake on Gelfilm substrate.This same flake is presented in Figure 1d of the main text after being deposited on a PC substrate.(c) Relative transmittance spectrum obtained from the spectra acquired at the sample and gelfilm substrate spots marked (red circles) in panel b.The solid lines correspond to the simulated results from the transfer-matrix model.

Figure S3 :Figure S4 :
Figure S3: Graphical comparison of critical temperature and transition width for all PC-based devices with thickness as indicated in the legend: (a) A zoomed-in view of the two-terminal resistance around the superconducting transition temperature.Dots represent experimental data.Solid lines represent a step function fit of the data points.Curves have been normalized by the height (R0) of the step function fit.(b)Derivative of the step function fit with respect to temperature (dR/dT), emphasizing a notable difference in superconducting transition between all devices, primarily attributed to the induced strain transferred from the PC substrate.

Figure S5 :
Figure S5: Thickness-dependent evolution of the broadening of the superconducting phase transition, extracted from the step-function fits applied to resistance experimental data points for Si/SiO2-based devices (blue markers) and PC-based devices (red markers).The vertical error bars correspond to the statistical uncertainty of the fit, while the horizontal error bars represent the AFM uncertainty of ±2 nm.