in Fig.\[figure:device\]c from 4 to 292K. b) Shift in the position of the E$_{2g}^1$ mode as a function of T for as-prepared bulk flake (black circles), a 4L-MoS$_2$ flake (blue circles), and a 4L-MoS$_2$ device with Au contacts (red circles).[]{data-label="fig:strain"}](Figure_strain_2.eps){width="\columnwidth"}

Strain can arise due to a mismatch in the thermal expansion coefficients (TECs) of MoS$_2$[@GanPRB2016], SiO$_2$ substrate[@NIST_standard1991] and Au electrodes[@NixPR1941]. Upon cooling, MoS$_2$, SiO$_2$ and Au would normally undergo a contraction. However the flake is also subject to a tensile strain due to TEC mismatch[@YoonNL2011]. The strain, $\epsilon_{MoS_2}$, due to the MoS$_2$-SiO$_2$ TEC mismatch is: $$\epsilon_{MoS_2}=\int_{T}^{292K} [\alpha_{MoS_2}(T)-\alpha_{SiO_2}(T)] dT$$ whereas the strain, $\epsilon_{Au}$, due to the Au contacts is: $$\epsilon_{Au}=\int_{T}^{292K} [\alpha_{Au}(T)-\alpha_{SiO_2}(T)] dT$$ $\epsilon_{MoS_2}$ and $\epsilon_{Au}$ are$\sim$0.1% and$\sim$0.3% at $\sim$4K, respectively[@YoonNL2011].

Any FL-MoS$_2$ on SiO$_2$ will be subject to $\epsilon_{MoS_2}$ at low T. When the flake is contacted, an additional contribution is present due to $\epsilon_{Au}$. This can be more reliably estimated performing T-dependent Raman scattering and comparing the spectra for contacted and un-contacted flakes[@YoonNL2011; @MohiPRB2009]. Figs.\[fig:strain\]a,b show how a T decrease results in the E$_{2g}^1$ mode shifting to higher frequencies for both as-prepared and contacted 4L-MoS$_2$, due to anharmonicity[@KlemPR1966]. However, in the as-prepared 4L-MoS$_2$, the up-shift is$\sim$1cm$^{-1}$ larger with respect to 