Measurement of the shear modulus of single-layer graphene

We have measured the shear modulus of large area (2mm $\times$ 5mm), single-layer (90--95\%) polycrystalline graphene sheets and found values consistent with theoretical predictions of $G$=200 GPa. The graphene was grown by chemical vapor deposition onto a copper foil and subsequently transferred onto a mechanical resonator known as a double-paddle oscillator (DPO). DPOs are fabricated from single-crystal, 0.3mm thick silicon wafers, and have a torsional vibratory mode at 5500 Hz which has a very large quality factor, $Q=5\times10^7$, at low ($<10$ K) temperatures, giving the DPO a high sensitivity to a film deposited on its torsional element. Such a film increases the (lumped-element) spring constant of the resonator, and the film's shear modulus can be deduced from the subsequent resonant frequency shift.