Measuring the height-to-height correlation function of corrugation in suspended graphene

Nanocorrugation of 2D crystals is an important phenomenon since it affects their electronic and mechanical properties. The corrugation may have various sources; one of them is flexural phonons that, in particular, are responsible for the thermal conductivity of graphene. A study of corrugation of just the suspended graphene can reveal much of valuable information on the physics of this complicated phenomenon. At the same time, the suspended crystal nanorelief can hardly be measured directly because of high flexibility of the 2D crystal. Moreover, the relief portion related to rapid out-of-plane oscillations (flexural phonons) is also inaccessible by such measurements. Here we present a technique for measuring the Fourier components of the height-height correlation function H(q) of suspended graphene which includes the effect of flexural phonons. The technique is based on the analysis of electron diffraction patterns. The H(q) is measured in the range of wavevectors q≈0.4-4.5 nm−1. At the upper limit of this range H(q) does follow the T/κq4 law. So, we measured the value of suspended graphene bending rigidity κ=1.2±0.4 eV at ambient temperature T≈300 K. At intermediate wave vectors, H(q) follows a slightly weaker exponent than theoretically predicted q−3.15 but is closer to the results of the molecular dynamics simulation. At low wave vectors, the dependence becomes even weaker, which may be a sign of influence of charge carriers on the dynamics of undulations longer than 10 nm. The technique presented can be used for studying physics of flexural phonons in other 2D materials.