Measuring flexural phonon spectrum of graphene by electron diffraction
Authors: Kirilenko Demid 
Published: December 2016

Graphene surface is always corrugated, particularly, when it is not in contact with a substrate [1]. The main reason for this are out‐of‐plane thermal oscillations (so‐called flexural phonons) whose amplitude reaches rather high values in suspended 2D crystals [2]. Studying suspended sheets is of great importance because crystal relief gives us information on the very intrinsic properties of graphene in this case. Contrariwise, graphene being in contact with a substrate either repeats its relief [3] or demonstrates a stationary state of the flexural phonon spectrum frozen by the interaction with the substrate and, thus, significantly distorted [4]. At the same time, suspended graphene shows the highest charge carriers mobility values and, thus, is the most prospective material for high‐speed electronics [5]. And flexural phonons are the main limiting factor of free‐standing graphene conductance [6], thus, investigating them is crucial for the developments in this field.
However, it is almost impossible to measure a suspended 2D crystal relief by direct techniques like atomic‐force or scanning tunneling microscopies because of the very high flexibility of the ultrathin sheets. Particularly, it is so for the dynamic, rapidly varying relief caused by the thermal oscillations.

Here we present a technique for measuring flexural phonon spectrum in suspended graphene. The technique is based on analysis of electron diffraction patterns [7]. The technique utilizes intensities variations around a diffraction spot observed when crystal is being tilted with respect to the incident electron beam. The measured intensities in the vicinity of a diffraction spot (Fig. 1) vary in different manner depending on the distance from the spot (Fig. 2). A mathematical model of a corrugated 2D crystal representation in the reciprocal space is developed. This model shows a way to extract flexural phonons spectral amplitudes from the experimental data. The main difficulty arises from the relatively broad tails of the microscope point‐spread‐function, which is defined by the spatial coherency of the electron beam in the case of diffraction imaging. A couple of spectrum reconstruction techniques are suggested which allow measuring the spectrum in a relatively wide range of wave‐vector lengths (0.4‐4 nm
‐1
is achieved in our case).

The directly measured spectrum and reconstructed by different techniques are shown in Fig. 3. The most remarkable is the q
‐4
dependence at the right hand part, which is in agreement with the theoretical predictions. This allows direct measuring of the bending rigidity of suspended graphene. At the same time, the dependencies at smaller wave‐vectors are considerably weaker than the predicted. This may be a sign of a more significant influence of charge carriers interaction with the lattice distortions on the flexural phonons dynamics. We performed a simulation of the point‐spread‐function effect on the measurements and also checked the validity of the suggested reconstruction techniques (Fig. 4) and found it all in agreement with our experimental data.

Finally, we introduce a technique for measuring flexural phonon spectrum in suspended graphene. The obtained dependence of the spectral amplitude on the wave‐vector can be directly compared with the results of theoretical simulations, which are quite abundant in this field (while, there is a lack of such straightforward experimental data). The found spectrum profile raises questions on the role of different mechanisms involved in the flexural phonon dynamics. Moreover, the technique is applicable to studying of suspended 2D crystals of any other types that can be of great importance in the corresponding fields.