Investigation of the graphene thermal motion by rainbow scattering

Abstract

The thermal motion of graphene atoms was investigated using angular distributions of transmitted protons. The static proton-graphene interaction potential was constructed applying the Doyle-Turner's expression for the proton-carbon interaction potential. The effects of atom thermal motion were incorporated by averaging the static proton-graphene interaction potential over the distribution of atom displacements. The covariance matrix of graphene displacements was modeled according to the Debye theory, and calculated using Molecular Dynamics approach. Proton trajectories were used for construction of angular yields. We have found that there are lines, called rainbows, along which the angular yield is very large. Their evolution in respect to different sample orientation was examined in detail. Further we found that atom thermal motion has negligible influence on rainbows generated by protons experiencing distant collisions with the carbon atoms forming the graphene hexagon. On the other hand, rainbows generated by protons experiencing close collisions with the carbon atoms can be modeled by ellipses whose parameters are very sensitive to the structure of the covariance matrix. Numerical procedure was developed for extraction of the covariance matrix from the corresponding rainbow patterns in the general case, when atoms perform fully anisotropic and correlated motion.

Describing the scattering of keV protons through graphene

“…For larger scattering angles, the signal continuously decreases in intensity with no discernible features, as shown in the Supplementary Material . Thus, we do not observe the outer rainbow feature reported previously in the literature [ Ćosić et al (2018) ; Ćosić et al (2019) ; Hadžijojić et al (2021) ]. We can, however, artificially reproduce these features by thermally averaging the interaction potential ( Supplementary Material ).…”

“…To capture the behavior described previously in the literature [ Ćosić et al (2018) ; Ćosić et al (2019) ; Hadžijojić et al (2021) ], we consider the transmission of protons with a kinetic energy of 5 keV through single-layer graphene. These particles propagate along the z -direction with velocity v z until they interact with the 2D membrane where they are scattered, as shown in Figure 1 .…”

“…In this context transmission of protons at 5 keV through graphene has been proposed as an analytic tool [ Ćosić et al (2018) ; Ćosić et al (2019) ; Hadžijojić et al (2021) ; Ćosić et al (2021) ]. Describing the interaction of protons with the membrane via a thermally averaged potential, two rainbow features are predicted: an “inner” and an “outer” one.…”

“…The outer one stems from a maximum in the deflection function close to a carbon atom, leading to a signal at around 170 mrad [ Ćosić et al (2018) ]. The authors argue that in combination these give very detailed insights into the internal temperature [ Ćosić et al (2019) ] and the interaction potential [ Ćosić et al (2021) ], the defect concentration [ Hadžijojić et al (2021) ] as well as the membrane’s orientation and inclination [ Ćosić et al (2018) ; Ćosić et al (2021) ]. However, there are several conceptual issues with the underlying theoretical description.…”

“…Additionally, we account for the membrane’s corrugation as well as the experimental resolution and quantify their impact on the pattern at the detector. Based on this, we evaluate the predictions from the literature [ Ćosić et al (2018) ; Ćosić et al (2019) ; Hadžijojić et al (2021) ; Ćosić et al (2021) ] and test whether classical scattering through membranes can be used as an analytic tool. In our analysis we find that the predicted outer rainbow is an artifact from an improper description of the scattering process.…”