Sultan Akhtar, Stefano Rubino, Klaus Leifer
We present a simple and fast method for thickness characterization of suspended graphene flakes that is based on transmission electron microscopy (TEM) techniques. For this method, the dynamical theory of electron diffraction (Bloch-wave approach in two-beam case approximation) was used to obtain an analytical expression for the intensity of the transmitted electron beam I0(t), as function of the specimen thickness t for thin samples (t<< {\lambda}; where {\lambda} is the absorption constant for graphite). We show that in thin graphite crystals the transmitted intensity is a linear function of the thickness. To obtain a more quantitative description of I0(t), high resolution (HR) TEM simulations are performed using the Bloch wave approach of the JEMS software. From such calculations, we obtain {\lambda} for a 001 zone axis orientation, in a two-beam case and in a low symmetry orientation. Subsequently, HR (used to determine t) and bright-field (to measure I0(0) and I0(t)) images were acquired to experimentally determine {\lambda}. We obtain that the experimental value in the low symmetry orientation is close to the calculated value (i.e. {\lambda}=225 nm for 300 kV accelerating voltage and 3 mrad collection angle). The simulations also show that the linear approximation obtained from the analytical expression is valid up to a sample thickness of several ten nanometers, depending on the orientation. When compared to standard techniques for thickness determination of graphene/graphite, the method we propose has the advantage of being relatively simple and fast, requiring only the acquisition of bright-field images.
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http://arxiv.org/abs/1210.2307
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