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Pretreatment processes including physical polishing (PP), wet-chemical polishing (WP), and thermal annealing for smoothing a surface of copper (Cu) foil were optimized for high-quality graphene growth. In this work, the cu foil with a small thickness of 25 µm has been used as a substrate. Preliminarily, a substrate surface was smoothed by the PP process using Brasso as a polishing substance. The substrate was next etched by the WP process using phosphoric acid $(H_{3}PO_{4})$ as an etchant in order to further reduce a surface roughness and remove surface contamination. For the condition of WP process, an etchant concentration was varied from 30 to 60%, and an etching period was adjusted to 60, 90, and 120 s. Finally, the substrate was thermally annealed with a nitrogen gas for 10 min before growing graphene. An annealing temperature was varied from 860 to 940 ºC. After the pretreatment processes, graphene was grown on the prepared cu foil substrate by direct-liquid-injection chemical-vapor deposition (DLI-CVD) technique using cyclohexane $(C_{6}H_{12})$ as a carbon precursor and nitrogen as a carrier gas. A growth temperature was fixed constantly at 920 ºC, and $C_{6}H_{12}$ was injected into the reactor with a flow rate of 0.5 g/min and pulse frequency of 0.5 Hz for 6 min. The surface morphology of Cu foil observed by optical microscopy exhibits that the substrate treated by PP and WP with the conditions that $H_{3}PO_{4}$ concentration is 45% and etching period is 90 s provides the smoothest surface without being damaged by the etchant. When the annealing temperature is assigned to 880 ºC, the room-temperature Raman spectra measured with a 473-nm excitation shows that an intensity ratio of 2D to G peak $(I_{2D}/I_{G})$ is enhanced above 2. This result indicated that a monolayer graphene could be successfully formed at this optimized temperature.