We present both numerical and analytical studies of graphene roughness with a crystal structure including $500\times 500$ atoms. The roughness can effectively result in a random gauge field and has important consequences for its electronic structure. Our results show that its height fluctuations in small scales have a scaling behavior with a temperature dependent roughness exponent in the interval of $0.6<\chi <0.7$. The correlation function of height fluctuations depends on temperature with a characteristic length scale of $\approx 90\mathrm{\AA }$ (at room temperature). We show that the correlation function of the induced gauge field has a short-range nature with a correlation length of about $\simeq 2–3\mathrm{\AA }$. We also treat the problem analytically by using the Martin-Siggia-Rose method. The renormalization group flows did not yield any delocalized-localized transition arising from the graphene roughness. Our results are in good agreement with recent experimental observations.

• Received 27 May 2007

DOI:https://doi.org/10.1103/PhysRevB.76.195407