Roughness of undoped graphene and its short-range induced gauge field

We present both numerical and analytical studies of graphene roughness with a crystal structure including 500x500 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 < 0.7. The correlation function of height fluctuations depends on temperature with a characteristic length scale of approximate to 90 angstrom (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 similar or equal to 2-3 angstrom. 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.