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

We present both numerical and analytical study 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 scaling behavior with a temperature dependent roughness exponent in the interval of $ 0.6 < \chi < 0.7 $. The correlation function of height fluctuations depends upon temperature with characteristic length scale of $ \approx 90 {\AA}$ (at room temperature). We show that the correlation function of the induced gauge field has a short-range nature with correlation length of about $\simeq 2-3 {\AA}$. 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.