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.
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