Height correlation of rippled graphene and Lundeberg-Folk formula for   magnetoresistance
  Application of an in-plane magnetic field to rippled graphene will make the system be a plane with randomly distributed vector potentials. Massless Dirac fermions carrying charges on graphene are scattered by the vector potentials and magnetoresistance is induced proportional to the square of amplitude of in-plane magnetic field $B_{\parallel}^2$. Recently, Lundeberg and Folk proposed a formula showing dependence of the magnetoresistance on carrier density, in which the coefficient of $B_{\parallel}^2$ is given by a functional of the height-correlation function $c(r)$ of ripples. In the present paper, we give exact and explicit expressions of the coefficient for the two cases such that $c(r)$ is (i) exponential and (ii) Gaussian. The results are given using well-known special functions. Numerical fitting of our solutions to experimental data were performed. It is shown that the experimental data are well-described by the formula for the Gaussian height-correlation of ripples in the whole region of carrier density. The standard deviation $Z$ of ripple height and the correlation length $R$ of ripples are evaluated, which can be compared with direct experimental measurements. 
Kazuyuki Genma, Makoto Katori
cond-mat.mes-hall cond-mat.stat-mech math-ph math.MP