Interplay between geometrical structure and electronic properties in rippled free-standing graphene

It has been argued that the electron-hole puddles formed on graphene are mostly due to substrate-induced charged impurities [J. Martin et al., Nature Phys. 4, 144 (2008), Y. Zhang et al., Nature Phys. 5, 722 (2009)]. Here, using first-principles ab initio calculations, we show that the existence of ripples and electron-hole puddles is indeed an intrinsic property of graphene at finite temperatures. We found a relatively large correlation between the electronic charge density distribution on the surface of graphene and its local geometrical properties, such as local mean curvature and average bond length. We show that the electron and hole puddles appear in places where curvatures are large and small, respectively. We also determined the average sizes of the observed electron-hole puddles and have reported their percolating nature.

Figure 1

(a) Configuration of a graphene sheet consisting of 1250 carbon atoms at 300 K, obtained by ab initio molecular dynamics simulation, without carrying out the electronic structure calculation. Colors indicate the height of the atoms. (b) Projected and averaged total density of states for one of the atoms in a graphene sheet consisting of 450 carbon atoms. Blue diamonds (red triangles) correspond to $\pi$ orbitals ($\sigma$ orbitals). Total density of states divided by the number of atoms is shown with black circles. The presented results suggest that the contribution of the $\sigma$ orbitals to the total charge density is much more than that of the $\pi$ orbitals.

Figure 2

(a) Twice the absolute value of the mean curvature of the graphene surface, i.e., $|\nabla {}^{2}Z(x,y)|$, and (b) the average bond length. (c) and (d) The $\pi$ and the $\sigma$ orbital charge density distributions subtracted from their average value in each case, respectively. (e) Color map of the spatial density variations of the $\pi$ orbital charge density. The green dashed lines show the zero-density contours separating the electron puddles from their adjacent hole puddles. (f) The histogram of the normalized $\pi$ orbital charge density, i.e., the charge density divided by its variance, and the red line is a Gaussian function with variance equal to 1, which is shown for comparison purposes. (g) shows the distributions of the absolute value of the mean curvature for the hole and the electron puddles. (h) The bond length distribution for the electron and the hole puddles.