Deformation and Scattering in Graphene over Substrate Steps

The electrical properties of graphene depend sensitively on the substrate. For example, recent measurements of epitaxial graphene on SiC show resistance arising from steps on the substrate. Here we calculate the deformation of graphene at substrate steps, and the resulting electrical resistance, over a wide range of step heights. The elastic deformations contribute only a very small resistance at the step. However, for graphene on SiC(0001) there is strong substrate-induced doping, and this is substantially reduced on the lower side of the step where graphene pulls away from the substrate. The resulting resistance explains the experimental measurements.

Figure 1

Illustration of graphene over a substrate step. Here $h_s$ is the step height and $h_{\mathrm{eq}}$ is the equilibrium distance of graphene from the substrate surface due to van der Waals interaction. ${\ell }_d$ is the length of graphene detachment from the substrate.

Figure 2

(a) Graphene geometry $h(x)$ for various step heights as indicated. Symbols are the full numerical calculation. Solid lines are best fit using Eq. (1). The respective step profile are illustrated by dashed lines. (b) Corresponding curvature $\kappa$ for geometries shown in (a), with lines and symbols as in (a). The minimum radius of curvature (inverse of maximum $\kappa$) is indicated.

Figure 3

(a) Numerically calculated maximum curvature ${\kappa }_{\max }$ for the upper edge (solid circles) and lower edge (open circles) of the step as function of step height $h_s$. Note that the often used approximation for curvature, $\kappa \approx {\partial }_x^{2}h$, is only reasonable for $h_s$ of few angstrom. Over the range of $h_s$ of our interests, it grossly overestimates the real curvature due to the large gradients in our graphene geometries. (b) Dependence of graphene deformation on step height $h_s$. Graphene step width $d_s$ and lateral shift $x_s$ are obtained by fitting Eq. (1) to the relaxed geometry $h(x)$ obtained numerically. We also show the detachment length ${\ell }_d$, i.e., the length of graphene separated from the substrate by $2h_{\mathrm{eq}}$. Note that ${\ell }_d\approx 1.2h_s$ for large $h_s$. The step width $d_s$ depends only weakly on step height.

Figure 4

(a) Resistance through graphene due to substrate step, $\mathcal{R}$, as function of step height $h_s$. Two electrostatic models for the doping variations are plotted, as described by Eq. (2) (model 1) and Eq. (3) (model 2). Experimental data [9] are plotted as circles, with statistical error bar indicated. (b) Potential profile $V_d(x)$ derived from Eq. (2) (model 1) for $h_0=0.5$, $1.0$, and 1.5 nm. Dashed horizontal line indicates the Fermi level position. (c) Same as (b), but derived from Eq. (3) (model 2). The small hump observed for $V_d(x)$ near the top is an artifact of the polynomial form of ${\Delta }_c(h)$.