Electron-hole puddles in the absence of charged impurities

It is widely believed that carrier-density inhomogeneities (“electron-hole puddles”) in single-layer graphene on a substrate such as quartz are due to charged impurities located close to the graphene sheet. Here we demonstrate by using a Kohn-Sham-Dirac density-functional scheme that corrugations in a real sample are sufficient to determine electron-hole puddles on length scales that are larger than the spatial resolution of state-of-the-art scanning tunneling microscopy.

Figure 1

Three-dimensional plot of the corrugated graphene sample studied in this work [experimental data are a courtesy of Geringer (Ref. 20)]. The color/grayscale coding of the surface labels the local value of the induced carrier density $\delta n\left(\mathit{r}\right)$ as calculated from the Kohn-Sham-Dirac self-consistent theory, Eqs. (5, 6, 7, 8, 9). The data in this figure have been obtained by setting $g_1=3\mathrm{eV}$, ${\alpha }_{\mathrm{ee}}=0.9$, and ${\overline{n}}_{\mathrm{c}}\approx 2.5\times {10}^{11}{\mathrm{cm}}^{-2}$ (see text).

Figure 2

Left: Color/grayscale plot of the scalar potential $V_1\left(\mathit{r}\right)$ (in units of meV) calculated using Eq. (4) with $g_1=3\mathrm{eV}$. Center: The $\widehat{\mathit{x}}$-component $A_x\left(\mathit{r}\right)$ of the vector potential (in units of meV) calculated using Eq. (4). Right: Same as in the central panel but for the $\widehat{\mathit{y}}$-component $A_y\left(\mathit{r}\right)$ of the vector potential.

Figure 3

Top: Fully self-consistent induced carrier-density profile $\delta n\left(\mathit{r}\right)$ (in units of ${10}^{12}{\mathrm{cm}}^{-2}$) in the corrugated graphene sheet shown in Fig. 1. The data reported in this figure have been obtained by setting $g_1=3\mathrm{eV}$, ${\alpha }_{\mathrm{ee}}=0.9$, and an average carrier density ${\overline{n}}_{\mathrm{c}}\approx 2.5\times {10}^{11}{\mathrm{cm}}^{-2}$. The thin solid lines are contour lines of the height map $h\left(\mathit{r}\right)$. Note that there is no simple correspondence between topographic out-of-plane corrugations and carrier-density inhomogeneity. Bottom: Same as in the top panel but for $g_1=20\mathrm{eV}$.