Nonlinear screening of charges induced in graphene by metal contacts

To understand the band bending caused by metal contacts, we study the potential and charge density induced in graphene in response to contact with a metal strip. We find that the screening is weak by comparison with a normal metal as a consequence of the ultrarelativistic nature of the electron spectrum near the Fermi energy. The induced potential decays with the distance from the metal contact as $x^{-1/2}$ and $x^{-1}$ for undoped and doped graphene, respectively, breaking its spatial homogeneity. In the contact region, the metal contact can give rise to the formation of a $p\text{−}{p}^{\prime }$, $n\text{−}{n}^{\prime }$, and $p\text{−}n$ junction (or with additional gating or impurity doping, even a $p\text{−}n\text{−}{p}^{\prime }$ junction) that contributes to the overall resistance of the graphene sample, destroying its electron-hole symmetry. Using the work functions of metal-covered graphene recently calculated by Khomyakov et al. [Phys. Rev. B 79, 195425 (2009)], we predict the boundary potential and junction type for different metal contacts.

Figure 1

(Color online) Schematic picture of a metal strip (electrode) on graphene. $V\left(x,z\right)$ is the metal-induced electrostatic potential, where $V\left(0,z\right)$ is fixed by the work functions $W_{\text{M}1}$, $W_{\text{M}2}$, $W_{\text{G}}$, and $W$, of the bottom and top metal layers, graphene and graphene-covered metal; $h_1$, $h_2$, and $H$ are the thicknesses of the bottom and top metal electrodes and the insulating substrate, and $L_0$ and $L$ are the lateral dimensions of the electrode and the free part of the graphene sample. ${\Phi }_{\text{g}}$ is the backgate voltage; ${\kappa }_0$ and ${\kappa }_1$ are the permittivities of the dielectric media above and below the graphene sheet. A typical experimental setup has $L_0\lesssim L\sim 1\mu \text{m}$, $H\sim 300\text{nm}<L$, and $h_1<h_2⪡H$.

Figure 2

The screening potential for undoped graphene: numerically exact solution (solid line), variational solution Eq. (13) (bold dots), $V\left(x\right)=V_{\text{B}}{\left(x/l_{\text{s}}\right)}^{-1/2}$ (dotted line) and $V\left(x\right)=V_{\text{B}}{\left(1+x/l_{\text{s}}\right)}^{-1/2}$ (dashed line). Inset: the charge density $\sigma \left(x\right)=e\lambda V\left(x\right)\left|V\left(x\right)\right|$, Eq. (10), for the Ti/Au electrode for ${\Phi }_{\text{g}}=-15\text{V}$ (dashed line), 0 V (solid line), and 15 V (dotted line); $V_{\text{B}}=0.19\text{eV}$, $\kappa =2.5$, $\lambda =8.69\times {10}^{-3}{\left(\text{eV}\text{Å}\right)}^{-2}$, $H=300\text{nm}$, and $l_{\text{s}}\sim 1\text{nm}$.

Figure 3

(Color online) The screening potential for doped graphene, Eq. (12): ${\mu }_{\text{F}}/V_{\text{B}}=-0.12$ (dotted-dashed green/gray line) and $-0.25$ (dotted-dashed black line); ${\mu }_{\text{F}}/V_{\text{B}}=0.12$ (dashed green/gray line) and 0.25 (dashed black line). As a reference we also show the screening potential for undoped graphene, ${\mu }_{\text{F}}=0$, (solid line). Inset: the charge density for doped graphene.