Nonlinear Screening in Multilayer Graphene Systems

Electrostatic screening in multilayer graphene is highly nonlinear due to the vanishing density of states at the Fermi level. Using a discrete model we study the charge screening normal to the layers. Our model shows a strong charge and temperature dependence and has a simple continuum limit at $T=0$ for undoped systems. Doped systems can exhibit more complex behavior due to minority-carrier screening. Most importantly we find that the screening length can vary more than an order of magnitude depending on the experimental conditions, reconciling the large range of screening lengths reported in previous experiments. This has important consequences for technological applications of multilayer graphene used in electrodes or transistor channels.

Figure 1

(a) Schematic showing MLG configured as in a field-effect transistor. (b) The electric field shifts the DP away from the Fermi level. (c) The electric field is screened by the charge $q_i$ accumulated at each layer. (d)–(e) Normalized charge distribution profiles at $T=0$ (solid symbols) induced in 4 (d) and 10 (e) layer systems for gate charge densities $Q_0={10}^{11}{\mathrm{cm}}^{-2}$ (squares), ${10}^{12}{\mathrm{cm}}^{-2}$ (diamonds), and ${10}^{13}{\mathrm{cm}}^{-2}$ (circles). Open symbols in Fig. 1e correspond to $T=300\mathrm{K}$.

Figure 2

Normalized charge distribution for a doped ($|q_D|=3\times {10}^{11}{\mathrm{cm}}^{-2}$) 10 layer system for $Q_0$ values of ${10}^{11}$ (squares), ${10}^{12}$ (diamonds), and ${10}^{13}{\mathrm{cm}}^{-2}$ (circles) under (a) forward gating and (b) reverse gating. The solid line in the upper figure corresponds to the high-doping limit $|Q_0|\ll |q_D|$ [Eq. (3)]. The horizontal dashed lines in the bottom figure denote the inversion density threshold (i.e., $q_i=-q_D$) for the two doping values indicated.

Figure 3

Effective screening length computed using the discrete (symbol) and continuum (solid line) models corresponding to: (a) 4 and (b) 10 layer systems at $T=0$. For comparison the screening length of the semi-infinite system is also plotted (dashed line). (c) Effective screening length ${\lambda }_{\mathrm{eff}}$ [Eq. (9)] between the first and second layers as a function of $Q_0$ for: semi-infinite MLG system at $T=0$ (squares), and 10 MLG at $T=30\mathrm{K}$ (diamonds) and $T=300\mathrm{K}$ (circles), respectively. Arrows indicate the screening length ${\lambda }_T$ [Eq. (10)].