Signatures of Klein tunneling in disordered graphene $p\text{−}n\text{−}p$ junctions

We present a method for obtaining quantum transport properties in graphene that uniquely combines three crucial features: microscopic treatment of charge disorder, fully quantum-mechanical analysis of transport, and the ability to model experimentally relevant system sizes. As a pertinent application we study the disorder dependence of Klein tunneling dominated transport in $p\text{−}n\text{−}p$ junctions. Both the resistance and the Fano factor show broad resonance peaks due to the presence of quasi-bound-states. This feature is washed out by the disorder when the mean free path becomes of the order of the distance between the two $p\text{−}n$ interfaces.

Figure 1

(Color online) (a) A schematic of the top gated setup. Graphene is represented by the red layer and the top-gate dielectric by a brown one. (b) Profile of the screened potential $V_{\text{sc}}$ in the absence of disorder. The back gate density is fixed at $n_{\text{bg}}=5\times {10}^{11}{\text{cm}}^{-2}$ and (from bottom to top) $\Delta n_{\text{tg}}=-5\times {10}^{12}{\text{cm}}^{-2}$ to $5\times {10}^{12}{\text{cm}}^{-2}$ in steps of $2\times {10}^{12}{\text{cm}}^{-2}$. All results in this Rapid Communication were obtained for square samples of size $W=L=160\text{nm}$ assuming $L_{\text{tg}}=30\text{nm}$, $d_{\text{tg}}=10\text{nm}$, $d=1\text{nm}$, ${ϵ}_1=1$ (air), and ${ϵ}_2=4\left({\text{SiO}}_2\right)$.

Figure 2

(Color online) Gate voltage dependence of (a) resistance and (b) Fano factor of a clean device. A single trace of (c) resistance and (d) Fano factor at a fixed back gate density $n_{\text{bg}}=5\times {10}^{11}{\text{cm}}^{-2}$. The smooth curves are obtained by averaging over boundary conditions. The arrows in (c) indicate the positions of the resistance minima obtained using Eq. (7). The insets show a close up of one of the narrow resonances obtained for transverse periodic boundary condition.

Figure 3

(Color online) (a) $V_{\text{sc}}\left(\mathbf{r}\right)$ for a disordered junction with $n_{\text{bg}}=5\times {10}^{11}{\text{cm}}^{-2}$, $\Delta n_{\text{tg}}=2\times {10}^{12}{\text{cm}}^{-2}$, and $n_{\text{imp}}=5\times {10}^{11}{\text{cm}}^{-2}$. (b) Resistance and (c) Fano factor in the presence of disorder (single realization) for a fixed back gate density $n_{\text{bg}}=5\times {10}^{11}{\text{cm}}^{-2}$ and several values of the impurity density (from bottom to top $n_{\text{imp}}=0$, 1, 2.5, 5, 10, and $15\times {10}^{11}{\text{cm}}^{-2}$). The disordered curves have been shifted for clarity, with dashed lines showing their zero. (d) The top-gate density dependence of the resistance at a narrow resonance for different values of the impurity density.

Figure 4

(Color online) Disorder averaged resistance (a) and Fano factor (b) as a function of $\Delta n_{\text{tg}}$ for different values of $n_{\text{imp}}$ [parameters and color coding the same as in Fig. 3b]. (c) The resistance difference $\Delta R$ where we have subtracted the resistance at $n_{\text{imp}}=1.5\times {10}^{12}{\text{cm}}^{-2}$ where the transport is diffusive, to emphasize the survival of broad oscillations for $n_{\text{imp}}$ as high as ${10}^{12}{\text{cm}}^{-2}$. (d) Disorder averaged $R$ as a function of $\Delta n_{\text{tg}}$ for different values of $n_{\text{bg}}$.