Contact resistance and shot noise in graphene transistors

Potential steps naturally develop in graphene near metallic contacts. We investigate the influence of these steps on the transport in graphene field effect transistors. We give simple expressions to estimate the voltage-dependent contribution of the contacts to the total resistance and noise in the diffusive and ballistic regimes.

Figure 1

(Color online) (a) Transmission probability ${\mathcal{T}}_{\text{step}}$ across the metal/graphene interface as a function of transverse momentum $k_y$ for several lengths $d$ of the potential rise. The ratio between Fermi wave vectors in bulk graphene and below the metal is set to $k_F^{\left(m\right)}/k_F^{\left(g\right)}=3$ (unipolar contact). A schematic of the device and corresponding Fermi wave-vector profile $k_F\left(x\right)$ are shown in the inset. (b) Same curves for a bipolar contact with $k_F^{\left(m\right)}/k_F^{\left(g\right)}=-3$.

Figure 2

Contact conductance (left) and Fano factor (right) associated with a single-potential step as functions of the Fermi wave vector in bulk graphene $k_F^{\left(g\right)}$ for several values of the dimensionless parameter $k_F^{\left(m\right)}d$. The maximal value of the conductance is given by $G_0=4e^{2}N/h$, where $N=k_F^{\left(m\right)}W/\pi$.

Figure 3

Conductance $G$ of a ballistic Fabry-Pérot cavity as a function of the dimensionless parameter $k_F^{\left(g\right)}L$. The maximal conductance $\left(G_0=4e^{2}N/h\right)$ is controlled by the number of available propagating modes in the leads $N=k_F^{\left(m\right)}W/\pi$. Here $k_F^{\left(m\right)}L=1.5$, which means $L=27\text{nm}$ for a density $n^{\left(m\right)}={10}^{11}{\text{cm}}^{-2}$, and the aspect ratio $W/L$ is large [in practice $W/L\gtrsim 4$ is sufficient (Ref. 15)]. For definiteness, we have assumed that the graphene is $n$-doped underneath the metal $\left(k_F^{\left(m\right)}>0\right)$. In this case, the minimal conductivity is reached at a nonzero negative density in the central channel. For other metals that produce $p$-doped graphene after charge transfer, one should obtain the symmetrical curves (with respect to vertical axis $k_F^{\left(g\right)}L=0$) of the ones presented here, and the shift of the minimal conductivity would occur at positive density.

Figure 4

Fano factor for a ballistic graphene Fabry-Pérot cavity as a function of the dimensionless parameter $k_F^{\left(g\right)}L$ for various values of $d/L$. Same parameters as in Fig. 3, in particular $k_F^{\left(m\right)}L=1.5$, while the aspect ratio $W/L$ is large (Ref. 15), namely, $W/L\gtrsim 4$.