Numerical study of negative nonlocal resistance and backflow current in a ballistic graphene system

In addition to the giant peak of the nonlocal resistance $R_{\text{NL}}$, an anomalous negative value of $R_{\text{NL}}$ has been observed in graphene systems, although its formation mechanism is not quite understood yet. In this work, utilizing the nonequilibrium Green's function method, we calculate the local-current flow in an H-shaped noninteracting graphene system located in a ballistic regime. Similar to the previous conclusions obtained from viscous hydrodynamics, the numerical results show that a local-current vortex appears between the nonlocal measuring terminals, and it induces a backflow current and a remarkable negative voltage drop at the probe. Specifically, the stronger the vortex is, the more negative $R_{\text{NL}}$ becomes. In addition, spin-orbital coupling is added as an additional tool to study this exotic vortex, although this coupling is not a driving force for the arising vortex at all. Moreover, a breakdown of the nonlocal Wiedemann-Franz law is obtained in this ballistic system, and two experimental criteria are provided to confirm the existence of this exotic vortex. Notably, it is shown that the vortex actually originates from a collision between the flowing current and the boundaries, due to the long electron mean free path and the resulting ballistic transport caused by the specific linear spectrum of graphene.

Figure 1

The schematic diagram of the H-shaped four-terminal system. The current is injected from lead 1 to lead 2. The voltage signal is detected on lead 12 for the local resistance, and on lead 34 for the nonlocal resistance. The rectangle circled by the black dashed line indicates the region where the local-current flow is calculated.

Figure 2

The local current flow in the black dashed rectangle marked in Fig. 1 when ${\lambda }_R=0.1$. (a) $E_F=-0.15$; (b) $E_F=-0.2$; (c) $E_F=-0.3$. The size of each arrow is proportional to the strength of the local current vector. It is obvious that a counterclockwise vortex emerges in panel (a) and gradually disappears with the increase of $|E_F|$ in panels (b) and (c). All three panels have sufficiently high resolutions that they can be zoomed in to show the vortex clearly.

Figure 3

(a) The local resistance $R_{\text{L}}$ (red) and nonlocal resistance $R_{\text{NL}}$ (blue) when ${\lambda }_R=0.1$. (b) The nonlocal resistance $R_{\text{NL}}$ with different Rashba strengths. The negative peak of $R_{\text{NL}}$ reaches its maximum at ${\lambda }_R=0.1$ (blue), and it gradually decreases with the increase of ${\lambda }_R$ (red and black). (c) and (d) The same results as those of panels (a) and (b) with double-sized parameters, where the oscillations do not seem evident now.

Figure 4

The local current flow in the black dashed rectangle marked in Fig. 1 when $E_F$ is fixed at $-0.15$. From panels (a)–(c), the Rashba strength increases from ${\lambda }_R=0.1$ to 0.15 and 0.2, respectively. The size of each arrow is proportional to the strength of the local current vector. We find that the vortex is evident with a smaller Rashba effect, and it gradually disappears with the increase of ${\lambda }_R$. Since the vortex in panel (c) is too weak, the size of the arrows in this panel has been doubled compared with those in panels (a) and (b). All three panels have sufficiently high resolutions that they can be zoomed in to show the vortex clearly.

Figure 5

(a) The local thermal conductance ${\mathrm{\Theta }}_{\text{L}}$ (red) as a function of the Fermi energy $E_F$, compared with the local WF law ${\mathcal{L}}_0^{-1}{R}_{\text{L}}/T$ (blue) at $T=500\text{K}$ and ${\lambda }_R=0.1$. (b)–(d) The nonlocal thermal conductance ${\mathrm{\Theta }}_{\text{NL}}$ (red) and the corresponding nonlocal WF law ${\mathcal{L}}_0^{-1}{R}_{\text{NL}}/T$ (blue) from ${\lambda }_R=0.1$ to 0.2 and 0.3, respectively.

Figure 6

The local current flow in the black dashed rectangle marked in Fig. 1 when a list of very strong Anderson disorder is added at the boundaries. All other parameters are the same as those in Fig. 2. It is obvious that a counterclockwise vortex remains, and it tends to move forward into the central region compared with Fig. 2.

Figure 7

The local current flow of a square lattice whose size and Hamiltonian are the same as those in Fig. 1. It is obvious that no vortex exists anymore.