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Recent experiments find the signal of giant nonlocal resistance $R_{NL}$ inH-shaped graphene samples due to the spin/valley Hall effect. Interestingly,when the Fermi energy deviates from the Dirac point, $R_{NL}$ decreases to zeromuch more rapidly compared with the local resistance $R_L$, and the well-knownrelation of $R_{NL}\propto R_L^3$ is not satisfied. In this work, based on thenon-equilibrium Green's function method, we explain such transport phenomena inthe H-shaped graphene with Rashba spin-orbit coupling. When the Fermi energy isnear the Dirac point, the nonlocal resistance is considerably large and is muchsharper than the local one. Moreover, the relationship between the Rashbaeffect and the fast decay of $R_{NL}$ compared with $R_L$ is furtherinvestigated. We find that the Rashba effect does not contribute not only tothe fast decay but also to the peak of $R_{NL}$ itself. Actually, it is theextremely small density of states near the Dirac point that leads to the largepeak of $R_{NL}$, while the fast decay results from the quasi-ballisticmechanism. Finally, we revise the classic formula $R_{NL}\propto R_L^3$ byreplacing $R_{NL}$ with $R_{Hall}$, which represents the nonlocal resistancemerely caused by the spin Hall effect, and the relation holds well.

Numerical study of the giant nonlocal resistance in spin-orbital coupled graphene