Figure 1

(a) The chiral edge modes in the QHE regime at a Landau filling factor $\nu =1.5$. Strong backscattering between the topmost channels occurs via the bulk states. (b) The helical edge modes in the QSHE state at $B=0$. (c) The counterpropagating chiral modes in an electron-hole system (and graphene) in the QHE regime at the charge neutrality point (CNP). Small arrows show the backscattering between edge modes.Reuse & Permissions

Figure 2

(a) Top view of the sample. The perimeter of the gate is shown by the rectangle. (b) The longitudinal $R_{xx}$ ($I=1,6$; $V=3,4$) and Hall $R_{xy}$ ($I=1,6$; $V=3,9$) resistances as a function of the gate voltage at zero and different nonzero magnetic fields, $T=1.4\mathrm{K}$. The $R_{xx}$ trace at $B=11\mathrm{T}$ should be multiplied by 10. (c) The nonlocal $R_{\mathrm{NL}}$ ($I=3,9$; $V=8,4$) resistance as a function of the gate voltage at zero and different nonzero magnetic fields, $T=1.4\mathrm{K}$. The trace at $B=11\mathrm{T}$ should be multiplied by 1000. Insert: Solid traces shows the local (black) and nonlocal (red) magnetoresistance at the CNP as a function of the magnetic field. Dashed traces show the local resistance predicted by the edge plus bulk state model.Reuse & Permissions

Figure 3

The local $R_{xx}$ (a) and nonlocal (b) resistances as a function of the gate voltage and magnetic field, $T=1.4\mathrm{K}$. Dashed lines are guides to the eye to show the evolution of the resistance peaks.Reuse & Permissions

Figure 4

(a) The local resistance $R_{xx}$ ($I=1,6$; $V=3,4$) as a function of charge density $N_s$ swept through the CNP point $N_{\mathrm{CNP}}$ at different temperatures, $B=11\mathrm{T}$ [$T(\mathrm{K})\mathrm{\text{:}}1.4,10,67$]. Dashed traces show $R_{xx}$ predicted by the model (see Ref. 20 and text). Parameter values for $T=1.4\mathrm{K}$: $A=10$, $B=3$, ${\gamma }^{-1}=1.42\mu \mathrm{m}$, and $g^{-1}=1\mu \mathrm{m}$. Insert: $R_{xx}$ and $R_{\mathrm{NL}}$ as a function of $1/T$ at $B=11\mathrm{T}$. The dashed and dotted lines in the insert are fits of the data with Arrhenius functions where $\Delta =32$ and 14 K, correspondingly. (b) The nonlocal resistance $R_{\mathrm{NL}}$ ($I=3,9$; $V=8,4$) as a function of the gate voltage at different temperatures, $B=11\mathrm{T}$ [$T(\mathrm{K})\mathrm{\text{:}}1.4,10,67$]. Dashed traces show $R_{\mathrm{NL}}$ predicted by the model. Parameter values are the same as for the local resistance. Insert: Parameters $A$ (black) and $B$ (red) as a function of $1/T$ at $B=11\mathrm{T}$. The solid traces in the insert are fits of the data $A$ and $B$ with the $T^{-1}$ dependence.Reuse & Permissions