Theoretical investigation of how edge states are destroyed in disordered mesoscopic samples

We report theoretical investigations on how edge states are destroyed in disordered mesoscopic samples by calculating a “phase diagram” in terms of energy versus disorder strength $\left(E,W\right)$, and magnetic field versus disorder strength $\left(B,W\right)$, in the integer quantum Hall regime. It is found that as the disorder strength $W$ increases, edge states are destroyed one by one if transmission eigenchannels are used to characterize the edge states. Near the insulating regime, transmission eigenchannels are closed one by one in the same order as edge states are destroyed. To identify those edge states which have survived disorder, we introduce a generalized current density that can be calculated and visualized.

Figure 1

(Color online) (a) and (b) are “phase diagram” of edge states of a zigzag graphene ribbon for energies near the Fermi level (Dirac electrons): (a) in the $\left(E,W\right)$ plane; (b) in the $\left(B,W\right)$ plane. (c) The order of closing of eigenchannels at large disorder. Color coding is for number of eigenchannels.

Figure 2

(Color online) Phase diagram of edge states in $\left(E,W\right)$ plane for: (a) holelike charge carriers in the zigzag graphene ribbon; (b) electrons in a square lattice. The main difference between the two phase diagrams is due to opposite charges of hole and electron.

Figure 3

(Color online) Current density of two edge states flowing from left to right along $x$ direction, for a square lattice of $100\times 40$ sites in the absence of disorder. Here $E=0.4$ and $\varphi =0.052$. Note that the color scale is different for the two panels. Insets: classical skipping orbits.

Figure 4

(Color online) Current density along $x$ direction for the square lattice model as that of Fig. 3, with disorder strength $W=1$, energy $E=0.4$, and magnetic field $\varphi =0.052$. (a) For an edge state that has survived disorder; (b) edge state that has been destroyed by disorder.

Figure 5

(Color online) Current density along $y$ direction for the square lattice model as that of Fig. 3, with disorder strength $W=1$, energy $E=0.4$, and magnetic field $\varphi =0.052$. (a) For an edge state that has survived disorder; (b) edge state that has been destroyed by disorder.