Transport through a disordered topological-metal strip

Features of a topological phase, and edge states in particular, may be obscured by overlapping in energy with a trivial conduction band. The topological nature of such a conductor, however, is revealed in its transport properties, especially in the presence of disorder. In this work, we explore the conductance behavior of such a system with disorder present, and contrast it with the quantized conductance in an ideal two-dimensional topological insulator. Our analysis relies on numerics on a lattice system and analytics on a simple toy model. Interestingly, we find that as disorder is increased from zero, the edge conductivity initially falls from its quantized value; yet, as disorder continues to increase, the conductivity recovers, and saturates at a value slightly below the quantized value of the clean system. We discuss how this effect can be understood from the tendency of the bulk states to localize, while the edge states remain delocalized.

Figure 1

A two-dimensional lattice model of a topological conductor. The model is based on the Kane-Mele model (Refs. 1 and 2) on the honeycomb lattice, which is denoted by dashed lines. The Kane-Mele model consists of nearest-neighbor hopping $t_1$ (green) between the two (A, B) sublattices of the honeycomb lattice (denoted by empty and filled circles, respectively), and a complex second-nearest-neighbor hopping (a spin-orbit coupling term) with opposite sign when clockwise ($+i\lambda$, red) and counterclockwise ($-i\lambda$, blue), as indicated by the curved arrows in the figure. In addition to sites of the honeycomb lattice, we include a new set of sites (C) at the centers of the hexagonal plaquettes of the honeycomb lattice, denoted by cross marks. The C sites form a triangular sublattice, and nearest-neighbor hopping between them $t_2$ (brown) forms a metallic band. To explore the interplay between the topological insulators helical surface states and the bulk metallic band we mix the two systems by allowing hopping between the C sites and the honeycomb lattice sites $t_3$ (purple).

Figure 2

(a) Energy spectrum of the topological conductor without disorder. Bands at the top and bottom are the bulk states of the honeycomb lattice. The band in the middle is the metallic bulk from the interstitial sites. The parameters in units of $t_1$ are $\lambda =0.1$, $t_2=0.1$, $t_3=0.1$, and the width of the strip is $M=30$ zigzag lines. (b) Zoom of the band structure in (a). Different Fermi-energy regimes are indicated. The region between II and III has the same properties as region I.

Figure 3

Lattice system with a width of $M$ zigzag lines. The disordered region of length $L$ is the conductor which is connected to a lead on either side. Dashed rectangles show the unit cells of the strip geometry.

Figure 4

Illustration of the “dip-and-recovery” feature of the edge-channel conductance. Conductance $g_n$ of different channels is plotted as a function of disorder strength $W$. The edge-channel conductance shows a sharp minimum $g_{\min }$ at finite disorder $W_0$ and a broad maximum $g_{\max }$. Parameters are $E_F=0.1$, $\lambda =t_3=0.1$, and $M=L=20$. The inset shows the total conductance. The dip is no longer visible, but one can see that the conductance remains at a value different from zero for a large range of disorder strengths. The dashed line marks the conductance $e^2/h$ of a single channel.

Figure 5

Dependence of the conductance dip and maximum on system size (Region I). (a)–(c) Dependence on length $L$ of the system for $M=20$. Note the log scale on the $y$ axis in (a); the dashed line is a guide to the eye to emphasize the exponential dependence. (d)–(f) Dependence on system width for (d), (e) $L=40$ and (f) $L=20$. Parameters: $E_F=0.1$, $\lambda =t_3=0.1$. While the observed feature is nearly independent of the system width, the conductance minimum decreases exponentially with system length. The maximum conductance is nearly independent of the system length and the deviation from the quantized value presumably results from backscattering at the interface between leads and disordered region.

Figure 6

Region I. Dependence of the edge-channel conductance on system parameters. Dependence on (a) hybridization $t_3$ ($\lambda =0.1$, $M=20$) and (b) second-nearest-neighbor hopping $\lambda$ ($t_3=0.1$, $M=10$) for $E_F=0.1$ and $L=20$. (c) Dependence on Fermi energy with $\lambda =t_3=0.1$, $M=10$, and $L=20$; (d) indicates the energy range from $E=0.1$ to $-0.3$. Decreasing the hybridization $t_3$ and increasing the second-nearest-neighbor hopping $\lambda$ increases the conductance over the entire disorder range. Moving the Fermi energy away from the metallic bulk has the same effect until $E_F$ is too close the honeycomb valence band when the characteristic feature is starting to vanish.

Figure 7

Region II. Conductance of most conducting channel for a Fermi energy deep within the metallic band. (a) Dependence on disorder strength for different hopping strengths $t_3$. $E_F=0.2$, $\lambda =0.1$, $M=20$, and $L=20$. (b) Conductance of each channel at different Fermi energies. For Fermi energies between $E=0.21$ and $0.22$, a single highly conducting channel exists possibly connected to surface resonances. Parameters are $W=1.3$, $\lambda =t_3=0.1$, $M=10$, and $L=20$. (c) Conductance as a function of disorder at $E_F=0.21$ for $\lambda =t_3=0.1$, $M=10$, and $L=20$. A single channel stands out, highly conducting at finite disorder. Dashed line indicates the disorder strength taken in (b). (c) shows the energy range from $E=0.18$ to $0.23$.

Figure 8

Region III. Conductance of the edge state for different Fermi energies. Parameters are $\lambda =t_3=0.1$ and $M=L=20$. Inset indicates the energy range from $E=0.35$ to $0.45$. At Fermi energies close to the metallic bulk ($E_F=0.35$), disorder-induced states above the bulk band still lead to a dip in the conductance. The closing of the honeycomb bulk gap due to disorder quickly decreases the conductance in the entire energy range.

Figure 9

Band structure with Rashba coupling (a) ${\lambda }_R=0.1$ and (b) ${\lambda }_R=0.3$. Parameters are $\lambda =t_3=0.1$ and $M=30$. Rashba coupling reduces the band gap of the honeycomb subsystem.

Figure 10

Region I with Rashba coupling. Conductance of one edge channel for (a) $t_3=0.05$ and (b) $t_3=0.1$ for different Rashba coupling strengths ${\lambda }_R$. Parameters are $E_F=0.05$, $\lambda =0.1$, $M=10$, and $L=20$. We plotted only a single-spin orientation for readability. Rashba coupling overall reduces the conductance. The effect is enhanced for larger hybridization between the honeycomb and metallic subsystems.

Figure 11

Illustration of the toy model. Bulk modes are independent of each other since we assume that the bulk channels describe the eigenmodes of the diffusive bulk. The edge mode couples to the bulk through $(N-2)$ scattering sites, giving in total $(N-2)M$ couplings $V_n\left(j\right)$. The coupling strength and phase are uniformly distributed.

Figure 12

Conductance contribution via different channels as a function of coupling strength ($x$ axis) for the system of a single chiral and bulk mode with a single coupling: from edge to edge $g_{e-e}$, edge to bulk $g_{b-e}$, bulk to edge $g_{e-b}$, and bulk to bulk $g_{b-b}$. The edge-to edge-conductance has a minimum at finite disorder before recovering to the quantized value. The conductance is in units of $\frac{e^2}{h}$.

Figure 13

(a) Sum of the conductances of chiral-to-chiral (b) and bulk-to-chiral (c) channels for $M=4$ and $N=5$ (blue), 7 (green), 9 (red), and 11 (cyan) for $k_e=\frac{1.4\pi }{a}$ and $k_b=\frac{1.2\pi }{a}$. The inset in (a) shows the exponential decay of the minimum conductance with system length in agreement with the numerical calculation in Fig. 5a. The conductance is in units of $\frac{e^2}{h}$.

Figure 14

(a) Sum of conductance of chiral-to-chiral (b) and bulk-to-chiral (c) channels for $N=5$ and $M=4$ (blue), 7 (green), 10 (red) for $k_e=\frac{1.4\pi }{a}$ and $k_b=\frac{1.2\pi }{a}$. The conductance minimum is unaffected by changes in the system width in agreement with the results of the numerical calculation. (c) Renormalizing the disorder strength by $\sqrt{M}$, $g_{e-b}$, which is the dominant contribution to the conductance in the region of the minimum, becomes independent of the system width. The conductance is in units of $\frac{e^2}{h}$.

Figure 15

Considering a superposition of physical modes in lead $R$, one can cancel the unphysical modes in lead $L$ by destructive interference. The coefficients ${\beta }_{k_m,n}$ have to be chosen such that, after backwards propagation, the exponentially increasing modes in lead $L$ have zero weight. The coefficients ${\beta }_{k_m,n}$ are given by Eq. (A6). The coefficients ${\gamma }_{q_n^{},q_m^{\prime }}$ of the resulting superposition (of only physical modes) in lead $L$ are given by Eq. (A7).