Distributed Current Injection into a One-Dimensional Ballistic Edge Channel

TL;DR

The paper generalizes Landauer transport to a 1D ballistic edge channel that is contacted not only at endpoints but via a distributed injection from a conducting 2D half-plane. A semiclassical Poisson-based framework is developed, yielding a nonlocal filling condition that equates the injected current density from the half-plane to the fully filled density of states in the ballistic channel, with the local energy window $[-\phi(x),\phi(x)]$ governed by the interface potential $\phi(x)$. A detailed numerical solution for a symmetric two-tip injection geometry is provided, alongside analytic results for classical quasi-1D Ohmic proxies, enabling clear criteria to distinguish ballistic edge channels from diffusive edge channels in multi-terminal measurements. The work also outlines how to extend the approach to other geometries and discusses experimental estimations for STM potentiometry, offering a practical route to identify topological ballistic transport in 2D/1D hybrid systems. Overall, this framework broadens the applicability of edge-state transport analyses to realistic distributed-injection scenarios relevant for topological materials and nanoscale probes.

Abstract

We generalize Landauer's theory of ballistic transport in a one-dimensional (1D) conductor to situations where charge carrier injection and extraction are not any more confined to electrodes at either end of the channel, but may occur along its whole length. This type of distributed injection is expected to occur from the two-dimensional (2D) bulk of, e.g., a quantum spin (or anomalous) Hall insulator to its topologically protected edge states. We apply our conceptual solution to the case of two metal electrodes contacting the 2D bulk, enabling us to derive criteria that discriminate ballistic from resistive edge channels in multi-terminal transport experiments.

Figures (7)

• Figure 1: 1D ballistic edge channel with distributed injection.(a) In the standard Landauer setup, charge carriers are injected and extracted locally at the ends of a 1D ballistic channel. The contact resistance at each contact (indicated by the teal interfaces) is $R_\mathrm{c} = 1/G_0$ per spin. (b) In the contact geometry considered here, current enters and exits a 1D ballistic edge channel in a distributed manner via a conducting half-plane with sheet conductivity $\sigma$ and symmetrically positioned source (S) and drain (D) contacts (e.g., STM tips). (c) Potential $\Phi(x,y)$ in the 2D half-plane, obtained by solving the Poisson equation with boundary condition $\Phi(x,y=0^+) = \phi(x)$ at the interface with the 1D ballistic channel. The interface potential $\phi(x)$, indicated as solid orange (blue) lines on the source (drain) side, is obtained by numerically solving the filling condition. (d) The antisymmetric $\phi(x)$ (see text for details). (e) Energy diagram of charge carrier transport in the standard Landauer setup shown in (a). (f) Energy diagram for the contact geometry shown in (b).
• Figure 2: Normalized interface potential $\phi_\mathrm{ball}$ of a 1D ballistic edge channel (solid orange line and orange circles), in comparison with several resistive proxies: quasi-1D Ohmic channels $\Omega_1$ with $\sigma_{\Omega_1}= (G_0/2) L/d = 3874\,\textnormal{\textmu} {S}\square^{-1}$ (dashed blue), $\Omega_2$ with $\sigma_{\Omega_2}= 0.7\,\sigma_{\Omega_1}$ (dash-dotted yellow), $\Omega_3$ with $\sigma_{\Omega_3}= 0.5 \,\sigma_{\Omega_1}$ (dotted purple), and Ohmic lower half-plane with $\sigma_\mathrm{lhp} = G_0/2$ (solid green), (a) as a function of $x$ with $L = 1\,\textnormal{\textmu} \mathrm{m}$, $W = 0.2\,\textnormal{\textmu} \mathrm{m}$, $\sigma = 0.375\,G_0/2= 14.53\,\textnormal{\textmu} {S}\square^{-1}$ in all cases. Note that for simplicity we show only one quadrant, since $\phi(-x)= -\phi(x)$. (b)-(d) The maximum as a function of (b) the sheet conductivity $\sigma$ in the 2D half-plane ($L = 1\,\textnormal{\textmu} \mathrm{m}$, $W= 0.2\,\textnormal{\textmu} \mathrm{m}$), (c) source-to-drain distance $L$ ($\sigma = 0.25 \, G_0/2= 9.69\,\textnormal{\textmu} {S}\square^{-1}$, $W = 0.2\,\textnormal{\textmu} \mathrm{m}$), and (d) distance from the ballistic channel $W$ ($\sigma = 0.25 \, G_0/2$, $L = 1\,\textnormal{\textmu} \mathrm{m}$). The widths of the quasi-1D Ohmic channels are $d=0.01\,\textnormal{\textmu} \mathrm{m}$.
• Figure 3: (a),(c)Peak position $x^\ast$ of the interface potential and(b),(d)corresponding FWHM, as a function of (a),(b) the sheet conductivity $\sigma$ in the 2D half-plane and (c),(d) the distance $y$ from the interface ($\sigma = 0.375 \, G_0/2= 14.53\,\textnormal{\textmu} {S}\square^{-1}$), with $L = 1\,\textnormal{\textmu m}$, $W = 0.2\,\textnormal{\textmu m}$ in all cases. For the latter, the potential $\Phi(x,y)$ was considered with $y>0$ fixed, instead of $\phi(x)$. Numerical solutions for the ballistic channel are shown by orange circles, analytic solutions of quasi-1D Ohmic channels (see Fig. \ref{['fig:2']} for conductivities) $\Omega_1$ (dashed blue), $\Omega_2$ (dash-dotted yellow), and $\Omega_3$ (dotted purple), and a conducting lower half-plane (lhp) with $\sigma_\mathrm{lhp} = G_0/2$ (solid green) are plotted for comparison.
• Figure S1: The inverse width $1/\Delta x$(a) as a function of $x$ and (b) as a function of $\phi$ for the interface potential of a ballistic channel that is presented in Fig. \ref{['fig:2']} of the Main Text, i.e., for a setup with $L=1\,$µ m, $W=0.2\,$µ m, and $\sigma=0.375 \, G_0/2$.
• Figure S2: (a) The interface potential profile for distributed injection as in Fig. \ref{['fig:2']} in the Main Text ($L=1\,$µ m, $W=0.2\,$µ m), with the sheet conductivity of the half-plane of injection reduced to $\sigma = 0.125 \, G_0/2$ and discrete parametrization indicated. (b) The injected current density profiles for the interface potential profiles depicted in (a), normalized by the current density at $x^\ast_{\sigma_\Omega \rightarrow \infty}$ of a perfectly conducting Ohmic channel (see \ref{['subsubsec:pc']}). We also show here the profiles for a perfectly conducting proxy ($\phi_{\sigma_\Omega\rightarrow\infty}$) and a standard Landauer treatment of a ballistic channel with local injection ($\phi_\mathrm{local}$), considering source and drain electrodes of width $L/3$ at constant potential $\phi^\ast$ and injection/extraction uniformly distributed over their widths (see text for details).