Meandering conduction channels and the tunable nature of quantized charge transport

Our second central result is implied by the first: the completely natural appearance of a broad conduction channel in the QAHE. It arises as the combination of the Chern bands of finite width separated by a gap on one hand, and the confining potential on the other, yielding a spatially inhomogeneous setup in the following way. Locally, the chemical potential can lie below both bands; within the lower (valence) band; between the bands; within the upper (conduction) band; or above both bands (Fig. 1). Most basically, the conduction properties in each of these situations are different, and hence changing the chemical potential will also change where the current flows.

Consider the horizontal line in Fig. 1C, where we sketched the local density of states of a disorder-free system as a function of the transverse position. Here, due to the parabolic potential, the energy at the intersection of the horizontal line with the bottom of the valence band lies within the gap between valence and conduction bands in the sample center. If one places the chemical potential at this energy, there must then be two spatially extended regions on either side of the center, where the chemical potential lies within the valence band. Quantum mechanically, these regions host equal and opposite equilibrium anomalous Hall currents (18), turning them into conducting channels with intrinsically finite widths. Near the boundary, no current carriers are available as the confining potential pushes the valence band above the chemical potential. At the same time, the middle region, where the chemical potential lies between the bands, is incompressible for the clean system, and can also carry current in presence of Coulomb interactions, as we discuss below. In the presence of disorder, this region hosts only localized states with a localization length of the order of a few lattice spacings.

We now turn to the discussion of the nonequilibrium current, i.e. the excess current distribution due to a chemical potential difference $\mathrm{\Delta }$ between the lateral ends of the sample (denoted C, D in Fig. 1A). This leads to redistribution of charge in the two meandering channels; see Materials and Methods. However, while in QHE systems, we have purely chiral edge states, this is different for the QAHE, where each Chern band contains states propagating in both directions (which we refer to as propagating and counterpropagating)—rendering current quantization more delicate. How then is the quantization achieved in the Chern insulator?

In establishing the quantized conductance it turns out that—as in the QHE—disorder plays a crucial role; see Materials and Methods Fig. 5. In a nutshell, disorder Anderson-localizes all the states which are not protected by their chiral nature. This, in particular, removes the ability of all counterpropagating states to carry current. Crucially, first, the number of states which survive localization on each side of the sample (separated by the incompressible region) is given by the Chern number; and, second, their geometrical appearance and spatial distribution can vary wildly depending on disorder. Indeed, the nonequilibrium current distribution depicted in Fig. 2 (before coarse-graining) has microscopic fluctuations which are greatly in excess of the mean current densities. For the more detail-oriented reader, we emphasize with a more expansive discussion in Materials and Methods how differently disorder asserts itself in the QAHE in comparison with the QHE.

Generically, the current-carrying region can have the same support as the region where the chemical potential resides in the Chern band: the appearance of a broad current-carrying region is entirely natural in this picture. Its width is eventually determined by disorder, and primarily by a combination of the bandwidth of the Chern band, and the steepness of the potential in the region—itself externally tunable—where the chemical potential intercepts the Chern band. Our results for the four-terminal resistance are consistent with a previous analysis of the two-terminal situation in a disordered Chern insulator in a transverse electric field (19). Note that this reference considered step-like electrostatic potentials, and also found disorder-broadened conduction channels along lines where the potential jumps occur.

The full disordered and interacting problem, as usual, does not offer a possibility for a theoretical analysis on the scale and detail of our above treatment of noninteracting disordered electrons. We can, nonetheless, make a number of robust qualitative observations, which turn out to be conceptually important.

In the absence of Coulomb interactions, the quantized nonequilibrium current is carried entirely by the additional electrons and holes (due to applied bias voltage) in the respective compressible strips corresponding to meandering channels. The effect of adding Coulomb interactions is then (at least) twofold—we analyze a simple illustrative phenomenological model in Materials and Methods, and here mention the results of importance for the following discussion of experiment.

First, within each channel, Fig. 6, the charge distribution flattens, while the peaks of the current distribution shift toward the center of the sample, with the inner side of the channels carrying an increasing amount of current on account of the large self-consistent electric field that develops at this position.

The second is qualitatively even more striking: the added electrons and holes in the two respective channels generate electric field in the incompressible region separating them, just like a charged capacitor with two parallel plates; thus inducing a quantized bulk current there, which is absent in the noninteracting model ($U=0$ vs. $U>0$ curves in Fig. 6). We point out that in the experimental setup (11) the current in the central incompressible region is strongly suppressed by the screening due to a nearby metallic top gate. With this in hand, we can discuss the following central question.

The question regarding the qualitative nature of the current distribution in the QHE or QAHE setting is often, but not always (6, 20), phrased in the form of a dichotomy: does the current flow in the incompressible bulk or along the compressible edges ?

Our answer to this question is twofold, and it negates the strict dichotomy: First, it depends on the detailed circumstances; and second, the current can do both at the same time. The only thing that matters for the quantization is the total electrochemical potential drop, whose detailed shape is fixed by electrostatics, disorder, and the Coulomb interactions. Only the distribution of the current is influenced in this way, not its total value. This is the topological censorship at work.

We are now in a position to connect our theoretical picture with the beautiful experimental results of refs. 10 and 11 on the QAHE in Cr-doped (Bi,Sb)${}_2$Te${}_3$ heterostructures. The Stanford experiment (10) measured the potential along the edge using local (lateral) voltage probes. It found a potential profile consistent with the current distribution arising from a translationally invariant and strongly anisotropic conductivity tensor, of the type discussed in ref. 21. This experiment seems to rule out the picture of the current flowing exclusively through a narrow edge channel, but it says little about the microscopic nature of the bulk and the conduction process through it, and indeed about the origin and the scales of the nonzero diagonal entries of the conductivity tensor. As usual, diagonal components of this tensor are associated with dissipation processes.

The Cornell experiment (11) used a micro-SQUID to measure the magnetic field distribution above the sample. From this, the current distribution was inferred for different experimental parameters, which is depicted in Figs. 1B and 3. The theory-experiment agreement in the latter figure is our final central result.

The most striking outcome of these measurements is the observation of a pair of broad regions in the bulk—rather reminiscent of our meandering channels—carrying the nonequilibrium current in the quantized regime. This is again at variance with the simple picture of charge transport in a narrow edge channel, while it also suggests considerably more spatial structure than the Stanford experiment. Given details of the sample geometry, preparation, gating are different between these experiments, our observation of the nonuniversality of the transport mechanism is plausibly supported by this discrepancy. The authors of ref. 11 suggest an interpretation in which this current is carried entirely in the incompressible regions. They support this claim by emphasizing that regions carrying most of the nonequilibrium current are also those in which the magnetic response to a modulation of the gate voltage reaches its maximum absolute value. The sequence of current profiles in the quantized regime is therefore interpreted as follows. The curve corresponding to the highest chemical potential on the quantized Hall plateau has two peaks, attributed to two incompressible strips. These strips gradually move toward the center of the sample as the chemical potential is reduced, and finally merge.

Based on our discussion of meandering channels, we do not see any particular reason why there should be no transport at all in the compressible regions, and at any rate, the experimental data are not precise enough to exclude this possibility. In fact, the current never seems to entirely vanish near the center of the sample, which, according to the proposed picture, is supposed to host a partially filled (hence compressible) conduction band in a finite range of chemical potentials. Also, our attempts to derive a mechanism of a purely incompressible bulk transport for realistic models, based on a Hartree–Fock treatment, were unsuccessful. For an explicit illustration of the possibility of simultaneous transport in compressible and incompressible regions, we refer to the above-mentioned Fig. 6 and discussions of the phenomenological model in Materials and Methods.

Our results on the quantized Hall response carried by compressible meandering channels then suggest an alternative interpretation. The large peaks are now attributed to such compressible meandering channels built from states lying inside the valence band. The residual current near the center might result from the transverse electric field across an incompressible region, due to charge imbalance between the two meandering channels, whose small value is compatible with the strong screening of Coulomb interactions due to the close proximity ($\sim$40 nm) of the metallic top gate. When the chemical potential is reduced, the meandering channels move gradually toward the center. As they do so, they encounter a weakening of the local electric field of the confinement potential, and hence broaden.

In passing, we note the asymmetry of the experimental nonequilibrium current, showing a sign-changing structure on one side, and a uniform sign on the other. This appears to involve heating, in other words, physics beyond the linear response regime (11). A sign-changing pattern is also promoted by the fact that the Berry curvature of each Chern band changes sign as a function of energy; see Fig. 7 in Materials and Methods. In presence of both confining potential and disorder, this translates into a tendency to carry current in opposite directions on opposite sides of the meandering channel, that appears to explain the red traces near the sample edge in the Right panel of Fig. 3.

In the next few paragraphs, we present an explicit account of how the genesis of the conductance plateau differs in basic ways between the QAHE and the QHE. This is a consequence of the presence of counterpropagating states in the Chern band, which can spoil quantization in the clean case, and need to be localized by disorder. This happens as follows. Consider a situation where the chemical potential $\mu$ lies below the bottom of the valence band near the edge, and above its top near the center of the sample. In the absence of disorder, and for an infinitely long strip, translation symmetry along the $x$ direction implies that the momentum $k_x$ is a good quantum number. The presence of a transverse confining potential along $y$ yields a discrete spectrum labeled with an integer quantum number $n$. Let us denote by ${ϵ}_n(k_x)$ the corresponding energy spectrum. In such a quasi 1D geometry, the equation ${ϵ}_n(k_x)=\mu$ has a finite number of solutions $(n_i,k_{n,i})$, each of which defines a single conducting channel. Recall that we have two spatial intervals, on either side of the center of the strip, where the chemical potential lies within the valence band. Assuming that quantum tunneling processes between these two intervals at energy $\mu$ can be neglected, we get half of the conducting channels with spatial support lying in the upper $y$ interval and the other half in the lower interval. Let us now concentrate on the former subset of channels. Their corresponding wave-functions are subject to a confining force directed toward the center of the strip. The simultaneous presence of this transverse force and the valence band with nonzero Chern number, $\mathrm{Ch}$, creates a left–right asymmetry among these channels. Denoting by $N$ (resp. $M$) the number of channels with a positive (resp. negative) group velocity $d{ϵ}_n(k_x)/d{k}_x$, we have the general relation $N-M=\pm \mathrm{Ch}$, where the sign depends on the direction of the transverse force along the $y$ axis. This relation follows from the existence of a spectral flow ${ϵ}_n(k_x+2\pi /a)={ϵ}_{n\pm \mathrm{Ch}}(k_x)$.

In order to evaluate the four-terminal Hall conductance of a finite clean strip, we need to specify the four-terminal transmission matrix (22). Let us label the four reservoirs by letters as indicated in Fig. 1. To simplify this discussion, we assume that these transmission matrix elements are arranged in four groups: $T_{\mathrm{CD}}\simeq T_{\mathrm{DC}}=(N+M)(1-ϵ)$ (direct transmission), $T_{\mathrm{AC}}\simeq T_{\mathrm{DA}}\simeq T_{\mathrm{BD}}\simeq T_{\mathrm{CB}}=Nϵ$ (chiral propagation), $T_{\mathrm{BC}}\simeq T_{\mathrm{DB}}\simeq T_{\mathrm{AD}}\simeq T_{\mathrm{CA}}=Mϵ$ (chiral counterpropagation), $T_{\mathrm{AB}}\simeq T_{\mathrm{BA}}\simeq 0$ (absence of direct scattering between lateral voltage probes for a large enough sample). The small parameter $ϵ$ indicates the probability that a wave packet injected in any of the $2N$ propagating channels or any of the $2M$ counterpropagating ones will tunnel into one of the two lateral voltage probes. Because here we focus on an ideal system, we neglect in this model the contact resistances at the Left (C) and Right (D) reservoirs. This implies that all incoming wave packets from these reservoirs are eventually transmitted to other reservoirs, and this translates into $\sum _{{\mathrm{R}}^{\prime }\ne \mathrm{R}}{T}_{{\mathrm{R}}^{\prime }\mathrm{R}}=M+N$ for $\mathrm{R}=\mathrm{C},\mathrm{D}$. Following standard procedure, we obtain the Hall resistance, in units of h/e${}^2$,

This formula is quite instructive. First, it shows that no quantization is expected in the presence of counterpropagating channels ($MN\ne 0$), because the Hall resistance depends on the value of the tunnel probability $ϵ$. The quantized value $R_{\mathrm{H}}=1/N$, independent of $ϵ$, is recovered only in the absence of counterpropagating channels. The ubiquity of such channels for clean topological insulators in the presence of a smooth confining potential is in sharp contrast with the situation for the QHE Landau levels in a strong magnetic field, where counterpropagating edge channels are entirely absent on this level of description for a smooth confining potential.

Therefore, the role of disorder in the stabilization of sharply quantized Hall plateaus is significantly different in these two situations. For the QAHE, we have just seen that disorder is necessary to eliminate counterpropagating channels via Anderson localization, and thus to ensure a quantized Hall response carried by meandering channels. In the QHE, the clean system exhibits perfectly quantized Hall conductance as long as the chemical potential lies between two consecutive Landau levels. For a fixed electronic density, this occurs only for a discrete set of magnetic field values, corresponding to an integer number of filled Landau levels. Therefore the quantized Hall plateaus have a vanishing width, as a function of external magnetic field for a clean system. The main role of disorder in the QHE, as has been realized for a long time, is to generate plateaux of a finite width by allowing the chemical potential to change smoothly inside the gaps as the magnetic field is varied.

As stated in the introduction, a disorder-based mechanism by which edge states in two-dimensional noninteracting electron systems in a strong magnetic field can spread deep inside the system bulk has been known for a long time (13–16). For such infinite systems, all states are localized, except at energies close to the centers of disorder-broadened Landau levels, corresponding to phase transitions between quantum Hall plateaus. For finite strips, almost extended edge states develop at most energies, except near plateau transitions, where the bulk localization length becomes comparable to the system width. When this happens, it has been shown that edge states spread into the bulk, thus forming broad conduction channels, as shown for example in figure 6C in ref. 16. It is however quite clear on this figure that such broad edge states exist only in a rather small energy range around the plateau transitions. By contrast, as we have shown, broad meandering conduction channels exist in disordered Chern insulators in the presence of a smooth confining potential for the whole energy range corresponding to a quantum Hall plateau. Anderson localization transitions in the same model as the one studied in this paper have been investigated in refs. 33 and 34. For moderate disorder, the phase diagram in the middle of figure 1 in ref. 33 shows that the Anderson transitions between Chern insulators and topologically trivial phases occur at energies inside the two bands of the clean system. This suggests that, as we have checked numerically in the absence of the smooth confining potential, the width of edge states remains very small, of the order of a few lattice spacings, when the energy lies inside the gap of the clean system, provided the disorder remains below the critical value where only the topologically trivial phase remains. This is also consistent with the conclusion of ref. 34. While the basic physics involved in the formation of broad QHE edge states and meandering conducting channels in Chern insulators may lead to the same universality class regarding Anderson localization transitions, we again emphasize that the combination of a wide band kinetic energy in the Chern band and of a smooth confining potential enhances the energy range in which such states exist in comparison to disorder-broadened Landau levels. This obviates the need to fine-tune the chemical potential near a plateau transition for their observation.

The confinement potential, in particular its interplay with the finite bandwidth of the Chern insulator, plays an important role for the current distribution. In order to show how a parabolic potential can appear in an electrostatic problem, and to estimate its size, we have calculated the electrostatic potential in a model of the experimental system using the finite element method. Materials and Methods Fig. 4 displays the results of this calculation, which yields a quadratic confining potential of the form utilized in our modeling. The difference between its maximum and minimum turns out to be of the order 1 eV for a backgate voltage of 100 V. We note that, depending on the device configuration, one can obtain either sign of the parabola.

Our above interpretation of current profiles in ref. 11 is based on a simple phenomenological model. Without Coulomb interactions, the quantized nonequilibrium current is carried by additional electrons and holes in the respective compressible strips hosting the meandering channels, as shown in Fig. 8. With Coulomb interactions, this pair of oppositely charged strips creates an electrostatic field across the central incompressible region, leading to an additional drop $\delta V_{\mathrm{I}}$ in the electrochemical potential between the two strips. A total current $I_{\mathrm{I}}$ then also flows in this central incompressible region, characterized by the same quantized Hall conductance as the meandering channels.

Based on this picture, we introduce an effective model, in which we assume translation invariance along the wire (obtained for example after averaging over disorder). Let us denote by $\varphi (y)$ the local electrostatic potential seen by electrons, the sum of the equilibrium potential ${\varphi }_0(y)$ and the induced potential, ${\varphi }_{\mathrm{ind}}(y)$ created by the charged compressible strips, that appears under finite dc voltage bias. We assume that the particle density at energy $E$ is a function of the kinetic energy in the valence band, which is given by $E+e\varphi (y)$. Specifically, $n(y,E)=f(E+e\varphi (y))/a_x{a}_y$, where $f$ is a positive function with unit total weight, and support lying in the $[0,W]$ interval, $W$ being the energy width of the valence band, and $a_x$ and $a_y$ are the lattice constants along $x$ and $y$ directions. A second key assumption is that the local current density $j(y,E)$ carried by a state at energy $E$ is given by $j(y,E)=(e^2/h)f(E+e\varphi (y)){\varphi }^{\prime }(y)$. In equilibrium, local particle density $n(y)$, and current density $j(y)$, are obtained by integrating the energy-resolved functions $n(y,E)$ and $j(y,E)$ over $E$ up to the equilibrium chemical potential $\mu$. The incompressible regions are defined within the regions which are obtained from the equation $\mu +e{\varphi }_0(y)>W$, and the corresponding local density equals to $n_0(y)=1/a_x{a}_y$ in these regions, as expected for a filled valence band.

In the presence of a finite voltage bias, we integrate over energies up to ${\mu }_b=\mu +\delta \mu$ for $y$ corresponding to the bottom half of the strip, and to ${\mu }_t=\mu -\delta \mu$ for $y$ in the top half. This model gives a quantized Hall response for the total current $I$, obtained by integrating $j(y)$ over the whole width of the sample, as $I=(e/h)({\mu }_b-{\mu }_t)$, or equivalently $I=(e^2/h)({\varphi }_t-{\varphi }_b)$ in terms of the total electrochemical potential difference across the sample. We denote by $\delta n(y)$ and $\delta j(y)$ the change in local particle and current densities after introducing the bias $\delta \mu$.

The induced electrostatic potential ${\varphi }_{\mathrm{ind}}(y)$ reads

where we assume an infinitely long strip, and the presence of a perfectly conducting top electrode located at distance $d$ above the sample. In the Cornell experiment $d=40$ nm. To the first order in $\delta \mu$,

To solve the coupled Eqs. 4 and 5, one may eliminate either $\delta n(y)$ or ${\varphi }_{\mathrm{ind}}(y)$ from one of these equations, yielding an inhomogeneous linear integral equation for the remaining variable, which we have solved numerically. After this step, the nonequilibrium current in the incompressible region can be obtained from the equation

whereas in the lateral compressible strips

The physical interpretation of this expression is quite simple. The first term arises from the change in the local electron density in the nonequilibrium state as compared to the equilibrium situation, in the presence of the unperturbed confining potential and hence unperturbed local anomalous velocity field. The second term may be regarded as the response (in the spirit of the Kubo formula) of a locally partially filled (compressible strips) or completely filled (incompressible strips) valence band to the additional electrostatic potential created by the change in the charge distribution.

To estimate the strength of the Coulomb interaction, we first neglect ${\varphi }_{\mathrm{ind}}(y)$ in Eq. 5, and compute the maximal absolute value $\kappa (\mu )$ of the ratio $e{\varphi }_{\mathrm{ind}}(y)/\delta \mu$. It is reached when $y$ lies near the centers of the compressible strips. Assuming that they are much larger than the distance $d$ from the top gate, we estimate this ratio to be of the order of $4(R_y/W)(a_{0}d/a_x{a}_y)$, where $R_y$ is the Rydberg energy and $a_0$ is the Bohr radius. We get a rather large number (on the order of ${10}^3$) for the Cornell experiment.

Coulomb interactions modify the current profile. Most notable is a reduction of the nonequilibrium charge density $\delta n(y)$ for a given $\delta \mu$ because the Coulomb repulsion among particles on a given compressible strip increases their total potential energy. Assuming for simplicity that only the overall scale of $\delta n(y)$ is modified, but not its shape, this can be captured by a renormalization of the bias $\delta \mu$ into a much smaller $\delta {\mu }^{\ast }$, defined by $\delta {\mu }^{\ast }=\delta \mu -\kappa (\mu )\delta {\mu }^{\ast }$, so $\delta {\mu }^{\ast }=\delta \mu /(1+\kappa (\mu ))$. The new maximal value of $e{\varphi }_{\mathrm{ind}}(y)/\delta \mu$ is now equal to $\kappa {(\mu )}_{\mathrm{eff}}=\kappa (\mu )/(1+\kappa (\mu ))$, which is very close to, but smaller than 1, in the physically relevant case where $\kappa (\mu )$ is large. Note that $\kappa (\mu )$ is linear in the strength $R_y/W$ of the Coulomb interaction. So $\delta {\mu }^{\ast }$ decreases as $W/R_y$ when interactions are large, but the $e{\varphi }_{\mathrm{ind}}(y)/\delta \mu$ ratio scales to a finite limit value at large $R_y/W$. This value depends mostly on geometric parameters such as the widths of the compressible strips, their separation, and the effective range $d$ of the screened Coulomb potential.

The reduction of the nonequilibrium charge density is not uniform across the compressible strips, so that $\delta n(y)$ becomes more uniform, and hence effectively wider, than the noninteracting envelope function $f(\mu +e{\varphi }_0(y))$. These effects are clearly seen in the numerical results shown in Fig. 6.

B.D., D.K., and R.M. designed research; performed research; analyzed data; and wrote the paper.

The authors declare no competing interest.