Coherent-Incoherent Crossover of Charge and Neutral Mode Transport as Evidence for the Disorder-Dominated Fractional Edge Phase

Couplings between topological edge channels open electronic phases possessing nontrivial eigenmodes far beyond the noninteracting-edge picture. However, inelastic scatterings mask the eigenmodes’ inherent features, often preventing us from identifying the phases, as is the case for the quintessential Landau-level filling factor $\nu =2/3$ edge composed of the counterpropagating $\nu =1/3$ and 1 ($1/3–1$) channels. Here, we study the coherent-incoherent crossover of the $1/3–1$ channels by tuning the channel length in situ using a new device architecture comprising a junction of $\nu =1/3$ and 1 systems, the particle-hole conjugate of the $2/3$ edge. We successfully observed the concurrence of the fluctuating electrical conductance and the quantized thermal conductance in the crossover regime, the definitive hallmark of the eigenmodes in the disorder-dominated edge phase left experimentally unverified.

Popular Summary

In topological materials—exotic systems with properties impervious to perturbation—the manipulation of electronic states confined to their edges is key to the development of future quantum information technologies. In systems where multiple edge channels coexist, strong interaction can lead to disorder-dominated couplings among those channels, transforming elementary edge excitations into entirely different ones. While such a disorder-dominated phase has been extensively studied theoretically, it has never been experimentally verified. Here, we report the first experimental hallmark of this phase.

One type of system that embodies the essence of edge channels is the fractional quantum Hall liquid, a quantum fluid with quasiparticles whose electrical charges are a fraction of the electron’s. The disorder-dominated phase appears when an edge channel with one kind of fractional charge interacts strongly with a counterpropagating channel with another kind. This phase and its weak-interaction counterpart often show similar transport characteristics, making it hard to distinguish them.

Theoretically, one way to clearly identify the disorder-dominated phase is to measure the electrical and thermal conductance when the system is in a “crossover regime,” where the transport behavior is between coherent and incoherent. We employ a new device architecture comprising a junction of the two counterpropagating edge channels, which enables us to tune the channel length to span the coherent and incoherent regimes. In the crossover regime, we observe a fluctuating electrical conductance at the same time as a quantized thermal conductance—the definitive hallmark of the disorder-dominated phase.

Identifying the edge phase and its elementary excitations is vital for establishing novel quantum technologies by manipulating edge excitations. Our work demonstrates a virtue of a junction of topologically distinct systems for this purpose, stimulating and advancing extensive studies in various 2D electronic systems.

Figure 1

(a) Phase diagram of the $2/3$ edge at zero temperature [1]. Red (blue) region is the strong-coupling (weak-coupling) regime. (b) Summary of electrical ($G_{\mathrm{E}}^{(2)}$) and thermal ($G_{\mathrm{T}}^{(2)}$) conductance of conventional $\nu =2/3$ device in units of $G_{\mathrm{e}}$ and $G_{\mathrm{Q}}$, respectively [3]. Downstream (DS) and upstream (US). (c) DMRG calculation results of $n_{\mathrm{e}}(x)$ profiles across the $2/3–0$ and (d) $1/3–1$ junctions, plotted in units of filling factor ${\nu }^{*}$. Both profiles show the counterpropagating $1/3–1$ channels, demonstrating the particle-hole symmetry between the junctions. The profile in (d) enables us to estimate the compressible $\nu =1/3$ ($W_{1/3}$) and $\nu =1$ ($W_1$) strip widths and their separation ($W_0$). We calculated channel capacitances using the 2D finite-element method and obtained $\mathrm{\Delta }\cong 1.2$, which indicates that our $1/3–1$ channels are in the strong-coupling phase (see Appendix pp2). (e) Schematic of conventional $\nu =2/3$ device with two edges and (f) our $1/3–1$ junction with a single QH interface mimicking the $2/3$ edge. Interacting $1/3–1$ channels (length $L$) are connected to Ohmic contacts via noninteracting leads (length $L_0$).

Figure 2

(a) Schematic of experimental setup. The interacting $1/3–1$ channels appear at the narrow junction, forming downstream charge (denoted by “C”) and upstream neutral (“N”) modes. We measured differential conductance $g=d{I}_1/d{V}_{1/3}$ and current noise $S_{1/3}$. (b) False-color scanning electron micrograph of $L_{\mathrm{g}}=0.6\mathrm{\mu }\mathrm{m}$ sample. The left (red) and right (blue) regions are set at $\nu =1/3$ and $\nu =1$, respectively, while the split gate (yellow) is energized to form a narrow $1/3–1$ junction. (c) $V_{\mathrm{S}}$ dependence of $g$ and (d) $\mathrm{\Delta }{S}_{1/3}$ at several $V_{1/3}$ values of $L_{\mathrm{g}}=0.6\mathrm{\mu }\mathrm{m}$ sample. The upper horizontal axis shows the corresponding $1/3–1$ channel length $L$. A narrow junction is formed below $V_{\mathrm{S}}=-0.55\mathrm{V}$ (indicated by dashed black lines). Zero-bias conductance oscillations and almost $V_{\mathrm{S}}$-independent $\mathrm{\Delta }{S}_{1/3}$ at $-2.4<V_{\mathrm{S}}<-0.7\mathrm{V}$ are signatures of coherent-incoherent crossover. (c) Inset: zero-bias $g$ data converted into $G_{\mathrm{E}}^{(2)}$ using Eq. (2) with $G=g$. (d) Inset: schematic of noise-generation process. Mixing of $\nu =1/3$ and 1 channels with different chemical potentials causes heating at the hot spot, leading to the heat flow toward the noise-generating spot.

Figure 3

(a) $V_{1/3}$ dependence of $g$ and (b) $\mathrm{\Delta }{S}_{1/3}$ of the $0.6\text{−}\mathrm{\mu }\mathrm{m}$ sample measured at several $V_{\mathrm{S}}$ values. Change in $V_{\mathrm{S}}$ significantly varies the conductance behavior but varies $\mathrm{\Delta }{S}_{1/3}$ only slightly. The data are symmetrized against $V_{1/3}$ to improve the measurement resolution and compensate for minor self-gating effects (see Appendix pp4). (c) Noise temperature $T_{\mathrm{N}}$ estimated from $\mathrm{\Delta }{S}_{1/3}$ as a function of the injected Joule power $\mathrm{\Delta }P$. Solid blue line is the $T_{\mathrm{meso}}$ curve obtained from Eq. (10). The light blue region indicates the possible $T_{\mathrm{N}}$ range $T_{\mathrm{e}}\le T_{\mathrm{N}}\le T_{\mathrm{meso}}$. The data approach the $T_{\mathrm{meso}}$ curve at low bias, demonstrating dissipationless charge and neutral mode transport and quantized thermal conductance $G_{\mathrm{Q}}/2$ of the $1/3–1$ channels in the mesoscopic regime. Inset: $V_{1/3}$ dependence of $T_{\mathrm{N}}$ at $V_{\mathrm{S}}=-1.2\mathrm{V}$. The data depart from the $T_{\mathrm{meso}}$ curve (blue dotted line) above $V_{1/3}\cong 50\mathrm{\mu }\mathrm{V}$, signaling the dissipation due to bias-induced inelastic processes. Results for wider junctions are available in Appendix pp6. (d) $\mathrm{\Delta }P$ dependence of $G_{\mathrm{T}}$ in units of $G_{\mathrm{Q}}$ estimated from the data at $V_{\mathrm{S}}=-1.2\mathrm{V}$. Inset: schematic of the heat-flow input-output relationship.

Figure 4

(a) Zero-bias conductance and (b) $\mathrm{\Delta }{S}_{1/3}$ at $V_{1/3}=90\mathrm{\mu }\mathrm{V}$ as a function of the $1/3–1$ channel length $L$ for several samples with different $L_{\mathrm{g}}$. The $g$ and $\mathrm{\Delta }{S}_{1/3}$ data for the $0.6\text{−}\mathrm{\mu }\mathrm{m}$ sample are the same as those in Fig. 2. Narrow junctions ($L<1\mathrm{\mu }\mathrm{m}$) show the signatures of coherent-incoherent crossover, namely the mesoscopic electrical-conductance fluctuations and almost constant $\mathrm{\Delta }{S}_{1/3}$. The blue dashed line in (b) is a visual guide for constant $\mathrm{\Delta }{S}_{1/3}$. Wide junctions ($L<1\mathrm{\mu }\mathrm{m}$) show the suppression of electrical-conductance fluctuations and the decrease in $\mathrm{\Delta }{S}_{1/3}$ with increasing $L$, indicating the inelastic processes in the junction. The blue solid line is a visual guide $\mathrm{\Delta }{S}_{1/3}\propto 1/\sqrt{L}$ for the length dependence of $\mathrm{\Delta }{S}_{1/3}$ in the incoherent regime [47]. The crossover between the narrow and wide junction regions suggests $L_{\mathrm{in}}\sim 1\mathrm{\mu }\mathrm{m}$ inelastic scattering length.

Figure 5

(a) Schematic of the positive-charge configurations used for DMRG calculations of $2/3–0$ and (b) $1/3–1$ junctions. Positive background charges (density $n_{\mathrm{p}}$) locate at a vertical distance $d$ from the 2DES. Abrupt changes in $n_{\mathrm{p}}$ induce the electrostatic potentials in the 2DESs to form the QH junctions, gently varying near the junctions. (c) Full-range ${\nu }^{*}$ profile of the $1/3–1$ junction obtained by the DMRG calculation. (d) Schematic cross section of the $1/3–1$ junction with capacitive couplings. Red and blue regions, respectively, depict the compressible strips of the $\nu =1/3$ and $\nu =1$ channels, while pink and cyan regions are the incompressible bulk regions. Yellow regions denoted as ${\mathrm{G}}_{\mathrm{L}}$ and ${\mathrm{G}}_{\mathrm{R}}$ are left and right surface-gate electrodes, respectively. (e) Distributed channel capacitances estimated by a 2D finite-element method. While the DMRG calculation result shows $W_1\cong 10.5\mathrm{nm}$ and $W_0\cong 13\mathrm{nm}$, the oscillations of ${\nu }^{*}$ in the $\nu =1/3$ region make it difficult to estimate $W_{1/3}$. Therefore, we calculated the capacitances for several $W_{1/3}$ values. (f) Bare interchannel interaction parameter Δ estimated using the capacitance values in (e). The $\mathrm{\Delta }$ values stay at $\mathrm{\Delta }\cong 1.2$ over the entire range of $W_{1/3}$ used for the calculation, indicating that the $1/3–1$ channels are in the strong-coupling phase.

Figure 6

(a) Schematic of the whole device and measurement setup. Dotted lines indicate the geometries of surface-gate electrodes. The 2DESs below the right surface gate ($V_{\mathrm{R}}=0\mathrm{V}$) and outside the gated area are in the $\nu =1$ state (blue regions) at $B=10\mathrm{T}$. The $\nu =1/3$ state (red region) is formed by decreasing the electron density below the left surface gate with $V_{\mathrm{L}}\cong -0.45\mathrm{V}$ [see data in (b)]. The long $1/3–1$ channels across the $80\text{−}\mathrm{\mu }\mathrm{m}$-wide Hall bar, located immediately to the left of the $\nu =1/3$ region, are fully equilibrated to produce a unidirectional channel of electrical conductance $2e^2/3h$. In the standard Landauer-Büttiker edge transport picture, the setup can be expressed simply as that in Fig. 2 [42]. (b) $V_{\mathrm{L}}$ dependence of zero-bias conductance $g$ at $V_{\mathrm{R}}=V_{\mathrm{S}}=0\mathrm{V}$. (c) Color plot of $g$ as a function of $V_{\mathrm{S}}$ and $V_{\mathrm{L}}$ at $V_{\mathrm{R}}=0\mathrm{V}$. (d) Same $g$ data as (c) plotted with different ranges of $g$ and $V_{\mathrm{L}}$: upper panel: $e^2/2h<g<e^2/h$ and $-0.3\mathrm{V}<V_{\mathrm{L}}<0\mathrm{V}$; lower panel: $0<g<e^2/2h$ and $-0.6<V_{\mathrm{L}}<-0.3\mathrm{V}$. The conductance oscillations signaling the mesoscopic electrical-conductance fluctuations are observed for the $1/3–1$ junction at $V_{\mathrm{L}}\cong -0.45\mathrm{V}$ and not for the $2/3-1$ junction at $V_{\mathrm{L}}\cong -0.2\mathrm{V}$. The kink structure of the $g\cong e^2/3h$ region at $V_{\mathrm{S}}\cong -0.55\mathrm{V}$, highlighted by the white arrow in the lower panel, indicates the formation of the narrow $1/3–1$ junction.

Figure 7

The $T_{\mathrm{MC}}$ dependence of raw noise amplitude of the $1/3–1$ junction measured at zero bias. The blue curve is a linear fit to the data at and above $T_{\mathrm{MC}}\cong 50\mathrm{mK}$. The noise amplitude at the base temperature indicates $T_{\mathrm{e}}\cong 30\mathrm{mK}$.

Figure 8

(a) $V_{1/3}$ dependence of $g_{1/3}$ and (b) $\mathrm{\Delta }{S}_{1/3}$ of the $0.6\text{−}\mathrm{\mu }\mathrm{m}$ sample measured at $V_{\mathrm{S}}=-0.75\mathrm{V}$. (c) $V_1$ dependence of $g_1$ and (d) $\mathrm{\Delta }{S}_1$. (e) Symmetrized $d{I}_1/d{V}_1$ at $V_{\mathrm{S}}=-0.75\mathrm{V}$, measured by applying $V_1$. Back reflection from the $1/3–1$ junction gives $d{I}_1/d{V}_1\cong 2e^2/3h$, signaling the charge mode transport with conductance $2e^2/3h$ from the down side to the up side of the junction. (f) Symmetrized $\mathrm{\Delta }{S}_{1/3}$ data measured simultaneously with the data in (e).

Figure 9

Transmission probability dependence of $\mathrm{\Delta }{S}_{1/3}$ at $V_{1/3}=90\mathrm{\mu }\mathrm{V}$.

Figure 10

(a) $V_{\mathrm{S}}$ dependence of $g$ at $V_{1/3}=0\mathrm{\mu }\mathrm{V}$ and (b) $\mathrm{\Delta }{S}_{1/3}$ at $V_{1/3}=90\mathrm{\mu }\mathrm{V}$ of 4.8- and $40\text{−}\mathrm{\mu }\mathrm{m}$ samples. (c) $T_{\mathrm{N}}$ of the $4.8\text{−}\mathrm{\mu }\mathrm{m}$ sample plotted as a function of $\mathrm{\Delta }P$, measured at $V_{\mathrm{S}}=-0.75\mathrm{V}$. Inset: $V_{1/3}$ dependence of $T_{\mathrm{N}}$ at $V_{\mathrm{S}}=-0.75\mathrm{V}$.

Figure 11

$\mathrm{\Delta }P$ dependence of $T_{\mathrm{N}}$ measured for $0.6\text{−}\mathrm{\mu }\mathrm{m}$ junction at several $V_{\mathrm{S}}$ values [the same data as in Fig. 3] plotted with the $T_{\mathrm{b}}$ curve obtained from Eq. (g1) with $\alpha =15\mathrm{pW}/{\mathrm{K}}^3$ (blue dashed line).

Figure 12

(a) Example of 2DES shapes used for the finite-element method calculation of the electron-density profile ($V_{\mathrm{S}}\cong -1\mathrm{V}$ for the $0.6\text{−}\mathrm{\mu }\mathrm{m}$ sample). (b) Calculated electron-density profiles at $V_{\mathrm{S}}=-1.03\mathrm{V}$ along the $x=0$ (upper) and $y=0$ (lower) lines in (a). (c) Channel length $L$ plotted as a function of $V_{\mathrm{S}}$. Red circles are data points obtained from the finite-element method calculation. Green and blue curves are the fits using Eqs. (h1) and (h2), respectively. Inset: pinch-off trace of the $0.6\text{−}\mathrm{\mu }\mathrm{m}$ sample at $B=0\mathrm{T}$. The surface-gate and back-gate voltages are set at the same values as those to form the $1/3–1$ junction at $B=10\mathrm{T}$.

Figure 13

(a) Magnified view of the zero-bias electrical-conductance fluctuations of the $0.6\text{−}\mathrm{\mu }\mathrm{m}$ sample shown in Fig. 4. (b) Fourier transform spectrum of the conductance fluctuations in (a) in $400\le L\le 740\mathrm{nm}$. (c) $L$ dependence of the zero-bias conductance and (d) $\mathrm{\Delta }{S}_{1/3}$ at $V_{1/3}=90\mathrm{\mu }\mathrm{V}$ of $L_{\mathrm{g}}=0.3$-, 0.6-, and $0.9\text{−}\mathrm{\mu }\mathrm{m}$ junctions plotted on a linear scale of $L$.