Probing the breakdown of topological protection: Filling-factor-dependent evolution of robust quantum Hall incompressible phases

The integer quantum Hall (QH) effects characterized by topologically quantized and nondissipative transport are caused by an electrically insulating incompressible phase that prevents backscattering between chiral metallic channels. We probed the incompressible area susceptible to the breakdown of topological protection using a scanning-gate technique incorporating nonequilibrium transport. The obtained pattern revealed the filling-factor- ($\nu$-) dependent evolution of the microscopic incompressible structures located along the edge and in the bulk region. We found that these specific structures, respectively, attributed to the incompressible edge strip and bulk localization, show good agreement in terms of $\nu$-dependent evolution with a calculation of the equilibrium QH incompressible phases, indicating the robustness of the QH incompressible phases under the nonequilibrium condition. Further, we found that the $\nu$ dependency of the incompressible patterns is, in turn, destroyed by a large imposed current during the deep QH effect breakdown. These results demonstrate the ability of our method to image the microscopic transport properties of a topological two-dimensional system.

Figure 1

(a) Schematic of the experimental setup of the SGM. $V_{\mathrm{x}}$ is recorded as the tip is scanned at $V_{\mathrm{tip}}$ with $I_{\mathrm{sd}}$ and $B$, respectively, applied in the $x$ and $z$ directions. (b) Schematic of inter-LL tunneling (marked by red arrows) between two LL subbands near the edge of the higher ${\mu }_{\mathrm{chem}}$ side under the nonequilibrium condition, which is derived from the deviation between the Fermi energies in the edge ($E_{\mathrm{f},\mathrm{edge}}$) and bulk ($E_{\mathrm{f}}$) compressible regions. Tip-induced LL bending, indicated by the blue dashed line, enhances inter-LL tunneling. The incompressible and compressible strips are indicated by “IS” and “CS,” respectively. “Edge” and “bulk” indicate the edge strips and 2DES bulk region, respectively. The magnitudes of ${\nu }_{\mathrm{L}}$ with respect to the exact integer $i$ are also shown for each region. (c) SGM image of a tip-induced $V_{\mathrm{x}}$ change ($\mathrm{\Delta }V$) at $\nu =2.27, I_{\mathrm{sd}}=3.1\mu \mathrm{A}$, and $B=4$ T; dashed lines denote the Hall bar edges. The line noise was removed using 2D Fourier filtering.

Figure 2

SGM measurements near $\nu =2$ and 4. (a), (b) Representative $\mathrm{\Delta }V$ images at different $\nu$ tuned by $n$ at (a) $B=4\mathrm{T}$ near $\nu =2$ and (b) $B=2\mathrm{T}$ near $\nu =4$ (see the Supplemental Material [40] for the corresponding longitudinal and Hall resistances at equilibrium in the QH regime). The scale bar is 4 $\mu \mathrm{m}$. Dashed lines denote the Hall bar edges. The line noise was removed using 2D Fourier filtering. As $\nu$ decreases toward integer $i$, the patterns are further enhanced, such that the full scale of contrast is appropriately optimized for clarity. (c), (d) Position (dot) and width (bar) of the $\mathrm{\Delta }V$ peak as a function of $\nu$ near (c) $\nu =2$ and (d) 4. The position and width are, respectively, determined by the distance from the Hall bar edge ($y_{\mathrm{k}}$) and the full width at half-maximum ($W_{\mathrm{FWHM}}$). These are extracted from the cross-sectional $\mathrm{\Delta }V$ profile spatially averaged over the $x$ region ($\overline{\mathrm{\Delta }V}$), as indicated in the inset in (c) [obtained from the image for $\nu =2.10$ in (a)]. Here, $\nu =2.0–2.4$ and 4.0–4.8 are selected to ensure the same $n_{\mathrm{s}}$ range $1.93\times {10}^{15}–2.32\times {10}^{15}{\mathrm{m}}^{-2}$. The gray area denotes the incompressible regions determined by LSDA calculation for (c) ${\nu }_{\mathrm{L}}=2$ and (d) 4.

Figure 3

SGM measurements near $\nu =1$. (a) Representative $\mathrm{\Delta }V$ images taken at different $\nu$ tuned by $n$ at $B=8\mathrm{T}$. The scale bar is 4 $\mu \mathrm{m}$. Dashed lines denote the Hall bar edges. The line noise was removed by 2D Fourier filtering. (b) Positions $y_{\mathrm{k}}$ (dots) and widths $W_{\mathrm{FWHM}}$(bars) of the $\overline{\mathrm{\Delta }V}$ profile as a function of $\nu$. Gray area: the calculated incompressible region for ${\nu }_{\mathrm{L}}=1$. (c) Longitudinal ($R_{\mathrm{xx}}$) and normalized Hall ($\mathrm{\Delta }{R}_{\mathrm{xy}}$) resistance curves measured for the same Hall bar at the equilibrium QH condition, $I_{\mathrm{sd}}=10$ nA, at $B=8\mathrm{T}$. Vertical dashed lines mark the $\nu$ positions at which the SGM images shown in (a) were captured.

Figure 4

Filling factor and electrical current evolution of SGM patterns taken near $\nu =1$ at $B=6\mathrm{T}$ (see the Supplemental Material [40] for corresponding $R_{\mathrm{xx}}$ and $\mathrm{\Delta }{R}_{\mathrm{xy}}$ at equilibrium in the QH regime). (a)–(c) Filling-factor-dependent structures of the ${\nu }_{\mathrm{L}}=1$ incompressible phase taken at low $I_{\mathrm{sd}}$ (0.6–1.0 $\mu \mathrm{A}$) for $\nu =1.05$, 1.01, and 1.00. (d), (e) SGM patterns obtained at high $I_{\mathrm{sd}}$ (2.2–2.5 $\mu \mathrm{A}$) for $\nu =1.05$ and 1.01. The scale bar is 4 $\mu \mathrm{m}$; dashed lines denote the Hall bar edges. The line noise was removed using 2D Fourier filtering. (f) Schematic density profile along a red line across a closed-loop pattern in (c). (g) $V_{\mathrm{x}}-I_{\mathrm{sd}}$ curves for $\nu =1.05$, 1.01, and 1.00. Crosses mark the measurement conditions for (a)–(e).