Imaging Fractional Incompressible Stripes in Integer Quantum Hall Systems

Transport experiments provide conflicting evidence on the possible existence of fractional order within integer quantum Hall systems. In fact, integer edge states sometimes behave as monolithic objects with no inner structure, while other experiments clearly highlight the role of fractional substructures. Recently developed low-temperature scanning probe techniques offer today an opportunity for a deeper-than-ever investigation of spatial features of such edge systems. Here we use scanning-gate microscopy and demonstrate that fractional features were unambiguously observed in every integer quantum Hall constriction studied. We present also an experimental estimate of the width of the fractional incompressible stripes corresponding to filling factors $1/3$, $2/5$, $3/5$, and $2/3$. Our results compare well with predictions of the edge-reconstruction theory.

Figure 1

(a) SGM scan at the center of a QPC in a ${\nu }_b=1$ QH system ($V_{\mathrm{tip}}=-6\mathrm{V}$). The map shows the transmitted differential conductance $G_T$ as a function of the tip position, together with contour lines at constant $G_T$. The blue arrow indicates a structure induced by a local potential fluctuation. On the right, a zoom of the $50\times 200\mathrm{nm}$ region corresponding to the dashed rectangle is displayed. (b) Profile of $G_T$ along the light blue line in (a), together with its derivative. (c) Scheme of the SGM experimental setup.

Figure 2

Histogram of the average occurrence of each $G_T$ value within 9 different SGM scans performed at different $V_{\mathrm{tip}}$ values. Peaks for $G_T=1/3$, $2/5$, $3/5$, and $2/3$ are visible.

Figure 3

Histogram plots of the occurrences of $G_T$ values for two SGM scans performed at 300 mK (red curve) and 4.2 K (blue curve), on a QPC fabricated on sample $C$. The increase of temperature completely washes out the fractional IS, so that the $1/3$ peak disappears.

Figure 4

IS width ${\delta }_{\mathrm{IS}}$ plotted as a function of the electron density gradient (scatter plots), together with the reconstruction picture predictions for $\Delta {\mu }_f=200$, 300, and $400\mu \mathrm{eV}$ (thin lines). (inset) Plot of ${\delta }_{\mathrm{IS}}$ for the $1/3$ fraction measured for different $V_{\mathrm{tip}}$ values (green spots), together with the ${\delta }_{\mathrm{IS}}$ values calculated with the formula of Chklovskii et al. (black squares). The latter have been determined by estimating the electron density gradient corresponding to each scan, and assuming that ${\Delta }_f=1\mathrm{K}$ (see text).