Structure of a Single Sharp Quantum Hall Edge Probed by Momentum-Resolved Tunneling

Momentum-resolved magnetotunneling spectroscopy is performed at a single sharp quantum Hall (QH) edge to probe the structure of integer QH edge modes. An epitaxially overgrown cleaved edge is shown to realize the sharp-edge limit with interchannel distances smaller than both the magnetic length and the Bohr radius where the Chklovskii soft-edge picture is no longer valid. The line shape of principal conductance peaks is explained, and an edge filling factor is determined from the peak position. A step in the dispersion is attributed to fluctuations in the QH ground energy.

Figure 1

The differential tunnel conductance $dI/dV$ versus magnetic field $B$ at zero dc bias voltage for sample I. The dotted and dashed curves indicate the expected resonance line shape with and without disorder broadening. Inset: two quantum wells are arranged in a $T$ shape separated by a tunnel barrier $b$; the field $B$ creates quantum Hall edges in ${\mathrm{Q}\mathrm{W}}^{\perp }$ probed by tunneling from ${\mathrm{Q}\mathrm{W}}^{\parallel }$.

Figure 2

Calculated dispersions $E_n^{\perp }(X)$ and $E^{\parallel }(X,k_z)$ vs orbit-center coordinate $X$ at $B$ fields of the experimentally observed $n=0$ (left) and $n=1$ (right) zero-bias conductance peaks. The peaks demonstrate a resonance between the Fermi point in the probe and the Landau dispersion. The occupied part of each Landau band is black, with the unoccupied part gray. The conduction band potential is shown as a shaded gray background. The wave functions of each mode at $E_{\mathrm{F}}$ are depicted with thin solid lines at the Fermi energy.

Figure 3

$\frac{dI}{dV}$ vs $V$ at different $B$ (0.05 T steps offset by $0.5\mu \mathrm{S}$.) The conductance peak for the outermost $n=0$ channel splits into two near 3.5 T, evidence that the $\nu =2$ jump in the bulk chemical potential is seen at the edge. Peaks for the inner $n=1,2,3$ channels remain continuous since the smoothly depleted edges screen the bulk $\nu =2$ jump (sample II).