Edge-State Wave Functions from Momentum-Conserving Tunneling Spectroscopy

We perform momentum-conserving tunneling spectroscopy using a GaAs cleaved-edge overgrowth quantum wire to investigate adjacent quantum Hall edge states. We use the lowest five wire modes with their distinct wave functions to probe each edge state and apply magnetic fields to modify the wave functions and their overlap. This reveals an intricate and rich tunneling conductance fan structure which is succinctly different for each of the wire modes. We self-consistently solve the Poisson-Schrödinger equations to simulate the spectroscopy, reproducing the striking fans in great detail, thus, confirming the calculations. Further, the model predicts hybridization between wire states and Landau levels, which is also confirmed experimentally. This establishes momentum-conserving tunneling spectroscopy as a powerful technique to probe edge state wave functions.

Figure 1

(a) Sample schematic (not to scale) with upper and lower quantum well (light blue) together with top and side Si dopants (red). A surface gate $G$ (green) of $2\mu \mathrm{m}$ width controls the local electron density. Source and drain ohmic contacts (brown) are labeled $S$ and $D$, respectively. (b) Wave function $\psi (y,z)$ for the ground state $H_0$ (${\mathrm{LW}}_1$) in the upper (lower) quantum well. Dashed lines indicate locations of cuts shown in (c) and (d). (c) Conduction band profiles along the $y$ direction in the upper (blue) and lower (green) quantum wells. Horizontal lines represent the lowest three eigenenergies. (d) Conduction band profile along the $z$ direction 20 nm (purple) and 590 nm (orange) away from the cleavage plane [dashed white in (a)].

Figure 2

Experimental (left and middle) and theoretical (right) tunneling conductance between various hybrid states and lower wire modes as a function of out-of-plane ($B_Z$) and in-plane ($B_Y$) magnetic fields. For the middle panel, a smooth background is subtracted from the experimental data and an oversaturated color scale is used to emphasize weak features. The tunneling conductance to the lower wire mode ${\mathrm{LW}}_2$ is strongest for the hybrid state $H_1$ and depends weakly on the magnetic field $B_Z$. In contrast, the tunneling conductance to the wire modes ${\mathrm{LW}}_{3,4,5}$ shows significant strength variations as magnetic field $B_Z$ is increased, see main text for details.

Figure 3

Evolution of the wave functions of the hybrid state $H_1$ (left column), $H_2$ (right column), and lower wire mode (${\mathrm{LW}}_3$) as a function of magnetic field $B_Z$. The cleaved edge is at $y=0$. The squared overlap normalized to its maximal value in Fig. 2 is indicated in each panel. Red and blue colors represent the signs of the probability amplitude.

Figure 4

(a) Dispersion of hybrid states ($H_0,H_1,\dots$, colors) in the upper system. Empty states above the Fermi level are shown in gray. (b) The wave function of the hybrid state $H_0$ (red), $H_1$ (dashed orange), $H_2$ (solid orange), and the total electrostatic confinement potential (blue) at $z=31\mathrm{nm}$ in the upper system for a range of values of the guiding center position $y_0$. The corresponding momenta $k_x$ are indicated with markers in (a).

Figure 5

Tunneling conductance between hybrid states in the upper system and the lower wire mode ${\mathrm{LW}}_1$. Insets qualitatively depict the dispersion relation of the hybrid states (red) and ${\mathrm{LW}}_1$ (black) for $B_Y$ and $B_Z$ which satisfy the resonant tunneling condition. Measurements extended for $B_Z<0$ are shown in Fig. S10 in the Supplemental Material [34].