Wave-function description of conductance mapping for a quantum Hall electron interferometer

Scanning gate microscopy of quantum point contacts (QPC) in the integer quantum Hall regime is considered in terms of the scattering wave functions with a finite-difference implementation of the quantum transmitting boundary approach. Conductance ($G$) maps for a clean QPC as well as for a system including an antidot within the QPC constriction are evaluated. The steplike locally flat $G$ maps for clean QPCs turn into circular resonances that are reentrant in an external magnetic field when the antidot is introduced to the constriction. The current circulation around the antidot and the spacing of the resonances at the magnetic field scale react to the probe approaching the QPC. The calculated $G$ maps with a rigid but soft antidot potential reproduce the features detected recently in the electron interferometer [F. Martins et al., Sci. Rep. 3, 1416 (2013)].

Figure 1

Sketch of the QPC system considered in this paper. The dots in the left corner show the finite difference mesh used in numerical calculations. The vertical dashed red line shows the path of the conductance scan discussed in the text. The gray area in the center of QPC indicates a local potential maximum introduced to model the quantum Hall island.

Figure 2

(a) Transfer probability through the QPC summed over the incident subbands [Eq. (7)] and (b) the number of transport modes $M$ in the leads as functions of Fermi energy $E_F$ and the perpendicular magnetic field $B$. The dashed lines on both plots show the energy value $E_{\mathrm{F}}=6\mathrm{meV}$, which is considered further in this paper. (c) Cross-section of (a) and (b) for $E_{\mathrm{F}}=6\mathrm{meV}$.

Figure 3

Conductance maps obtained for $B=0.9$ T and different values of $U_{\mathrm{tip}}$ (a) without QHI and (b)–(d) with QHI within QPC.

Figure 4

Conductance for $E_F=6$ meV as a function of the $y$ position of the tip and the magnetic field. The scans were performed along $x_{\mathrm{tip}}=1100$ nm line (see Fig. 1) with $d_{\mathrm{tip}}=60$ nm for different values of the tip potential $U_{\mathrm{tip}}$ [Eq. (2)]. (a) Scan of the QPC obtained with $U_{\mathrm{tip}}=3$ meV and $U_{\mathrm{QHI}}=0$ (in the absence of QHI). (b)–(e) Scans obtained for QHI present with: $U_{\mathrm{QHI}}=11$ meV and $d_{\mathrm{QHI}}=40$ nm [Eq. (3)] for $U_{\mathrm{tip}}=1,2,3$, and $-5$ meV.

Figure 5

(a) Transfer probability as a function of the amplitude $U_{\mathrm{QHI}}$ and magnetic field for $d_{\mathrm{QHI}}=40$ nm and the center of the potential maximum located at the center of QPC [see Fig. 1]. The number of fully transparent subbands $f_c$ is given. The vertical dashed line present the value of $U_{\mathrm{QHI}}$, which chose for further calculations. (b)–(g) Probability density current distribution (color scale shows the absolute value, and vectors the orientation of the current) for various points along resonances with locations indicated by arrows.

Figure 6

(a)–(c) Probability density current distribution (color scale shows the absolute value, and the arrows—the orientation of the current) corresponding to points marked by symbols in the conductance scan of Fig. 4. The position of the tip is marked by cross.

Figure 7

$\Delta B$ dependence on the position $y$ of the tip for different values of $U_{\mathrm{tip}}$ as extracted from Figs. 4 and 4. For the definition of $\Delta B_1$ and $\Delta B_2$ see Fig. 4.

Figure 8

Same as Fig. 4 only for the hard-wall profile of the QHI [Eq. (9)].