Current Flow in the Self-Consistent Microscopic Picture of the Quantum Hall Effect

6.2 Current Flow in the Self-Consistent Microscopic Picture of the Quantum Hall Effect

6.2 Current Flow in the Self-Consistent Microscopic

6 Two Parallel Quantum Point Contact Constrictions

flow inside both constrictions from the left to the right (black arrows). The potential of the compressible antidot strip is determined by the distribution of the net current between the upper and the lower constriction. Here, it is set to be symmetric and the potential resides midway between the source bias and the ground potential at drain.

This behavior stands in contrast to the description of the current distribution within the edge-state model. Although there are two paths for the Hall current around the antidot region in the self-consistent model, the origin of the current distribution is different. Here the Hall current is carried inside incompressible regions with fully occupied states below the Fermi level. It is determined solely by the Hall voltage drop between the upper and lower edge of the 2DES, independent of the details of the voltage drop along the cross-cut through the device. This renders the splitting of the Hall current into different paths in the incompressible strips unimportant for the conductance value of the device. Compressible droplets in the 2DES of a Hall bar, like the antidot resembles, have also been observed in the measurements using a scanning probe technique determining the Hall potential landscape [3]. Their occurrence does not change the quantized Hall conductance. In consequence, it is unlikely that the conductance modulations in the antidot system can be seen as a coherent interference effect of electrons propagating from the source to the drain of the device.

6.2.1 Interaction of the Compressible Antidot Ring with the Edges of the Two-Dimensional Electron System

In order to enable the antidot compressible strip to affect the transport through the de-vice, it has to interact with the potential carrying compressible edges of the device. The interaction can be introduced by changing the width of the constrictions close to the anti-dot. At first a well developed incompressible strip is present within the QPC constriction separating both compressible region. With a more negative voltage applied to the split-gate the compressible edge of the sample is pushed further towards the compressible strip around the antidot. As the distance in between shrinks, the overlap of the electron wave functions of both compressible regions increases. This enables electrons to be exchanged over the incompressible potential barrier and thus couples the compressible ring of the antidot to the compressible edge of the sample. The Hall voltage drop is thereby effec-tively diminished by electron scattering. This results at first in a reduction of the drift velocity of the electron states below the electro-chemical potential within the region of the QPC. In this regime of weak coupling the compressible/ incompressible landscape in the 2DES within the QPC is no longer well defined. However, a Hall potential drop can still be present. To denote this situation we will use the nomenclature of a local filling factor ν_l for the QPC, deviating from an integer value. Without coupling of the compressible annulus around the antidot to a compressible edge the local filling factor ν_l takes an integer value.

Let us first discuss the asymmetric case for the lower constriction remaining on the local filling factor ν_l = 1 while the upper one is set to ν_l < 1 by the gate voltage. The filling factor in the bulk of the leads remains unchanged. The 2DES strip pattern together with a potential cross-cut taken at the antidot is illustrated in figure 6.4. The width of the upper constriction reduces as the compressible regions of the edge and the antidot are pushed towards each other. A close up of the potential cross-cut of the upper constriction is illustrated in figure 6.4b. The overlap of the electron wave function in the compressible 58

6.2 Current Flow in the Self-Consistent Microscopic Picture of the Quantum Hall Effect

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antidot CR edge CR

QPC constriction overlap

VDS

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Figure 6.4 (a) Schematic illustration with a cross-cut of the potential bending due to the compressible/incompressible strip pattern around the antidot geometry. The upper constriction is set a local filling factor ofν_l<1and couples the antidot compressible ring to the upper compressible edge at source potential(azure blue connection). The lower constriction is set to a local filling factor of ν_l = 1.

The bulk filling factor of the leads remains unchanged. The potential of the compressible ring converges to the value of the source potential and the net current flows only in the lower constriction (black arrow). The dotted gray electro-chemical potential cross-cut serves as guide to the eye illustrating the previous situation with integer local filling factors in both constrictions as a reference. (b) Not-to-scale enlargement of the constriction region highlighted in orange in (a). The wave function of the electrons in the compressible regions (gray shaded areas) overlap (blue) and allow a transfer of electrons bridging the constriction in between. For simplicity the overlap is sketched for equal energy of the wave functions.

6 Two Parallel Quantum Point Contact Constrictions

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Figure 6.5 Schematic illustration with a cross-cut of the potential bending due to the compressible/incompressible strip pattern around the antidot geometry. Both constrictions are set to a local filling factor toνl<1and allow charge exchange across them (azure blue connection). The charge exchange from source to drain reduces the Hall voltage drop across the entire device (red curve) and the conductance of the device reduces. The dotted gray electro-chemical potential cross-cut serves as guide to the eye illustrating the previous situation with integer local filling factors in both constrictions as a reference.

region bridges the upper constriction allowing the charge transfer across it. While the upper constriction allows charge exchange between the upper compressible edge of the sample and the compressible antidot ring (azure blue connection in figure 6.4a) no charge exchange occurs via the lower one. This coupling to one edge lifts the electro-chemical potential of the compressible antidot strip and it will converge towards the potential of the upper compressible edge at source level. In consequence, the Hall voltage in the upper constriction drops to zero, whereas the full Hall voltage will drop across the lower constriction leading to a current flow only through this constriction (black arrow).

The inverted configuration with the coupling of the compressible antidot ring solely to the lower compressible edge at the drain potential shows the same behavior but with opposite characteristic. The electro-chemical potential of the antidot compressible strip converges towards the value of the lower edge and the Hall voltage drops entirely across the upper constriction. The net current in the lower constrictions reduces to zero and it fully flows in the upper one.

The situation becomes different if both constrictions deviate from the quantized con-ductance value, as it is illustrated in figure 6.5. Both constrictions are set to a local filling factorν_l<1 and coupling of the antidot compressible region to both compressible edges is obtained, while the bulk filling factor in the leads remains unchanged at ν = 1. Likewise to the previous scenario the electro-chemical potential of the antidot compressible ring increases towards the value of the upper edge, but the transfer of electrons to the lower 60

6.3 Interplay of two Parallel Constrictions Around the Local Filling Factor One: My Experimental Observations compressible edge occurring at the same time shifts the electro-chemical potential to the value of the lower edge. In consequence, the electro-chemical potential of the antidot will remain between the levels of both edges. The charge exchange across both constrictions permits a continuous draining of electrons from one compressible edge via the compress-ible antidot ring to the other edge of the device. In the cross-cuts of the potential bending is assumed that the constant bias of −V_DS is only applied to the source side of the device and both the compressible ring as well as the drain potential will rise in energy due to the coupling. Without loss of generality it is further assumed that the coupling to both edges to the antidot compressible region is symmetric and its electro-chemical potential will stay midway between the applied source and the lifted drain potential. The Hall voltage drop across the entire device reduces compared with case without charge transfer (gray potential cross-cut). This reduces at first the drift velocity of the electron states in both constrictions. In consequence, the Hall current reduces (smaller black arrows) and the conductance of the device decreases. The amount of scattered charge carriers between the compressible edges thereby determines how large the reduction in the measured two-terminal conductance will be.

Common to the edge-state picture and in the self-consistent ansatz, the conductance ues of both constrictions at the antidot have to deviate from a quantized conductance val-ues to alter the transport through the device. The current distribution and the underlying mechanism are completely different though. In the edge-state picture the coherent edge channel split and recombine giving rise to interference contributions in the transport as the reflection channel is modulated by the Aharonov-Bohm phase. In contrast, the self-consistent approach suggests that the external current is modulated by scattering of electrons between the upper and lower compressible edge region reducing the drift velocity of the electronic states below the electro-chemical potential. The origin of the conductance modulations with the magnetic flux change will be clarified in the following.

6.3 Interplay of two Parallel Constrictions Around