Based on the edge-state picture for the quantum Hall effect, different device layouts of quantum Hall samples where the paths of edge states split and remerge have been described in literature as Fabri-Pérot and Mach-Zehnder-like *electronic* interferometers, respectively [1]. As expected from the Aharonov-Bohm effect, indeed such devices have shown conductance modulations which are periodic in the magnetic flux penetrating the area encircled by the split edge-state paths. Periods of *h/e* and integer fractions *h/ne* have been identified – for *n > 2* usually by Fourier transformation analysis.

In contrast, scanning probe microscope investigations on quantum Hall samples which have been performed in our group for many years have lead to a microscopic picture for the current distribution in quantum Hall samples which contradicts the current-carrying edge-state model [2]. Depending on the local electron concentration, the two-dimensional electron system (2DES) under quantum Hall conditions becomes locally compressible (Fermi level lies *within* a Landau level) or incompressible (Fermi level lies *between* two Landau levels). Especially within the natural depletion regions along the 2DES edges, compressible and incompressible stripes appear which evolve in position and width with magnetic field. Within a quantum Hall plateau, we have experimentally found that the externally biased current is flowing without dissipation inside *incompressible* regions: The local Hall field *E*_{y}, pointing in *y-*direction, causes that all electronic states get a drift velocity in *x*-direction along the incompressible stripe which leads to a local current density *j*_{x}, carried by *all *occupied states below the local Fermi level, given by the local Landau level filling factor *n*: *j*_{x}* = **n **e²/h E*_{y}.

Based on our results imaging the Hall potential landscape and therefore non-equilibrium current distribution in quantum Hall samples, edge-state interference seems not to be the right approach to understand the magnetic flux periodic conductance modulations observed in the previously mentioned quantum Hall samples.

**Fig. 1:**(a) False colored scanning electron micrograph with a top view of the device. The dark blue areas are etched below the 2DES level, the golden parts are metal top-gates and bridges, respectively. (b)–(d) Schematic view of the device including a certain compressible/incompressible stripe pattern of the 2DES evolving at an applied high magnetic field. For simplicity the air-bridge connection to the metallic gates situated directly at the central groove, denoted as side-gates, are not displayed. Indicated by the light blue color are incompressible regions within the 2DES, while the dark blue corresponds to the compressible ones. (b) Configuration of two constrictions with the Hall current flowing in the incompressible regions (red arrows). (c) Configuration of two quantum dots with the source-drain current mediated by single-electron tunneling (green arrows). (d) Combination of a single quantum dot and a constriction.

By patterning the two-dimensional electron system in an (Al,Ga)As heterostructure, in the last years we have realized and investigated a versatile mesoscopic quantum Hall device (see **Fig. 1**(a)) which offers in-situ tunable different combinations of parallel paths from source to drain (sketched in Fig. 1(b)–(d): two parallel constrictions, two parallel quantum dots – each acting as single-electron transistors, and the mixed combination – a single quantum dot and a single constriction in parallel. Applying a strong magnetic field, certain compressible/incompressible landscapes can be imprinted as indicated by the dark (compressible) and light blue (incompressible) shading in Fig. 1(b)–(d). In any case of Fig. 1(b)–(d), compressible stripes along the 2DES edges carry the applied voltage difference between source and drain contact into the sample – here, due to the orientation of the magnetic field, the compressible stripe at top of the source side carries the potential of source, whereas the compressible stripe on the bottom of the drain side carries the drain potential. The voltage difference between upper left and lower right compressible stripe can act in two ways: (1) causing as potential gradient directed electron scattering between compressible regions if they are close to each other, and (2) driving as a local Hall voltage a dissipationless current. All configurations of Fig. 1 show under certain conditions conductance modulations, periodic in the applied magnetic flux penetrating the area enclosed by the two paths, i.e. the area of the depletion region defined by the etched hole in the center. Furthermore, periodic conductance modulations are also observed by tuning the depletion area around the center hole. The dependence of magnetic flux density and enclosed area hint together – in first view - on an Aharonov-Bohm-like interference effect, but detailed investigations on all configurations exclude interference as underlying mechanism.

## Results

**Fig. 2:**Color-coded graph of the conductance modulations – measured at a source-drain voltage bias of

*V*

_{DS}

*=*25 µV – in the parameter space of the applied magnetic flux density and the depleted area around the center hole, tuned via the side-gate voltage, for the configuration of two constrictions. (a) Modulations with

*h/e*periodicity for a conductance value of each constriction in between the

*1e²/h*conductance plateau and pinch-off. (b)

*h/*2

*e*periodicity for a conductance value of each constriction set in between the 2

*e²/h*and the 1

*e²/h*conductance plateau.

In case of two parallel constrictions, well quantized conductance values i*e*²*/h *are observed as long as scattering from the top compressible edge stripe to the bottom compressible edge stripe via the compressible ring around the etched hole is suppressed – the current flows dissipationless in the incompressible region through the constrictions’ centers with the integer-valued local Landau level filling factor i. This quantized value is even not affected if electron scattering from *one* compressible edge stripe to the compressible ring is allowed. Conductance modulations (**Fig. 2**) are observed only in case of conductance values between quantized conductance plateaus where both top and bottom edge stripes couple to the compressible ring allowing for electron scattering between these compressible regions. It was already reported in literature that in this regime, magnetic flux periodic conductance modulations can be understood as periodically shifting the resonance condition for single-electron tunneling via the closed compressible ring with increasing magnetic flux density [3]: An increase of magnetic flux density contracts the closed electronic wave functions around the hole enhancing the Coulomb energy for the electrons. By a rearrangement of the electrons, this electrostatic energy is relaxed. This happens periodically with roughly the magnetic flux change *h/e *penetrating the effective depletion area of the hole. We extended the model by spin-degeneracy explaining also the observed *h/*2*e *periodicity. Striking the observation that the conductance modulation *amplitude* obviously scales with the local Landau level filling factor we expect to be present in the constrictions, see **Fig. 3**. It is easily explained in our picture where a potential difference between compressible edge stripe and compressible ring – which is periodically modulated by the resonance condition for single-electron tunneling - causes a drift for *all *occupied states in between.

**Fig. 3:**Comparison of the periodic conductance modulations of

*h/e*and

*h/*2

*e*periodicity. Besides the different periodicity a scaling in the modulations magnitude is present. The scaling arises from the different number of occupied Landau levels present in the center of the constrictions.

The parallel arrangement of two quantum dots – where each dot behaves as a single-electron transistor - shows that the common conductance maximum is less than the sum of the conductance maxima of the individual quantum dots, but we have never observed that the common conductance maximum falls below the conductance maxima of the individual quantum dots which is expected in case of destructive interference. In cases where the common conductance maximum reaches *e*²*/h, no *conductance modulations with a period of the magnetic flux quantum *h/e* can be observed, while in the flanks of the common conductance peak these are visible. Tuning both quantum dots individually into the Kondo regime where a spin-degenerated correlated electronic state is formed between leads and quantum dot, the modulation period changes to *h/*2*e*, but falls back to *h/e* if one quantum dot is tuned off the Kondo regime – we break up the spin-degeneracy of the closed electronic states around the etched hole. For explanation we also assume here that the electronic states encircling the hole are contracted by an increase in magnetic flux density. An increase in electrostatic energy is periodically relaxed by a rearrangement of the electron occupation. This process affects the shallow potential profiles at the quantum dot’s tunneling barriers. In consequence, both the linewidth of the single-electron tunneling peaks and the peak conductance value change in a periodic fashion giving rise to the periodic conductance modulations. Taking this model, in case of maximum conductance equal to *e*²*/h*, the conductance modulations are absent, consistent with our experimental observation.

In the mixed configuration of a constriction and a single quantum dot we could observe again both the *h/e* and the *h/*2*e* periodicity of the conductance modulations. The latter is thereby observed only in the presence of a spin-degeneracy in *both* device parts, with a Kondo resonance present in the quantum dot and a conductance value of the constriction higher than *e*²*/h.* Obviously, a spin-degeneracy for the wave function around the ring has again to be present. Furthermore, within this mixed configuration we could actually demonstrate that phase coherence around the ring is of importance: By shifting the electrostatic potential of the quantum dot, we can tune through the single-electron tunneling resonance condition and indeed we observe a phase shift of the conductance modulations as a function of magnetic flux density by 180°.

In summary, we have demonstrated that interference effects can be excluded as the cause of the periodic conductance modulations for all three configurations of two-path configurations investigated at high magnetic fields. Instead, we found that treating the electrostatics of the system self-consistently accounts for all our experimental observations. In every configuration there is the possibility for closed and possibly spin-degenerate wave functions to be present. The wave functions contract while increasing the magnetic flux density in order to preserve the enclosed magnetic flux and consequently the electrostatic energy for the electron system increases. This surplus energy can be relaxed by rearranging electrons and occurs with the characteristic periodicity of *h/e* or *h/*2*e*, with respect to the area the wave functions enclose. It now depends on the device configuration how the source-drain current is affected by this periodic breathing process. In case of the two parallel constrictions it leads to a single-electron tunneling mediated scattering between the compressible edges that reduces the Hall voltage and therefore the Hall current. Whereas in case of two quantum dots a pure electrostatic influence on the tunneling barriers alters the tunneling rate of the individual quantum dots changing the overall transmission through the device. And consequently both effects can be present in the combined configuration of a single quantum dot and a constriction.