The energy spectrum of a two-dimensional electron system in quantizing magnetic fields (the so-called ladder of Landau levels) with taking into account the potential of edges of the sample. 7
A schematic diagram of implementation of electronic interferometers using edge states in the QHE regime: The electronic counterparts of interferometers of the Fabry-Perot type (a), the Mach-Zehnder type (b). Only those parts of the edge states at boundaries of the sample are shown, the transport over which is essential for transfer of an electron from the source to the drain. 1
A schematic diagram of implementation of the electronic interferometer using co-propagating edge states in the QHE regime. An electron moves along the inner edge state from the point A and can either pass to the outer edge state in the first interaction region and go to the ohmic contact (the path ABDE), or remain in the inner edge state until the second interaction region, where to go to the outer edge state (the path ABCE).
The realization of an interferometer using the quasi-Corbino geometry developed for studying the transport between co-propagating edge states. A gate (yellow) defines a geometry of the experiment, depleting a two-dimensional electron gas under it down to the filling factor 1. At the part of the sample which is not covered with the gate (green) the filling factor is equal to 2. Ohmic contacts (a rectangle with the number) are made to both the internal and external etched boundaries of the sample. The edge states appear at boundaries of the sample; one of them, following along a border of the gate, in a controlled manner connects the inner and outer boundaries, and makes it possible to study transport between co-propagating edge states in the gate-gap region on the outer boundary of the sample (not to scale).
A realization of the quasi-Fabry-Perot interferometer. 39 An area of the gate-gap with the width of 10 μm contains a set of additional gates of small width (200 nm each), separated by the interaction region of 400 nm. It was assumed that a translational repetition of the structure in Fig. 3 would enhance the visibility of the interference pattern: electrons which did not pass to the outer edge state in the second interaction region, go to the third, and so on, so that there is an analogue of the Fabry-Perot interferometer created with a number of reflections limited by a number of additional gates.
An example of the interference oscillations for the quasi-Fabry-Perot interferometer with the filling factor 2 in the gate-gap region, 1 under the gate. 39 The inset (a) shows a dependence of position of the oscillation on its number, which allows to determine the oscillation period 0.35 T. The inset (b) shows the oscillations with the same period for filling factors 3 and 1, i.e. for a similar configuration of the edge states. For measurements the current through the sample were set to I = 11.49 nA.
A snapshot of interferometer's working area (a region of the gate gap and an additional gate) obtained with a scanning electron microscope (from Ref. 41 ). An additional central gate is connected to the main one outside a region occupied by a two-dimensional gas (outside the mesa). The dotted lines show schematically a position of the edge states. The numbers 1-4 denote contacts where the corresponding edge states come to.
An example of interference oscillations in a regime of the integer QHE at filling factor 1 under the gate (from Ref. 41 ) at a minimum temperature of 30 mK. A change of magnetic field at a fixed gate voltage; a monotonous dependence V(B), reflecting a behavior of the Zeeman splitting on the edge of the sample, was subtracted 47 (a). A changing of the gate voltage at a fixed magnetic field for different values of magnetic field. A monotonous dependence was not subtracted. The measurement current is I = 4 nA (b). Plotting a dependence of the oscillation position on its number (the upper inset in (a)) one can reliably identify periods ΔB = 67 mT and ΔVg = 7.8 mV respectively for the sample with an additional 1-μm gate. An inset at the bottom shows an example of the sample oscillations with a different width of the additional gate (1.5 μm) with a period of 45 mT; a monotonous dependence V(B) was not subtracted. The measurement current I = 10 nA.
An example of independence of interference oscillations in a regime of the integer QHE at the filling factor 1 under the gate on the temperature (a) and the imbalance between the edge states (b) (from Ref. 41 ).
A structure of compressible (white) and incompressible (yellow - for a local filling factor 1 under the gate and inside the incompressible strip at the edge, green - for the filling factor 2 in the gate-gap region) regions of electron fluid in the working area of the interferometer (from Ref. 41 ). An additional central gate locally increases a width of the incompressible strip with a filling factor 1. The electron transport across the incompressible strip occurs on both sides of this region.
An energy profile across the sample edge in different parts of the interferometer (from Ref. 41 ). (a) Between the main and the additional gates in the absence of applied imbalance of electro-chemical potentials. Two Landau sublevels filled in the volume raise up by the edge potential when approaching the edge. Each of them is pinned to the Fermi level in a neighborhood of the intersection point (compressible strip), cf. Ref. 8 . (b) At the same place for the created imbalance of electro-chemical potentials of compressible strips equal to an energy gap in the incompressible strip between them (a flat-band situation), cf. Ref. 47 . (c) In different parts of the interferometer at imbalances exceeding an initial gap in the incompressible strip. The arrow shows the transport channel, appearing in this regime, across the edge to an excited state with the same energy and spin. It is this channel which is responsible for retaining the coherence. The transport across the edge is suppressed in the region of the central gate by local broadening of the incompressible strip.
A current-voltage characteristics of the transport between the edge states in samples with an additional gate in the gate-gap region for the integer filling factors 1 under the gate and 2 in the gate-gap region (from Ref. 41 ): (a) An individual curve in a wide range of currents; (b) initial parts of two curves for values of the field and the gate voltage corresponding to the maximum and the minimum of interference oscillations. It is seen that all changes take place under imbalances between the edge states exceeding the threshold, at which the transport channel, shown in Fig. 11c , appears.
An example of interference oscillations in a regime of the fractional QHE for the sample with a width of the additional gate 1.5 μm (from Ref. 42 ) at filling factors 1/3 under the gate and 2/3 in the gate-gap region: (a) change of a magnetic field at a fixed gate voltage, and (b) change of a gate voltage at a fixed magnetic field for different values of the magnetic field. Plotting a dependence of the oscillation position on its number one can reliably determine the period (see the insets), ΔB = 150 mT (a) and ΔVg = 10 mV (b) respectively. The measurement current I = 0.3 nA.
The suppression of interference oscillations in a regime of the fractional QHE upon increasing temperature and imbalance (from Ref. 42 ) for filling factors 1/3 under the gate and 2/3 in the gate-gap region. The magnetic field B = 9.876 T.
(a) Suppression of interference oscillations in a regime of the fractional QHE upon increasing imbalance for the sample with a 1.5 μm gate at the filling factor 3/5 in the region of the gate-gap. (b) - (d) Examples of interference curves for a regime of the fractional QHE for the sample with a 1-μm additional gate at filling factors 1/3 under the gate and 2/3, 3/5 in the gate-gap region: change of the gate voltage Vg at a constant magnetic field (b), (c); change of the magnetic field B at a constant gate voltage (d). The measurement current I = 1 nA for (b),(d), I = 0.3 nA for (c). The temperature T = 30 mK for (a)–(d) (from Ref. 42 ).
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