Quantum rings by local anodic oxidation with an AFMA simple, cheap yet versatile technique to fabricate nanostructures is local anodic oxidation with an atomic force microscope (AFM). By applying a voltage between a conducting AFM-tip and a substrate, in a humid environment, the surface of the substrate directly below the atomically sharp tip is oxidized (see bright lines in Fig 1a). Because the tip can be moved to any desired location it is possible to ‘write’ every structure imaginable. By oxidizing the surface of a shallow two-dimensional electron system (2DES) we can also create electronic structures. The oxide lines deplete the 2DES directly underneath the surface by locally changing the bandstructure. Thereby isolating regions are created that can be used to define electronic structures in the high mobility 2DES. As a substrate we use a GaAs/GaAlAs-heterostructure with a two-dimensional electron gas (2DEG) 55 nm below the surface. The 2DEG is depleted underneath the bright oxide lines in Fig. 1a, leaving highly mobile electrons in the darker regions in between, e.g. effectively cutting out a ring structure. Figure 1b shows a schematic picture of the ring structure where the 'golden dikes' are the barriers created by the oxide in the 2DEG. The blue parts indicate the source (S) and drain (D) resevoirs from which electrons enter and exit the ring. The two arms of the ring through which the electrons can move from the entrance to the exit of the ring are indicated in red and green.
An electron wave moving from source to drain (indicated by the red line in Fig. 1a) in a quantum ring enclosing a non-zero flux accumulates a phase change, known as the Aharonov-Bohm effect. The accumulated phase differs in both arms of the ring leading to a phase difference 'phi' at the exit of the ring (see Fig. 1b). The phase difference changes by 2PI as the flux through the ring is changed by one flux quantum 'phi_zero' = e/h. This leads to constructive and destructive interference, or respectively high and low conductance, that is periodic in the magnetic flux penetrating the ring. The resulting conductance oscillations are described by
dB = (h/e)A, (1)
where dB is the period of the oscillation and A the area of the enclosed flux.
A typical result of the Aharonov-Bohm oscillations observed in the quantum ring from Fig. 1a is shown in Fig. 1c. The period of the oscillations, dB = 60 mT, corresponds by (1) to an effective ring area of 7.1 10^4 nm2 (diameter of 300 nm), in reasonable agreement with the effective lithographically defined area 9.6 10^4 nm2 (diameter of 350 nm). The amplitude of the oscillations can be tuned by asymmetric gate voltages and has a maximum of 0.15 e^2/h at a temperature T < 100 K. This indicates that although the rings show phase coherent transport, a considerable amount of phase information is lost by inelastic scattering events inside the ring. At magnetic fields above 0.2 T the 1/B-dependent Shubnikov-de-Haas oscillations in the 2DEG leads are superimposed upon the Aharonov-Bohm oscillations (periodic in B).
Figure 1 | Aharonov-Bohm effect in a quantum ring. a, Atomic force micrograph of a quantum ring.
The bright lines and dot are the oxides that define the ring structure in the 2DEG underneath. With the
in-plane gates (VG1, VG4 and VG3, VG6) it is possible to tune the quantum point contacts at the entrance
and exit of the ring (indicated by the green lines). The middle in-plane gates (VG2 and VG5) make it
possible to tune the electron concentration in both arms of the ring. b, Schematic representation of a
quantum ring with a flux through the center and the two different phases accumulated in both arms of the
ring, leading to a phase difference 'phi' at the drain that is periodic with the flux. c, Typical
measurement of the Aharonov-Bohm oscillations for two quantum rings fabricated on different samples
(T = 50 mK). The period of the oscillations, dB = 60 mT for the solid blue curve and dB = 80 mT for the
dotted red curve, correspond to ring diameters of respectively d ~ 300 nm and d ~ 260 nm, in reasonable
agreement with the lithographically defined diameters. The faster oscillations at higher magnetic fields
are the 1/B-dependent Shubnikov-de-Haas oscillations of the 2DEG leads superimposed on the Aharonov-Bohm
Quantum Hall effect in grapheneThe quantum Hall effect (QHE) observed in two-dimensional electron systems (2DESs) is one of the fundamental quantum phenomena in solid state physics. Since its discovery in 1980 by K. von Klitzing it has been important for fundamental physics and application to quantum metrology. Recently a new member joined the family of 2DESs: graphene, a single layer of carbon atoms. Graphene displays a unique charge carrier spectrum of chiral Dirac fermions and enriches the QHE with a half integer QHE of massless relativistic particles observed in single-layer graphene and a novel type of integer QHE of massive chiral fermions in bilayers. Moreover, the band structure of graphene even allows the observation of the QHE up to room temperature. Since localization in conventional quantum Hall systems is already fully destroyed at moderate temperatures, no QHE has been observed at temperatures above 30 K until very recently. Therefore, understanding a room temperature QHE in graphene goes far beyond our comprehension of the traditional QHE.
In order to access this intriguing phenomenon in more detail we preformed systematic measurements of the inter Landau level activation gap in graphene for magnetic fields up to 32 T and temperatures from 4 K to 300 K. We observed that the gap between the zeroth and the first Landau level approaches the bare, unbroadened Landau-level separation for high magnetic fields and we explain these findings by a much narrower lowest Landau level compared to the other ones. In contrast, for higher Landau levels, the measured activation gap behaves as expected for equally broadened states.
Figure 2 | Graphene. An artistic impression of a piece of graphene with its rippeled atomic structure
blended through. |
Quantum resistance metrology in grapheneThe Hall resistance in two-dimensional electron systems (2DESs) is quantized in terms of natural constants only, R_H = h/(ie^2) with i an integer number. Due to its high accuracy and reproducibility this quantized Hall resistance in conventional 2DESs is nowadays used as a universal resistance standard. Graphene, the purely two-dimensional form of carbon, is a system fundamentally different from conventional 2DESs displaying a novel type of half-integer quantum Hall effect which remains visible up to room-temperature. This makes graphene a promising candidate for a high-temperature quantum resistance standard and may give further evidence that the quantum Hall resistance is indeed given by the relation h/(ie^2). In close collaboration with the NMi van Swinden Laboratory and the University of Manchester, we have performed the first metrological characterization of the quantum Hall resistance in an only 1 micrometer wide graphene Hall-bar. The quantization of the Hall plateaus is within (-5 ± 15) ppm equal to that in conventional semiconductors, R_H = (12,906.34 ± 0.20)Ohm. The principal limitation of the present experiments is the relatively high contact resistances in the quantum Hall regime, inducing measurement noise and local heating. Extrapolating our results to samples with lower resistance contacts for both electrons and holes and using wider samples with high breakdown currents, would most probably allow precision measurements of the quantum Hall effect in graphene with accuracies in the ppb range.
Figure 3 | Graphene metrology. (left) Hall resistivity (red) and longitudinal resistivity (blue) in graphene at 14 Tesla and 0.4 Kelvin showing the exact quantization of the v = ± 2 Hall-plateaus and the accompaniing zero minimas in the longitudinal resistivity. The inset show a false color scanning electron image of the graphene Hall-barr sample.
(right) Presision measurements in the middel of the v = 2 plateaus for different contacts and their average, displaying the Hall-resistivity equal to conventional semiconductors within (-5 ± 15) ppm leading to R_H = (12,906.34 ± 0.20)Ohm. |