# Quantization of space - Flux-periodic high-field conductance oscillations in quantum rings

**Magneto-quantum oscillations in solid-state systems are based on either discrete energy levels (Landau levels) or the interference of electron waves propagating through a closed loop. The former lead to a 1/ B–periodic modulation of e.g. the magneto-conductance (Shubnikov-de Haas effect), the latter cause B-periodic oscillations when electrons propagate phase-coherently through a ring (Aharonov-Bohm effect). We report on a novel type of B-periodic high-field magneto-conductance oscillations in semiconductor quantum rings which are caused by the spatial discreteness of Landau orbits, the quantum mechanical analogue of classical cyclotron orbits.**

The electronic eigenstates of a two-dimensional electron system in a high magnetic field consist of Landau orbits with quantized energies. They are situated at discrete spatial positions and occupy a quantised area given by the magnetic flux quantum *h/e* divided by the applied magnetic field. Combined with the repelling Coulomb interaction, this discreteness of space leads to a discretisation of the electronic size of any mesoscopic structure in a quantizing magnetic field which changes periodically whenever a flux-quantum enters the structure.

Recent experiments at HFML Nijmegen have shown the spatial discreteness of the Landau orbits in high magnetic fields by means of magnetoconductance oscillations in semiconductor quantum rings. The structures were fabricated from high-mobility shallow two-dimensional electron systems in GaAs/AlGaAs heterojunction using local anodic oxidation with an atomic force microscope. The period of these new high-field oscillations is determined by the number of flux quanta penetrating the conducting area of the structure, i.e. the rim of the quantum rings. In this respect, they are distinctively different from the well-known Aharonov-Bohm oscillations where the period is governed by the number of flux quanta penetrating the entire ring.

Our experiments show explicitly that high magnetic fields do not only quantise the energy but also the size and position of the electronic states, an effect known implicitly since the early times of Landau in the thirties of the last century.

*(a) Schematic representation of a semiconductor quantum ring. The period of the semi-classical Aharonov-Bohm-effect is determined by the magnet flux penetrating the total area A of the ring. (b) High-field oscillations for three different quantum rings. The oscillation period is governed by the magnetic flux penetrating the rim of a ring (hatched areas).*

This work is published in:

A. J. M. Giesbers, U. Zeitler, M. I. Katsnelson, D. Reuter, A. D. Wieck, G. Biasiol, L. Sorba and J. C. Maan *Correlation-induced single-flux-quantum penetration in quantum rings, *Nature Physics, advance online publication, 31 January 2010 (DOI 10.1038/nphys1517)