Quantum physics in quantum rings

T he symmetry of a ring system is crucial for classical and quantum effects. Mathematically speaking a ring is a nonsingly connected geometry. In quantum mechanics the ring symmetry of the benzene molecule gives rise to its delocalized electronic states [1]. In ring geometries strongly connected to external leads the electron wave packets can take two different paths around the ring which gives rise to interference. This is reminiscent of Young’s double-slit experiment for photons. The use of charged particles in a ring geometry rather than neutral photons allows the relative phase of the electronic wave function in the two arms of the ring to be manipulated by a magnetic field perpendicular to the plane of the ring. Aharonov and Bohm proposed such a set-up to test experimentally the significance of the magnetic vector potential in quantum mechanics [2]. They predicted that the phase difference of the alternative paths changes by 2π as the flux through the ring is changed by one flux quantum h/q (q is the charge of the particle). Many experiments over the last three decades have demonstrated magnetic field periodic resistance oscillations in ring structures with a phase coherence length longer than or comparable to the perimeter. In mesoscopic physics the Aharonov-Bohm (AB) effect has become a standard tool to quantitatively investigate the phase coherence of transport in metallic [3] and semiconducting systems. In closed systems with fixed electron number a characteristic magnetic flux-periodic energy spectrum evolves. Such a spectrum can be detected experimentally by measuring electron transport through a lithographically defined quantum ring in the Coulomb blockade regime. In ring-shaped confined quantum systems the angular momentum becomes a good quantum number. In the case of a single mode ring the single-particle energy levels are given by: