Quantum physics in quantum rings

Abstract

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 signifi cance of the magnetic vector potential in quantum mechanics [2]. They predicted that the phase difference of the alternative paths changes by 271 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 mag netic 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 spec trum 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 momen tum becomes a good quantum number. In the case of a single mode ring the single-particle energy levels are given by:Here l is the angular momentum quantum number and m is the magnetic quantum number, i.e. the number of flux quanta penetrating the ring at a given external magnetic field. The only material-dependent parameter is the effective mass, m*. The ring radius is denoted by r. At zero magnetic field, m = 0, the ground state has angular momentum l = 0 , the next two degenerate states are characterized by 1 = ± 1 and so forth. At finite magnetic field the ground state develops a finite angular momentu n which can be translated into a persistent current [4]. In this model the wave functions are plane waves extended around the ring independent of magnetic field. Fabrication of quantum ringsIn this article we describe recent experiments on quantum ring: realized in semiconductor heterostructures. Figure 1 shows the Aharonov-Bohm effectAn electron wave coming from the left travels through the ring. The wave propagates in the upper as well as in the lower arm and interferes with itself. The phase accumulation in the upper arm is (ϕ1, in the lower arm (ϕ 2. The phase difference ϕp is determined by a constant depending on the details of the wave propagation in the two arms (ϕ geom) ...

Aharonov-Bohm quantum rings in high-Qmicrocavities

“…For a two-electron ring, placing it in a suitable cavity will lead to optical detection of the triplet-singlet crossover discussed in Ref. 23. The results for a three-electron ring should be somewhat similar to those for a single-electron ring due to Pauli blocking and apparent electron hole symmetry in a four-level system with three electrons.…”

“…23 However, spin is by far less important for optical transitions in the single-electron QR studied here. Since spin is conserved in optical transitions, our results in the absence of a magnetic field or in a weak magnetic field producing negligible Zeeman splitting (there is no enhanced exchange-induced splitting for a single-electron ring), remain completely unchanged, as the S MC SQR FIG.…”

“…It should be mentioned that the spin degree of freedom, which plays an important role for the transport properties of few-particle QRs, 23 is less important for single-electron spinconserving dipole transitions studied in this work. Therefore, electron spin is neglected for the most of the paper and its role is only discussed briefly near the end of Sec.…”

Electric dipole moment oscillations in Aharonov-Bohm quantum rings

“…[1][2][3] The fascination with quantum rings is partially caused by a wide variety of purely quantum-mechanical effects, which are observed in ringlike nanostructures (for a review, see Refs. [4][5][6]. The star among them is the Aharonov-Bohm effect, 7,8 in which a charged particle is influenced by a magnetic field away from the particle's trajectory, resulting in magnetic-flux-dependent oscillations of the ring-confined particle energy.…”

Magnetic-field-modulated terahertz absorption spectra of a quantum ring

“…[6][7][8] The energy levels of carriers can be controlled by the structural design of a QR, for example, variations in the radius and width of the QR. A magnetic field can result in the oscillatory behavior of carriers' energy levels.…”