# Localization in small metal rings: Current saturation without dissipation

#### Abstract

We consider an electron in a small 1-dimensional conducting ring subject to a large electric field induced by a linearly ramped magnetic flux. In the absence of inelastic scattering this constitutes a Hamiltonian system. The adiabatic eigenstates form a complete set of mini-bands which are separated by gaps determined by the static scattering potential in the ring. In high fields Zener tunneling transitions between these mini-bands promote the electrons into higher energy states and an increase of the induced current is expected. In a recent letter D. Lenstra and W. van Haeringen (Phys. Rev. Lett.57, 1623 (1986)) reported a saturation of the current at a value proportional to the electromotive force, i.e., the system showed "resistive" behavior. We have studied several model systems to investigate the quantum coherence effects in the time evolution of the single particle wave function. None of our models shows "resistive" behavior in agreement with the nonlinear character of the Zener tunneling transitions. Instead we find that the wave function becomes localized in energy space, preventing the particles from extracting energy from the induced electric field. As a consequence the current carried by the electrons returns back to zero. We expect that the inclusion of weak dissipation will destroy the quantum coherence effects and re-enable the electrons to attain large energies in the electric field. © 1989 IOP Publishing Ltd.

*This paper has been withdrawn.*