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Physical Review Letters

Berry’s phase, motive forces, and mesoscopic conductivity

Ady Stern

Published Feb 17, 1992• 222 citations
Publisher

Abstract

A ring in a magnetic field whose direction varies in space is considered. It is shown that the Berry phase accumulated by the spins of electrons encircling the ring affects the conductance of the ring in a way similar to the Aharonov-Bohm effect. A time-dependent Berry phase is shown to induce a motive force in the ring. This motive force couples to the electron's spin, similar to the way Faraday's law couples an electromotive force to the electric charge.

DOI
10.1103/physrevlett.68.1022

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Journal of Physics Condensed Matter•Mar 12, 2004•3 citations
A ring in a magnetic field whose direction varies in space is considered. It is shown that the Berry phase accumulated by the spins of electrons encircling the ring affects the conductance of the ring in a way similar to the Aharonov-Bohm effect. A time-dependent Berry phase is shown to induce a motive force in the ring. This motive force couples to the electron's spin, similar to the way Faraday's law couples an electromotive force to the electric charge.
The quantum orbital motion of electrons in mesoscopic normal-metal rings threaded by a magnetic flux produces striking interference phenomena such as persistent currents due to the Aharonov-Bohm effect. Similarly, when a quantum spin adiabatically follows a magnetic field that rotates slowly in time, the phase of its state vector acquires an additional contribution known as the Berry phase. We explore the combination of these two quantum phenomena by examining the interplay between orbital and spin degrees of freedom for a charged spin-1/2 particle moving in a mesoscopic ring embedded in a classical, static inhomogeneous magnetic field, i.e, a texture. As a consequence of its orbital motion through the texture, the spin experiences, via the Zeeman interaction, a varying magnetic field. This results in a Berry---or geometric---phase, leading to persistent (i.e., equilibrium) currents of charge and spin. These mesoscopic phenomena are related to (but should be distinguished from) the conventional persistent currents that result from magnetic flux through a ring. We develop a path-integral approach to decouple the orbital and spin motion and, by using an adiabatic approximation, we compute the equilibrium expectation values of the persistent charge and spin currents and the magnetization. We find that the persistent currents depend on the texture in a striking manner through a geometric phase (related to a surface area characterizing the texture) and a geometric vector (related to the projections of this area). In the special case of a cylindrically symmetric texture we use a spectrum obtained by Kuratsuji and Iida to obtain exact results that confirm, independently, the validity of the path-integral approach in the adiabatic limit. We discuss the connection between the geometric vector and quantum-mechanical correlations, and examine quantum fluctuations and the zero-point energy.
We calculate the oscillations of the dc conductance across a mesoscopic ring, simultaneously tuned by applied magnetic and electric fields orthogonal to the ring. The oscillations depend on the Aharonov-Bohm flux and of the spin-orbit coupling. They result from mixing of the dynamical phase, including the Zeeman spin splitting, and of geometric phases. By changing the applied fields, the geometric phase contribution to the conductance oscillations can be tuned from the adiabatic (Berry) to the nonadiabatic (Ahronov-Anandan) regime. To model a realistic device, we also include nonzero backscattering at the connection between ring and contacts, and a random phase for electron wave function, accounting for dephasing effects.
Simple analogues of two fundamental physical phenomena, Faraday's law of induction, and the Aharonov–Bohm effect, have been predicted by Ady Stern of Tel Aviv University, Israel, in the behaviour of electrons in a ring in a magnetic field (Phys. Rev. Lett. (1992) 68 1022). The surprising twist is that it is the electron's spin, rather than its charge, which plays the central role. These effects are associated with the quantum geometric (or Berry) phase, a discovery in elementary quantum mechanics which has generated great excitement in the last eight years.
Based on a gauge-theoretic approach, an equivalence between the Aharonov-Bohm effect of a solenoid and the Aharonov-Casher effect of a charged rod is explicitly demonstrated. It is shown that a time-dependent Aharonov-Casher phase can induce a motive force via the SU(2)spin{)}_{\mathrm{spin}})spin​ gauge field. The intrinsic non-Abelian nature of the SU(2)spin{)}_{\mathrm{spin}})spin​ gauge field predicts the non-Abelian Faraday law, subject to further experimental verification.
Electron spin transport through an Aharonov–Bohm ring driven by time-dependent inhomogeneous magnetic fields is treated. The system possesses an su (2) l × su (2) s dynamical symmetry in both orbital angular momentum space and spin space, and is thus proved to be integrable according to algebraic dynamics. Based on the analytical solutions, the relevant physical quantities such as electric current, spin current, magnetization and conductance are calculated. It is found that for a magnetic field with π/2 twist angle, the direction of spin-polarization will be reversed at zero magnetic flux. In the resonant rotating magnetic field, the spin transmission is oscillating with time t , and can reach unity, so that a complete spin flip can also be induced. The results obtained may be of practical significance for the design of nano-electromagnetic spin devices, such as a spin switch, in a controllable way.