Berry Phase of Electrons in a Ring

IRG 1: J.-B. Yau, E. De Poortere, and M. Shayegan

If we rotate an object (e.g. a bottle lying on its side) once and return it exactly to its starting position, we expect to recover the original state in principle. In the quantum-mechanical realm, however, the wave function describing the object is altered in a subtle way. Electrons, it seems, can never go home again after a trip.

We may grab hold of an electron’s spin S by applying a tilted magnetic field H (S aligns with H). As we slowly precess H so that it describes a cone, S follows suit and returns to its initial direction. In 1984, Michael Berry showed that the electron’s wave function, instead of recovering its initial form, acquires a phase shift (the Berry phase). In recent years, the existence of the Berry phase for photons has been demonstrated in laser experiments. However, an explicit observation of the Berry phase for electrons in solids is much more challenging.

Shayegan and students have recently achieved a particularly clean demonstration of the Berry phase for electrons in the two-dimensional gas trapped in a micron-sized conducting ring (left figure). Electrical current is injected into the ring at an electrode 1 and extracted at electrode 2. At 1, the electrons fork into two streams which go around the ring and re-combine at 2. The wave functions of electrons in the two streams interfere when they come together at 2. If the interference happens to be constructive, the transmission probability is enhanced, and the voltage measured between the electrodes is low. Conversely, destructive interference produces a high voltage. Previous experiments showed that a magnetic field B_ext applied parallel to z (normal to the plane of the ring) engenders phase shifts of opposite signs in the two beams. The phase shift is known as the Aharonov-Bohm or AB effect. If we increase B_ext, the AB phase shift grows in proportion, periodically changing the interference at 2 from constructive to destructive and back again. The observed voltage oscillates periodically with B_ext as a pure sine wave (rapid oscillation in right figure).

To observe the Berry phase, Shayegan’s group applied a fixed electric field E antiparallel to z, in addition to B_ext (see figure). In the rest frame of the moving electrons, E appears as an effective magnetic field B_eff that is directed radially in the plane of the ring (this is the origin of the spin-orbit coupling in an atom). The combination of the original B_ext and the new field B_eff produces a resultant field tilted at an angle q to z. Moreover, as the electron moves around the ring, the tilted field describes an inverted cone. As discussed above, the precession of each spin around the cone forces the wave function to acquire a Berry phase in addition to the original AB phase. The simultaneous presence of the two phase shifts is analogous to combining two distinct musical tones. Instead of a sine wave, the voltage acquires a beat pattern versus B_ext. Shayegan’s group measured the beat pattern (right figure) and showed that it agrees qualitatively with the prediction of Berry’s theorem.

Related publication:
J-B. Yau, E. De Poortere, and M. Shayegan, “Aharonov-Bohm Oscillations with Spin: Evidence for Berry’s Phase,” Phys. Rev. Lett. 88, 146801 (2002).