An electrostatic potential V affect

An electrostatic potential V affects the phase of the electron wave function through the term (e/¯h) RV dt in much the same way as a vector potential does. If the two arms of the ring have a potential difference V , and an electron traverses an arm in a time t, then the acquired phase shift would lead to oscillations in the resistance with periodicity
V = h/et. The electrostatic Aharonov-Bohm effect has a periodicity that depends on the transit time t, and is not a geometrical property of the ring, as it is for the magnetic effect. A distribution of transit times could easily average out the oscillations. Note that the potential difference effectuates the phase difference by changing the wavelength of the electrons (via a change in their kinetic energy), which also distinguishes the electrostatic from the magnetic effect (where a phase shift is induced by the vector potential without a change in wavelength). An experimental search for the electrostatic Aharonov-Bohm effect in a small metal ring was performed by Washburn et al.202 An electric field was applied in the plane of the ring by small capacitive electrodes. They were able to shift the phase of the magnetoresistance oscillations by varying the field, but the effect was not sufficiently strong to allow the observation of purely electrostatic oscillations. Unfortunately, this experiment could not discriminate between the effect of the electric field penetrating in the arms of the ring (which could induce a phase shift by changing the trajectories) and that of the electrostatic potential. Experiments have been reported by De Vegvar et al.203 on the manipulation of the phase of the electrons by means of the voltage on a gate electrode positioned across one of the arms of a heterostructure ring. In this system a change in gate voltage has a large effect on the resistance of the ring, primarily because it strongy affects the local density of the electron gas. No clear periodic signal, indicative of an electrostatic Aharonov-Bohm effect, could be resolved. As discussed in Ref.203, this is not too surprising, in view of the fact that in that device 1D subband depopulation in the region under the gate occurs on the same gate voltage scale as the expected Aharonov-Bohm effect. The observation of an electrostatic Aharonov-Bohm effect thus remains an experimental challenge. A successful experiment would appear to require a ring in which only a single 1D subband is occupied, to ensure a unique transit time.198,200