1. Aharonov-Bohm Ring Oscillations
Rui Liu

2. Outline
Aharonov-Bohm Effect
Ring Oscillations
Applications

3. A-B Effect
Schematic of double-slit experiment in which Aharonov-Bohm effect can be observed: electrons pass through two slits, interfering at an observation screen, with the interference pattern shifted when a magnetic field B is turned on in the cylindrical solenoid.
Illustration of interference experiment for Aharonov-Bohm effect

4. A-B Effect Comparson of non-B and B simulations
The following animation shows the diffraction of a Gaussian wave packet by two slits. In this case there is no magnetic field. 
The next animation illustrates the Aharonov-Bohm effect. The magnetic field is confined to the red area and is chosen such that the shift in the interference pattern is as large as possible.

5. A-B Effect Formulations (Magnetic A-B Effect) (Electric A-B Effect)
A-B effect implies that a particle with charge q travelling along some path P in a region with zero magnetic field ( ) must acquire a phase φ .
with a phase difference Δφ between any two paths with the same endpoints therefore determined by the magnetic flux Φ through the area between the paths
2. From the Schrödinger equation, the phase of an eigenfunction with energy E goes as exp(-iEt/hbar). The energy, however, will depend upon the electrostatic potential V for a particle with charge q. In particular, for a region with constant potential V (zero field), the electric potential energy qV is simply added to E, resulting in a phase shift: del fi = -qVt / hbar, where t is the time spent in the potential.
(Electric A-B Effect)

6. Ring Oscillations Ring Oscillation without E/B Field
A simple version of the double-slit experiment for electrons consists in a metal ring where an electron can travel either the upper or the lower arm (see the picture). This has the advantage that the phase-difference between the two paths can be adjusted by applying a magnetic field which threads the ring. Even when the field is confined to the interior of the opening in the ring and cannot affect the electron's motion directly, the Aharonov-Bohm effect will make the phase-difference dependent on the magnetic flux inside the ring. In the experiments, the diameter of such a ring is typically only a few micrometers.

7. Ring Oscillations Ring Oscillations with E/B Field
The Aharonov-Bohm effect. Electrons enter and leave the ring structure as indicated by the horizontal arrows. The quantum wave associated with each electron in the entrance region splits into two wave packets that go around the ring, as indicated by the curved arrows, and interfere in the exit region. Homogeneous magnetic field B and electric field E are applied normal to the plane of the ring. Whether the interference is constructive or destructive, and hence whether the current is maximum or minimum, depends on the values of B and E. Varying B therefore leads to oscillations of the current.
Oscillations in the ring. .R is the resistance (in ohms, after subtracting a smooth background resistance) in the presence of an externally applied magnetic field B (in teslas). As B is varied (with the externally applied electric field E fixed), .R oscillates because of the oscillation of the current. [Adapted from (1)]

8. Ring Oscillation Formulation of A-B Ring ΔR is the resistance change,
r is the radius of the ring,
N is the density of current carrying electrons in the current ring cross section,
W is the width of the current ring,
t is the thickness of the ring.

9. A-B Ring Applications A-B Ring in Semiconductor
The AB effect has been observed both in normal metal rings (first by Sharvin & Sharvin, JETP Lett. 34, 272 (1981)) and in semiconductor rings (first time convincingly by G. Timp et al., Phys. Rev. Lett. 58, 2814 (1987)). AB rings are of fundamental interest as probes of electron phase coherence, and as test devices for the concept of 1D channels as electron wave guides. We employ the fabrication technique of A. Kristensen et al., J. Appl. Phys. 83, 607 (1998) to make AB rings in GaAs/GaAlAs heterostructures. A Scanning Electron Microscope image of a device (before deposition of top gate) is shown below (left).

10. A-B Ring Applications A-B Ring in Metal
Figure 1 Quantum interference of electron–hole pair in mesoscopic ring. a, Energy diagram modelling a tunnelling process. At the moment of tunnelling, an electron (e) with energy ee is injected into the right part of the sample leavinga hole (h) with energy eh in the left part. b, Typical electron–hole trajectories which are sensitive to an Aharonov–Bohm flux. At the moment of a tunnel event an electron–hole pair is created. The hole and the electron diffuse separately to the other tunnel barrier where they recombine and interfere. The interference of the electron and hole wave is both sensitive to a magnetic field B piercing the ring (magnetic Aharonov–Bohm effect) and a voltage V across the ring (electrostatic Aharonov–Bohm effect). c, Scanning electron microscope image of the Aharonov–Bohm ring which is interrupted by two small tunnel junctions.
Figure 2 Conductance G versus magnetic field B at T ¼ 20mK and V ¼ 500 mV.

11. A-B Ring Applications
A-B oscillation in a Ring with a QD connected in series
The device contains a QD connected in series with an AB ring. The number of electrons in the QD is controlled by the voltage p V applied to the plunger gate. The two-terminal conductance between the source (S) and drain (D) is measured.
AB oscillation near a CB peak. (a) the conductance of the device as a function of the plunger gate voltage, p V , shows a CB peak. (b) Magneto-conductance of the device for 5 different points along the CB peak.

12. A-B Ring Applications A-B interferometer
The Aharonov-Bohm interferometer, or Mach-Zehnder interferometer as it is called in optics, works by splitting an electron wave into two halves. After propagating a certain distance the two waves are made to rejoin. If everything is symmetric the phases of the waves will be in phase and the electron will continue undisturbed. If, on the other hand, the length of the two branches is different the waves may be out of phase. In this case the electron will be reflected instead.
While it is impractical to change the physical length of the device a similar effect may be caused by a magnetic or electrical field. This will instead change the phase velocity of the electron which also will cause the waves to have different phase when rejoined.

13. References
Johnson, S.G. (July 2003). Illustration of interference experiment for Aharonov-Bohm effect. Wikipedia.
Michielsen, K. & De Raedt, H. (2006). Aharonov-Bohm.
Wikipedia. Aharonov-Bohm effect. Wikipedia.
Marquardt, F. (August 2001). The Aharonov-Bohm ring: “which-way” information, energy transfer and the Pauli principle. An introduction to the basics of dephasing.
Anandan, J. (September 2002). Putting a Spin on the Aharonov-Bohm Oscillations. Science, 297.
BlackLight Power, Inc. (2006). Aharonov-Bohm Effect.
Pedersen, S.V., Hansen, A.E., Kristensen, A., Sorensen, C.B. & Lindelof, P.E. (May 1999). Aharonov-Bohm rings. Mesoscopic electronics and clusters. Nano Physics Research Group, University of Copenhagen, Denmark.
Van Oudenaarden, A., Devoret, M.H., Nazarov, Y.V. & Mooij, J.E. (February 1998). Magneto-electric Aharonov-Bohm effect in metal rings. Nature, 391.
Palm, T. (2006). Simulations of Quantum Interference Devices. Laboratory of Photonics and Microwave Engineering, Department of Electronics, KTH, Sweden.
Israel Institute of Technology. (2006). Aharonov - Bohm Oscillation in a Ring with a Quantum Dot Connected in Series.

14. Thank You
Questions and Comments?