Breaking of phase symmetry in nonequilibrium Aharonov-Bohm oscillations through a quantum dot

Linear-response conductance of a two-terminal Aharonov-Bohm (AB) interferometer is an even function of magnetic field. This phase symmetry is not expected to hold beyond the linear-response regime and in simple AB rings the phase of the oscillations changes smoothly (almost linearly) with voltage bias. However, in an interferometer with a quantum dot in its arm, tuned to the Coulomb blockade regime, experiments indicate that phase symmetry seems to persist even in the nonlinear regime. In this paper we discuss the processes that break AB phase symmetry and show that breaking of phase symmetry in such an interferometer is possible only after the onset of inelastic cotunneling, i.e., when the voltage bias is larger than the excitation energy in the dot. The asymmetric component of AB oscillations is significant only when the contributions of different levels to the symmetric component nearly cancel out (e.g., due to different parity of these levels), which explains the sharp changes in the AB phase. We show that our theoretical results are consistent with experimental findings.

Figure 1

(Color online) (a) Schematic representation of the device studied in this paper. Solid red and dash blue arrows show cotunneling processes and direct lead-to-lead tunneling, respectively. (b) Example of a lowest-order cotunneling process contributing to AB oscillations: the dot electron tunnels to the right lead and an electron from the left tunnels to the dot, interfering with the process where an electron moves from left to right through the open arm.

Figure 2

(Color online) Examples of different third-order processes, whose real parts contribute to the current: (a) and (b) [or (c) and (d)] are two processes whose contributions to odd AB oscillations mutually cancel out; processes (a) and (c) are elastic, whereas (b) and (d) are inelastic. (e) [or (g)] is an example of an elastic (inelastic) third-order process which gives nonzero contribution to the odd AB oscillations. The other process constructed from the same matrix elements and beginning from the same initial state, (f) [or (h)], does not contribute to the current.

Figure 3

(Color online) Color plots of the differential conductance obtained from the theoretical model presented here (left panels) and from the experimental data of Ref. 11 (right panels). The upper and lower panels show, respectively, full and asymmetric components of the conductance.

Figure 4

(Color online) Power of asymmetric AB oscillations and differential conductance (rescaled) for theoretical model (left) and for experimental data (right).