Magnetic-Field-Dependent Transmission Phase of a Double-Dot System in a Quantum Ring

The Aharonov-Bohm effect is measured in a four-terminal open ring geometry. Two quantum dots are embedded in the structure, one in each of the two interfering paths. The number of electrons in the two dots can be controlled independently. The transmission phase is measured as electrons are added to or taken away from the individual quantum dots. Although the measured phase shifts are in qualitative agreement with theoretical predictions, the phase evolution exhibits unexpected dependence on the magnetic field. Phase lapses are found only in certain ranges of the magnetic field.

Figure 1

(a) AFM micrograph of the ring structure. Bright oxide lines fabricated by AFM lithography lead to insulating barriers in the 2DEG. (b) Schematic arrangement of the four terminal ring. (c) AB oscillations in $V_{\mathrm{n}\mathrm{l}}$. (d) Conductance through a quantum dot induced in segment 1 as a function of $V_{\mathrm{p}\mathrm{g}\mathrm{1}}$.

Figure 2

Plots of (a) the nonlocal voltage $V_{\mathrm{n}\mathrm{l}}$, (b) the currents $I_{\mathrm{B}}$ and (c) $I_{\mathrm{D}}$ as functions, respectively, of $V_{\mathrm{p}\mathrm{g}\mathrm{1}}$ and $V_{\mathrm{p}\mathrm{g}\mathrm{2}}$ tuning the dots in segments 1 and 2.

Figure 3

AB oscillations in $V_{\mathrm{n}\mathrm{l}}$ as a function of the plunger gate voltages tuned along trace linear 2. (a) The currents $I_{\mathrm{D}}$ and $I_{\mathrm{B}}$ at $B=95\mathrm{m}\mathrm{T}$. (b) A plot of $V_{\mathrm{n}\mathrm{l}}$ as a function of plunger gates and magnetic field. (c) Cross sections along dashed lines in (b) giving $V_{\mathrm{n}\mathrm{l}}$ vs $B$. The dashed lines mark constant values of $B$ where a $\pi$ phase shift of the AB oscillations can be seen, as gate voltages are tuned across the conductance peak in dot 2.

Figure 4

AB oscillations along the different traces indicated in Fig. 2. (a) Sweep linear 2 (b) sweep diag 1 (c) sweep linear 1, and (d) sweep diag 2. Vertical lines are guides to the eye. Dashed horizontal lines indicate where maxima in the magnetic field averaged contribution to $V_{\mathrm{n}\mathrm{l}}$ occur.