Temperature- and current-dependent dephasing in an Aharonov-Bohm ring

We investigate the dependence of the dephasing rate in a ballistic Aharonov-Bohm ring on the temperature $T$, the bias current, and the probe configuration. We find that the probe configuration influences the conductance but not the dephasing rate. Rather, averaging of the transmission phase, carried by thermally excited or current-induced electrons results in dephasing. We find that the appropriate energy window responsible for the dephasing is set by the drift velocity of the interfering electrons and the asymmetry of the ring path.

Figure 1

(Color online) (a) An optical micrograph of one of the devices used in the experiment. (b) SEM micrograph of the ring structure. (c) The two-probe conductance $G_{52}$ vs $V_g$. Since $V_g$ is less than $\sim -0.16\text{V}$, contact 5 is isolated from the device. (d) A diagram illustrating our notation of currents and voltages in an example of a five-terminal local measurement. Here $R_{14,23}=V_{23}/I_{14}$. The current drain is connected to the circuit ground.

Figure 2

(Color online) Magnetoresistance of the ring as measured in different probe configurations. (a) Four-terminal local setup ($V_g=-0.18\text{V}$, current: $1\to 4$, voltage: $2\to 3$) (b) Five-terminal local setup ($V_g=0\text{V}$, current: $1\to 4$, voltage: $2\to 3$) (c) Five-terminal nonlocal setup ($V_g=0\text{V}$, current: $1\to 2$, voltage: $3\to 4$) (d) Five-terminal mixed setup ($V_g=0\text{V}$, current: $5\to 4$, voltage: $1\to 2$). The dashed red line through the oscillations on each trace is the background resistance, $R_{back}$. (e) The Fourier spectrum of the data in Figs. 2a, 2b, 2c, 2d, after subtracting $R_{back}$. The bars indicate the ranges of inverse field which include $h/e$ and $h/2e$ peaks.

Figure 3

The evolution of AB amplitudes going from a five-terminal to a four-terminal ring by applying $V_g=0$ to $-0.19\text{V}$ in the local configuration. The data were taken in different cooling from Fig. 2. (a) Gray-scale plot of the AB oscillation component $R_{\text{AB}}$ as a function of $V_g$ and $B$. (b) Gray-scale plot of the subtracted background resistance $R_{back}$. (c) The gate dependence of AB oscillation amplitudes $\Delta R_{\text{AB}}$, obtained by FFT analysis of $R_{\text{AB}}$.

Figure 4

Temperature dependence of the AB amplitude $\Delta R_{\text{AB}}$ for various probe configurations. The solid lines are the results of fitting the data using exponential relationship as in Eq. (5). The dashed line marks the onset of linear fit. $\Delta R_{\text{AB}}$ saturates for the temperatures below the dashed line.

Figure 5

Temperature dependence of the AB amplitude $\Delta R_{\text{AB}}$ for different currents in a five-terminal configuration. (a) Local setup. (b) Nonlocal setup. The solid lines are fits using Eq. (5) in the temperature interval covered by the line. $\Delta R_{\text{AB}}$ approaches a saturation value for temperatures below the dashed lines. Values of the fit parameter $b$ are shown in the insets with error bars comparable to the symbol size.

Figure 6

The visibility of AB oscillations for various biasing currents in (a) the local setup, and (b) the nonlocal setup. The solid lines are least-squares fits of Eq. (8) to the data for $T>100\text{mK}$. (c) Voltage vs current characteristics for local ($V_{23}$ left axis) and nonlocal ($V_{34}$ right axis) measurements. The fitted parameters (d) $E_c$ and (e) ${\nu }_0$, where ${\nu }_{0\left(l\right)}$ and ${\nu }_{0\left(non\right)}$ represent the visibility for the local and nonlocal setups, respectively.