Energy-Dependent Tunneling in a Quantum Dot

We present measurements of the rates for an electron to tunnel on and off a quantum dot, obtained using a quantum point contact charge sensor. The tunnel rates show exponential dependence on drain-source bias and plunger gate voltages. The tunneling process is shown to be elastic, and a model describing tunneling in terms of the dot energy relative to the height of the tunnel barrier quantitatively describes the measurements.

Figure 1

(a) Electron micrograph of the gate geometry and schematic of the measurement circuit. Unlabeled gates are grounded. We measure the resistance of the QPC by sourcing a current through the QPC and monitoring the voltage $V_{\mathrm{qpc}}$ across the QPC. We have verified that the finite bandwidth of our measurement [24] does not significantly affect the results presented in this work using computer simulations. (b) When a voltage bias $V_{\mathrm{ds}}$ is applied across the quantum dot a small current flows and the charge on the dot fluctuates between 0 and 1 as shown in (c). We measure the time intervals $t_{\mathrm{on}}$ ($t_{\mathrm{off}}$) that the electron is on (off) the dot using an automated triggering and acquisition system described in Ref. 25. The offset in the trace is caused by the ac coupling of the voltage preamplifier. (d) Histogram of $t_{\mathrm{on}}$ times from data such as in (c). Fitting this histogram yields ${\Gamma }_{\mathrm{off}}$ as described in the text.

Figure 2

(a) ${\Gamma }_{\mathrm{on}}$ and ${\Gamma }_{\mathrm{off}}$ as a function of $V_{\mathrm{ds}}$ for large negative $V_{\mathrm{ds}}$. The solid line in the upper panel is based on a theoretical fit to the data discussed in the text. (b) Differential conductance vs $V_{\mathrm{ds}}$ and $V_g$, showing the 0 to 1 electron transition. The tunnel rates for this case are made large enough so that the differential conductance can be measured using standard transport techniques. The data shown in (a) are taken at the position of the dashed line.

Figure 3

${\Gamma }_{\mathrm{on}}$ (■) and ${\Gamma }_{\mathrm{off}}$ (○) as a function of $V_{\mathrm{ds}}$. The solid and dashed curves are calculations described in the text. The step features near $V_{\mathrm{ds}}=0$ result from the Fermi distribution.

Figure 4

(a) ${\Gamma }_{\mathrm{on}}$ and ${\Gamma }_{\mathrm{off}}$ as a function of $\Delta V_g$. Closed (open) circles are ${\Gamma }_{\mathrm{off}}$ (${\Gamma }_{\mathrm{on}}$) measured by observing spontaneous hopping caused by thermal broadening in the leads as depicted in (b). Closed (open) triangles are ${\Gamma }_{\mathrm{off}}$ (${\Gamma }_{\mathrm{on}}$) measured using a pulsed-gate technique. Solid lines are calculations as described in the text. (Inset) $p_{\mathrm{on}}$ (triangles) and $p_{\mathrm{off}}$ (circles) compared to $f(E,\eta )$ and $1-f(E,\eta )$ (solid lines), respectively, as described in the text. (b) Sketch of spontaneous hopping caused by thermal broadening in the leads. (c) Pulsed technique used to measure ${\Gamma }_{\mathrm{off}}$. The top panel shows the pulsed modulation of the one-electron state energy $E$. The bottom panel shows a sample time trace. The dashed vertical lines indicate when the gate pulse begins and ends: the QPC responds to the gate pulse because of direct capacitive coupling to LP2. When the electron energy level is brought near $E_F$ an electron tunnels on the device (indicated by a 1). When the electron level is brought back above $E_F$ the electron tunnels off the device (indicated by a 0).