Single-shot measurement and tunnel-rate spectroscopy of a Si/SiGe few-electron quantum dot

We investigate the tunnel rates and energies of excited states of small numbers of electrons in a quantum dot fabricated in a Si/SiGe heterostructure. Tunnel rates for loading and unloading electrons are found to be strongly energy dependent, and they vary significantly between different excited states. We show that this phenomenon enables charge sensing measurements of the average electron occupation that are analogous to Coulomb diamonds. Excited-state energies can be read directly from the plot, and we develop a rate model that enables a quantitative understanding of the relative sizes of different electron tunnel rates.

Figure 1

(a) Scanning electron micrograph of the quantum dot top gates, with gate labels shown. For pulsed-gate measurements, a pulse sequence was applied to gate $R$. (b) Chemical potential diagrams for tunneling between the left reservoir and the quantum dot. Loading (top diagram) and unloading (bottom diagram) are achieved by varying the gate voltages $V_M$ or $V_R$. (c) Typical oscilloscope traces of $I_{\mathrm{QPC}}$ versus time, showing real-time tunneling events. Downward edges indicate loading, upward edges indicate unloading. The three traces correspond to the three energy configurations shown in (b). (d) Fractional dot occupation as a function of $V_M$, obtained from traces like those in (c).

Figure 2

Pulsed gate voltage measurements. (a) A typical measurement of $I_{\mathrm{QPC}}$ (red curve), together with its contribution from cross talk (dashed black curve). The voltage pulse sequence applied to gate $R$ is shown as the solid black curve (arb. units). Insets: chemical potential diagrams for loading and unloading. (b) Cumulative counting statistics for loading events vs time with $V_{\mathrm{pp}}=8\hspace{0.5em}\mathrm{mV}$. (c) Average of 600 current traces (blue curve) and an exponential fit (red curve). (d) Average of 512 current traces (red & blue curves), for two different values of $V_{\mathrm{pp}}$ (the corresponding pulse sequence is shown in black with arbitrary units). (e) Loading time vs pulse amplitude, showing a strong increase in tunneling time as the quantum dot chemical potential is decreased.

Figure 3

(a) Coulomb diamond plot of current $I_{\mathrm{dot}}$ through the quantum dot in the many-electron regime. (b) Schematic representation of two Coulomb diamonds. (c) Energy-level diagrams showing the dot chemical potential and the Fermi level of the leads for two of the edges in (b). (d), (e) Expected transconductance (solid lines) of the charge sensing QPC when the $D$ and $S$ tunnel barriers, respectively, are the transport bottleneck. (f), (g) Color scale plot of the transconductance $g$ of the QPC obtained as a function of $V_M$ and $V_{\mathrm{SD}}$. Peaks in $g$ represent transitions where dot occupation changes. $V_R=-0.42\hspace{0.5em}$ and $-0.44\hspace{0.5em}\mathrm{V}$ in (f) and (g), respectively. $V_M$ differs in the two panels to compensate for the difference in $V_R$.

Figure 4

Excited-state spectroscopy. (a) Color-scale plot of $g$ as a function of $V_{\mathrm{SD}}$ and $V_M$. The pattern in the transconductance is analogous to a traditional Coulomb diamond plot of the differential conductance through a quantum dot. (b) Schematic representation of the “transition map” observed in (a), showing the role of excited states and how the energies of these states can be extracted in analogy with a Coulomb diamond plot. (c) Theoretical reconstruction $\partial f/\partial V_M$ of the transition map, based on fitting to a rate equation model described in the Appendix, demonstrating the robustness of the interpretation of the data. (d) Schematic energy-level diagrams of the dot at points A, B, and C in panel (b).

Figure 5

(a) Ground ($g$) and excited ($x,y$) state transitions in a quantum dot, between the electron occupations $(N-1)$ and $N$. The chemical potential describing transitions between the ground states is ${\mu }_g$. Transitions to excited states may occur in either direction when the dot loads (${\mu }_x$) or unloads an electron (${\mu }_y$). (b) The types of tunneling processes considered in this work. Fast decay of excited states essentially prohibits $x$-type unloading or $y$-type loading processes.

Figure 6

Cases considered for discussing the quantum dot filling, Eq. (A3).