Fast detection of single-charge tunneling to a graphene quantum dot in a multi-level regime
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(a) Schematic representation of the circuitry used for rf and dc measurements. The dotted black capacitor is the total stray capacitance, and the dashed red boxes encase on-chip bias tees allowing for separation of low- and high-frequency signals. (b) Atomic force microscopy image of the graphene device studied in our experiments. Dashed black lines mark regions of graphene etched away to form a nanoconstriction at the location of the single-sided arrow and a quantum dot encircled in dashed red. The nanoconstriction charge detector is attached to the resonant circuit as a resistive element R. (c) Conductance of the nanoconstriction measured via dc current (blue trace, left axis) and rf reflectometry (red trace, right axis).
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(a) Time-dependent rf signal, 8-th order Bessel low-pass filtered at 200 kHz and sampled at 500 kHz, revealing the times and the quantum dot is occupied or unoccupied by a single negative excess charge. (b) Sketched energy diagram corresponding to the multi-level regime studied here. A quantum dot with a closely spaced energy spectrum is connected to a lead with a tunnel rate . The Fermi function of the lead is softened by a temperature T which is much larger than the level spacing but much smaller than the charging energy . The difference between the Fermi level in the lead and the chemical potential of the quantum dot is denoted by .
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(a) Electronic occupancy of the lead as a function of dot-lead energy detuning . Every blue data point is the fraction of time the dot is occupied by N − 1 (instead of N) negative charges as extracted from traces as shown in Fig. 2(a). The solid black line is a fit to a Fermi distribution function in the lead, yielding a temperature of . The dashed black lines indicate the magnitude of temperature. (b) Number of tunneling events in a 10 ms long trace. The dashed green and solid red lines are calculated rates for a single and multi-level regime, respectively, using the temperature extracted in (a) and the height of the data points. (c) Tunneling rates determined through the average dwell times in (red crosses) and out (blue circles) of the quantum dot. The solid lines are calculations using Eqs. (4) and (5). For sake of clarity, the statistical error bars are only shown for one in 20 traces. (d) Measured (blue squares) and analytically determined (Eq. (6); solid red line) sum of the multi-level tunnel rates shown in (c). The dotted green line indicates the behavior in the single-level regime.
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