Nonequilibrium Probing of Two-Level Charge Fluctuators Using the Step Response of a Single-Electron Transistor

A. Pourkabirian, M. V. Gustafsson, G. Johansson, J. Clarke, and P. Delsing

Phys. Rev. Lett. 113, 256801 – Published 19 December 2014

Abstract

We report a new method to study two-level fluctuators (TLFs) by measuring the offset charge induced after applying a sudden step voltage to the gate electrode of a single-electron transistor. The offset charge is measured for more than 20 h for samples made on three different substrates. We find that the offset charge drift follows a logarithmic increase over 4 orders of magnitude in time and that the logarithmic slope increases linearly with the step voltage. The charge drift is independent of temperature, ruling out thermally activated TLFs and demonstrating that the charge fluctuations involve tunneling. These observations are in agreement with expectations for an ensemble of TLFs driven out of equilibrium. From our model, we extract the density of TLFs assuming either a volume density or a surface density.

Received 27 August 2014

DOI:https://doi.org/10.1103/PhysRevLett.113.256801

Authors & Affiliations

A. Pourkabirian¹, M. V. Gustafsson¹, G. Johansson¹, J. Clarke^1,2, and P. Delsing¹

¹Microtechnology and Nanoscience, Chalmers University of Technology, S-41296 Göteborg, Sweden
²Department of Physics, University of California, Berkeley, California 94720, USA

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Vol. 113, Iss. 25 — 19 December 2014

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Images

Figure 1
(a) Micrograph of a SET with typical junction size of $20 \times 50 {nm}^{2}$ . [(b),(c)] Schematic overview of the measurements. Starting from equilibrium (green), we apply a sudden voltage step $Δ V$ to the gate, bringing the TLF ensemble out of equilibrium (red). After the step, we use the SET to record the charge drift $Q (t)$ as the TLFs relax to their new ground states (blue).
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Figure 2
Step response measurements at $T = 30 mK$ on device 1. (a) Continuous measurement of $I_{SET}$ over a period of about 20 h. A step voltage $Δ V = 9.8 V$ was applied to the gate at $t = 0$ (red dashed lines). The inset shows the first 500 s after the step. (b) Charge drift extracted from (a). The inset shows the same data on a logarithmic time axis; the charge increases logarithmically with time.
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Figure 3
Normalized charge drift measured for four different devices. The curves have been offset vertically for clarity. For device 1, the charge drift was extracted from the current modulation of the SET. For devices 2–4, a PID loop was connected to the gate (see text). The measurements on different SETs on different substrates show a logarithmic charge drift with similar slopes. The charge drift per decade of time in Table 1 is extracted from a least-squares fit to each trace (solid black lines).
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Figure 4
Charge drift for different voltage step heights $Δ V$ for device 3 at 20 mK. The logarithmic slope of the charge drift $Δ Q (t) / Δ \log t$ increases with $Δ V$ . The inset shows $Δ Q (t) / Δ \log t$ vs $Δ V$ , and the line represents a least-squares fit constrained to pass through the origin.
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Figure 5
Microscopic model of a TLF and its influence on the SET. [(a),(b)] A charged particle in a double potential well with a distance $d$ between the two wells, barrier height $E_{b}$ , and energy difference $Δ E$ between the two states. Left: before applying the step, the switching time is $τ$ . Right: after applying the step, the energy difference between the two wells changes by $δ E$ , and the switching time changes to $τ^{'}$ . $E_{b}$ is defined with respect to the mean between the two states and does not change to first order. (a) Thermal activation. (b) Quantum tunneling. (c) Simplified geometry of the SET, gate, and TLFs. At the center, an individual TLF is shown schematically with an angle $θ$ between its displacement vector $\vec{d}$ and the gate field ${\vec{E}}_{G} (\vec{r})$ (solid lines). The dashed lines show the virtual field, ${\vec{E}}_{V} (\vec{r})$ (see text). The red and black circles represent the volume and surface distributions of TLFs, respectively.
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