Figure 1

Single-electron transistor. (a) Scanning electron microscope image showing the SET design. The area of each junction is approximately $60\times 30{\mathrm{nm}}^2$. The island to the far right is a result of the two-angle evaporation and plays no role in the SET operation. The gate electrode is located outside the picture, $600\mathrm{nm}$ to the right of the island. (b) $I$-$V$ characteristics of device S1, taken at four different gate voltages. Inset: Typical measured $I_{\mathrm{SET}}$ vs $Q_g$, the charge induced from the gate. The colored points correspond to those in the $I$-$V$ plot. The cross denotes the operating point used in Exps. A and B. (c) Solid blue curve: Typical noise power spectrum $S_Q\left(f\right)$ acquired at $T_0=50\mathrm{mK}$. The slope of this spectrum is $\alpha =1.24\pm 0.01$. The pilot signal, used for gain calibration, appears as a strong peak at $f_p=377\mathrm{Hz}$. The red dot marks ${\tilde{S}}_Q$, the average of $S_Q$ between $383\mathrm{Hz}$ and $401\mathrm{Hz}$. The dashed green line is the shot noise of the SET for this particular value of $I_{\mathrm{SET}}$. The solid black curve shows the amplifier noise, measured with open input, scaled to units of charge by the charge gain of the SET. The dashed red curve is an example of a spectrum (at $T_0=1.5\mathrm{K}$) with a substantial contribution from a single TLF at intermediate frequencies (see text). (d) Solid lines: Maximum and minimum SET current as a function of bias voltage. Blue square, red cross, and green triangle: Bias points used for the temperature sweep of Exp. B. At fixed $V_b$, the gate voltage $V_g$ was varied to adjust $I_{\mathrm{SET}}$ to the midpoint between the maximum and minimum SET current [panel (b) inset]. Dots: Measurement points where the charge noise was measured (at base temperature) in Exp. C. The bias voltage was fixed, and $I_{\mathrm{SET}}$ subsequently adjusted to the desired value by varying $V_g$.Reuse & Permissions

Figure 2

Different models for two-level fluctuators. (a) The TLFs that produce charge noise are usually modeled as charged particles, each moving stochastically between two potential wells in the dielectrics surrounding the SET. With symmetrical wells, and a single activation energy $E_{\mathrm{TLF}}$ which the particle must overcome by thermal excitation, the model predicts (Ref. 9) $S_Q\propto T/f$. (b) With two defining energy scales ($E_1$ and $E_2$), the model instead predicts (Ref. 20) $S_Q\propto T^2/f$. (c) The TLF model suggested by our experimental data consists of a single potential well which may be occupied (unoccupied) by an electron tunneling from (to) the SET electrodes. There should be approximately as many wells coupled to the leads as to the island, and both types of well contribute equally to the charge noise, assuming uniform temperature. $E_{\mathrm{TLF}}$ is the depth of each TLF with respect to the Fermi energy $E_F$ of the metal to which it is tunnel coupled. This model predicts a $S_Q\propto T/f^{\alpha }$ dependence (see text). (d) Only TLFs located inside the junctions or within a few barrier thicknesses of them can change the charge induced on the SET island sufficiently to contribute to the observed noise.Reuse & Permissions

Figure 3

Charge noise ${\tilde{S}}_Q$ as a function of refrigerator temperature $T_0$. (a) ${\tilde{S}}_Q$  for two SETs at fixed bias voltage (Exp. A): S1 (lower, blue dots) and S2 (upper, red dots). The solid black line is a guide to the eye, illustrating linear temperature dependence. The blue and red lines are linear fits in the logarithmic plot for temperatures above $0.5\mathrm{K}$, for S1 and S2, respectively. (b) ${\tilde{S}}_Q$ for S1, with bias voltage $V_b$ alternating between three different values (Exp. B). The solid line is a linear fit through zero to the noise for $T_0>1\mathrm{K}$; see text. Symbols correspond to those in Figs. 1d and 4c.Reuse & Permissions

Figure 4

Noise and temperature vs SET bias for S1, at refrigerator base temperature (Exp. C). (a) Charge noise ${\tilde{S}}_Q$ vs $V_b$. Points grouped by color and connected with lines were acquired with the same nominal value of $I_{\mathrm{SET}}$. Inset: Data for the lowest four values of $I_{\mathrm{SET}}$ with linear fits. For low values of $V_b$ (traces with the lowest ${\tilde{S}}_Q$), there is a clear increase in noise level with bias voltage. (b) The same ${\tilde{S}}_Q$ data as in (a) plotted versus power dissipated in the SET, $P_{\mathrm{SET}}=V_b{I}_{\mathrm{SET}}/2$. (c) Extracted TLF temperature $T_{\mathrm{TLF}}$ (points) versus $P_{\mathrm{SET}}$. The central line (dashed green) is fitted to the data of the logarithmic plot, yielding $T_{\mathrm{TLF}}\propto P_{\mathrm{SET}}^{0.29\pm 0.01}$. The square, cross, and triangle are the noise saturation levels for the three different bias points of Exp. B. The upper dashed line (black) is the calculated electron temperature of the SET island, and the black cross is the electron temperature extracted from the modulation curve of the SET [see text and Fig. 5b]. The lower solid line (red) is the calculated phonon temperature beneath the SET (see text).Reuse & Permissions

Figure 5

Extraction of the electron temperature from the $I$-$V$ characteristics of S1. (a) Maximum SET current versus refrigerator temperature $T_0$ for fixed $V_b$. The black curve represents experimental data and the red curve is a fit to the standard (orthodox) theory of the SET, for the regime $T_0>2\mathrm{K}$ where this model is accurate. The bias voltage $V_b$ and the charging energy $E_C$ were used as fitting parameters. (b) Minimum SET current versus refrigerator temperature for the values of $V_b$ and $E_C$ found in (a). The solid black curve shows experimental data, and the red curve is the theoretical (orthodox) calculation, with no fitted parameters. The experimental curve saturates at a value $I_{\mathrm{sat}}$ which is higher than the theoretical one, as a result of SET self-heating. The blue curve is a theoretical calculation with cotunneling included (Ref. 37), from which we can rule out that cotunneling is the reason for the high value of $I_{\mathrm{sat}}$. The crossover between the experimental and orthodox curves is marked with a black cross, and provides an experimental value for $T_e$. This data point is also plotted in Fig. 4c.Reuse & Permissions