Radio-frequency single electron transistors in physically defined silicon quantum dots with a sensitive phase response

Radio-frequency reflectometry techniques are instrumental for spin qubit readout in semiconductor quantum dots. However, a large phase response is difficult to achieve in practice. In this work, we report radio-frequency single electron transistors using physically defined quantum dots in silicon-on-insulator. We study quantum dots which do not have the top gate structure considered to hinder radio frequency reflectometry measurements using physically defined quantum dots. Based on the model which properly takes into account the parasitic components, we precisely determine the gate-dependent device admittance. Clear Coulomb peaks are observed in the amplitude and the phase of the reflection coefficient, with a remarkably large phase signal of ∼45°. Electrical circuit analysis indicates that it can be attributed to a good impedance matching and a detuning from the resonance frequency. We anticipate that our results will be useful in designing and simulating reflectometry circuits to optimize qubit readout sensitivity and speed.

Frequency dependence and fitting. First, we study the frequency dependence of the reflection coefficient at the device, Ŵ , without carrier accumulation using the back gate voltage V BG = 0 V (blue solid lines in Fig. 2a,b). Here, the charge sensor and all SGs are set to be floating for simplicity. A dip appears at a frequency which corresponds to the resonance of the circuit ( f r = 224.063 MHz). In a simple description of the LC resonance circuit, the load impedance, Z load , is proportional to device conductance at resonance; this would suggest that Z load should be close to zero in this case, which would result in almost complete reflection. This apparent discrepancy can be explained by the dielectric loss in the PCB which contributes as a conductance parallel to the device 12 . Given this, Z load has a finite value (at frequencies around the resonance) and Ŵ = (Z load − Z 0 )/(Z load + Z 0 ) can approach zero, where Z 0 = 50 Ω is the characteristic impedance of the external signal line.
As we will see below, the frequency dependence of observed reflection can be described by the equivalent circuit shown in Fig. 2c. The circuit mainly consists of an inductance, L , a parasitic capacitance, C p , and a QD impedance composed of a parallel circuit of a conductance G QD and a capacitance C QD ; however, each component has additional parasitic components in reality. Here, we take into account the parasitic capacitance and resistance Shadowed areas indicate unused floating electrodes: a side gate, a single QD charge sensor, and a reservoir. An LC resonance circuit for impedance matching, which comprises a surface mount wire-wound inductor with an inductance L = 680 nH and parasitic capacitance C p , is connected to the source of the QD. In addition, an RC bias tee (R b = 5 kΩ and C b = 4.7 μF) is connected to the LC circuit in series to apply a DC voltage V S . For RF measurement, a vector network analyzer (VNA) outputs the RF signal from port 1, which is applied to the device after attenuation to suppress heating up the QD. The signal reflected from the device is branched by directional coupler and input to port 2 of VNA after amplification. www.nature.com/scientificreports/ in the coil, C L and R L , and the parasitic conductance in the capacitor, G p . G p and R L are due to dielectric loss and skin effect, respectively, so that each has a frequency dependence: G p = ωC p tanδ and R L = ρ L √ ω , where ω = 2 πf is the angular frequency, tanδ is loss tangent, and ρ L denotes a coefficient 29 . In addition, the effects of external components are considered, such as coaxial cables, attenuators, and amplifiers. Attenuation and amplification offset the amplitude of the reflected signal, and the coaxial cable causes a linear phase shift as a function of frequency due to its propagation constant. Figure 2b,c show fitting results for the amplitude and the phase of the observed reflection at V BG = 0 V based on the equivalent circuit with following parameters (red dotted lines): L = 680 nH, ρ L = 190 µ�/ rad/s , C L = 348 fF, C p = 394 fF, and tanδ = 0.00614. Here, to reduce the number of fitting parameters, the nominal value of L is assumed, and ρ L is separately estimated from the measured transmission characteristic of an inductor nominally identical to the one used in these measurements (supporting information). The fittings have good agreements with the measurement results, proving the validity of the equivalent circuit model. The small deviation at offresonance frequencies can be attributed to a background frequency dependence due to interference between the reflected signal and the isolation leakage in the directional coupler 29 . The necessity for additional parasitic components can be confirmed from fitting with a simple LCR circuit without taking into account additional parasitic components, where an inductance 3.5 times larger than L is required for a good fitting (see supporting information for details).
Back gate voltage dependence. Next, we investigate the reflection dependence on back gate voltage, V BG , at a constant frequency, f = 223.464 MHz which corresponds to the resonance frequency at V BG = 5.4 V. Figure 3a,b show the DC QD current, I QD , and the amplitude of Ŵ , |Ŵ| , as a function of V BG . Hereafter, the amplitude and the phase of Ŵ as a function of a gate voltage are corrected in a similar manner described above by taking into account the external components. The QD device can also have parasitic components such as gate capacitance, C 2D , and its dielectric loss, G 2D . We derive the device admittance, Y dev = G dev + iωC dev , where G dev = G QD + G 2D and C dev = C QD + C 2D , from the V BG dependence by subtracting impedances of inductance and parasitic capacitance together with their additional parasitic components (Fig. 3c,d). As seen in Fig. 3c, C dev qualitatively agrees with a result in standard CV measurements of MOS capacitors 13 , which is reasonable because the only difference is the direction of signal: from gate to semiconductor (CV measurement of MOS capacitors) or from semiconductor to gate (RF reflectometry of QDs). The peak in G dev around V BG = 2.5 V is also similar to the one for CV measurements which can be explained by the effect of dielectric loss related to oxide (Fig. 3d) [30][31][32] . In addition to being useful for establishing a detailed device model, understanding this in-situ tunability can potentially provide an on-chip implementation of gate-tunable reflectometry circuits previously achieved via external components such as varactors [33][34][35] . At higher voltages, oscillations appear in |Ŵ| , corresponding to Coulomb peaks in I QD , which implies the successful realization of RF-SET. www.nature.com/scientificreports/ Side gate voltage dependence. We can expect the conductance sensitivity to further increase, when a SG voltage is swept instead of a back gate to suppress these changes in the device admittance related to the turn-on process. To perform such an experiment, we use another QD device with a nominally identical design and set the charge sensor and one of the side gates to be floating to avoid unintended RF paths. The measured RF reflectometry signals as a function of a SG voltage, V GR1 are shown in Fig. 4a,b. We find that peaks (dips) in amplitude (phase) of Ŵ nicely reproduce the Coulomb peaks of I QD in Fig. 4c. Strikingly, the observed phase shift ( ∼ 45 • ) is significantly larger than those observed in other RF-SETs 14,36 .

Discussion
To understand the origin of the large phase shifts, we simulate the V GR1 dependence of the reflected signal. In Fig. 4d, the simulation results are plotted in a Smith chart (orange dotted circle and purple dashed circles) together with the V GR1 dependence (blue solid line). For the one shown by the purple dashed circle, G QD is set to be higher than for the other one represented by the orange dotted circle by 3.7 μS, in order to approximately simulate the Coulomb peak conductance, with the other parameters obtained by fitting the frequency dependence. Both simulation results show constant resistance circles as expected from the equivalent circuit in Fig. 2c.
Remarkably, the orange one passes through almost the center of the Smith chart, meaning that the load impedance at resonance is closely matched to Z 0 . The results at given frequencies are indicated by the red triangles (223.464 MHz) and the green dots (224.157 MHz) in the Smith chart: the red ones are for the frequency used in the measurement (223.464 MHz) and the green ones for the resonance frequency for the orange circle condition (224.157 MHz). As expected, the red triangles agree with the V GR1 dependence experimentally observed. We also calculate the frequency dependence of the phase difference �ϕ expected for the same (3.7 μS) conductance change in Fig. 4e, with a red triangle and a green dot highlighting the same two frequencies as in Fig. 4d. It turns out that there are two maximums in the absolute value of �ϕ . We note that the resonance frequency (marked by the green dot) is located in between the two maximum points and has a small phase shift ( ∼ 2 • ). On the other hand, the frequency used in the measurement (the red triangle) is located close to one of the maximums, and the large phase shift ( ∼ 45 • ) agrees excellently with data. This large phase shift will not occur if the load impedance at resonance is away from impedance matching, where larger or smaller constant resistance circles would appear. Therefore, we conclude that a good impedance matching and a small frequency detuning from the resonance frequency are the necessary ingredients for the large phase shift caused by Coulomb oscillations. We note that this large phase shift is caused by a conductance change, rather than a change in quantum or tunnel capacitances, for which much smaller phase shifts are typically reported (e.g. Refs. 14,36 ). Observation of reflectometry phase shift due to a conductance change is scarce. However, in theory, the phase change can be as large as 180° when the system passes exactly across the matching condition. To the best of our knowledge, our result is one of the closest to this ideal situation in the literature. www.nature.com/scientificreports/

Conclusion
In conclusion, we have fabricated PD-QDs in SOI without the top gate structure and performed RF-SET measurements at liquid helium temperature. Based on an equivalent circuit for load impedance, the parasitic circuit parameters are estimated from the frequency dependence of RF reflection coefficient. Our method will be useful when one needs to modify the RF reflectometry technique to apply to new device structures. Furthermore, we find huge phase shifts corresponding to Coulomb peaks ( ∼ 45 • ), as a result of the combination of a good impedance matching and a detuning from resonance frequency. This is confirmed by simulation using the equivalent circuit model. We believe that our results will be helpful in designing reflectometry circuits through simulation in order to improve the sensitivity and speed of spin qubit readout.

Data availability
The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.  www.nature.com/scientificreports/ Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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