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Admittance and noise detection in mesoscopic systems via GHz impedance matching

V. Ranjan

Published Jan 1, 2017• 0 citations
Publisher

Abstract

Fast measurement of small currents and even smaller fluctuations in devices with large impedance is not trivial. Standard low frequency lock-in measurements suffer from stray capacitance of the order of nF reducing the bandwidth to merely kHz and a time resolution of ms. One approach of increasing the bandwidth is to connect the high-impedance device next to a low input impedance amplifier, though, the signal is decreased by the ratio of the impedances of the device and the amplifier, easily a factor of a thousand. Therefore, long averaging times are required to detect the small signal. Moreover, 1/f noise still affects the measurements. The issues of small bandwidth, impedance mismatch and 1/f noise can be simultaneously addressed using resonant LC circuits. The success of this impedance transformer lies in replacing the slow operation of applying voltages and measuring currents by applying fast radio frequency signals and measuring the reflectance. LC circuits remain appealing because of a rather simple assembly. Nonetheless, the challenge to reproducibly achieve the resonance frequencies in gigahertz (GHz) range while matching to high impedance still remains due to the large parallel stray capacitance of the measurement setup. Resonance frequency in GHz range is desirable for two reasons. Firstly, measurements are faster and 1/f noise smaller. Yet more importantly, one can profit from the wide range of measurement techniques developed in the context of circuit quantum electrodynamics, such as FPGA based hardware and quantum limited Josephson parametric amplifiers. \n \nIn this thesis, a matching circuit namely stub tuner based on coplanar transmission lines is designed, fabricated and integrated with mesoscopic devices. Owing to the simple design, we could predict the frequency response of a stub tuner in terms of the standard circuit model (Chapter 3). We initiated certain necessary fabrication adaptations to achieve integration of niobium circuits with devices. In particular, a reliable stamping of carbon nanotubes from the growth substrate to the target substrate was accomplished, improving both the device quality and yield (Chapter 4). RF reflectometry is employed to perform admittance measurements on CNT quantum dots. We could reliably deduce all circuit and device properties using the resonance response. The measurement bandwidth is shown to be in MHz range which is three orders of magnitude larger than that for DC measurements. In particular, using the phase response of the circuit, double-dot qubit energy and dephasing rate are studied (Chapter 5). We could even get rid of conventional contacts that could degrade the device properties, and still extract quantum capacitance and dissipation in graphene p-n junctions. The residual doping is shown to affect not only the mean free path but also the Fermi velocity. We further see that at small doping, electron-electron interactions could modify the Fermi velocity and that the dispersion relation could deviate from linearity (Chapter 6). \n \nIn the second half of the scientific results, real strength of stub tuner is tested in transmission measurements. The large bandwidth and signal to noise ratio is utilized to measure extremely clean noise power densities at high speed and draw comparisons with the corresponding average current. Clear changes in the shot noise spectrum are seen when a transport channel is added or removed from the bias window of the quantum dot. We especially observe noise enhancement outside the Coulomb diamonds due to blocking states (Chapter 7). Finally, we investigate noise properties of quantum dots when leads are superconducting. Distinct charge transfers due to multiple Andreev reflections are visible in the shot noise. The main finding, however, was the thermal noise at zero bias. We observed its periodic suppression and emergence across the quantum dot resonance. In another device, shot noise due to the Kondo effect is found to be enhanced in the superconducting state when the Kondo temperature exceeds the value of the superconducting gap (Chapter 8). Simple design and reliable calibration of the GHz matching circuit in both reflectance and transmission pave a way for fast and non-invasive measurements of mesoscopic conductors. \n \n \n \n \n

DOI
10.5451/unibas-006699807

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Fast measurement of small currents and even smaller fluctuations in devices with large impedance is not trivial. Standard low frequency lock-in measurements suffer from stray capacitance of the order of nF reducing the bandwidth to merely kHz and a time resolution of ms. One approach of increasing the bandwidth is to connect the high-impedance device next to a low input impedance amplifier, though, the signal is decreased by the ratio of the impedances of the device and the amplifier, easily a factor of a thousand. Therefore, long averaging times are required to detect the small signal. Moreover, 1/f noise still affects the measurements. The issues of small bandwidth, impedance mismatch and 1/f noise can be simultaneously addressed using resonant LC circuits. The success of this impedance transformer lies in replacing the slow operation of applying voltages and measuring currents by applying fast radio frequency signals and measuring the reflectance. LC circuits remain appealing because of a rather simple assembly. Nonetheless, the challenge to reproducibly achieve the resonance frequencies in gigahertz (GHz) range while matching to high impedance still remains due to the large parallel stray capacitance of the measurement setup. Resonance frequency in GHz range is desirable for two reasons. Firstly, measurements are faster and 1/f noise smaller. Yet more importantly, one can profit from the wide range of measurement techniques developed in the context of circuit quantum electrodynamics, such as FPGA based hardware and quantum limited Josephson parametric amplifiers. \n \nIn this thesis, a matching circuit namely stub tuner based on coplanar transmission lines is designed, fabricated and integrated with mesoscopic devices. Owing to the simple design, we could predict the frequency response of a stub tuner in terms of the standard circuit model (Chapter 3). We initiated certain necessary fabrication adaptations to achieve integration of niobium circuits with devices. In particular, a reliable stamping of carbon nanotubes from the growth substrate to the target substrate was accomplished, improving both the device quality and yield (Chapter 4). RF reflectometry is employed to perform admittance measurements on CNT quantum dots. We could reliably deduce all circuit and device properties using the resonance response. The measurement bandwidth is shown to be in MHz range which is three orders of magnitude larger than that for DC measurements. In particular, using the phase response of the circuit, double-dot qubit energy and dephasing rate are studied (Chapter 5). We could even get rid of conventional contacts that could degrade the device properties, and still extract quantum capacitance and dissipation in graphene p-n junctions. The residual doping is shown to affect not only the mean free path but also the Fermi velocity. We further see that at small doping, electron-electron interactions could modify the Fermi velocity and that the dispersion relation could deviate from linearity (Chapter 6). \n \nIn the second half of the scientific results, real strength of stub tuner is tested in transmission measurements. The large bandwidth and signal to noise ratio is utilized to measure extremely clean noise power densities at high speed and draw comparisons with the corresponding average current. Clear changes in the shot noise spectrum are seen when a transport channel is added or removed from the bias window of the quantum dot. We especially observe noise enhancement outside the Coulomb diamonds due to blocking states (Chapter 7). Finally, we investigate noise properties of quantum dots when leads are superconducting. Distinct charge transfers due to multiple Andreev reflections are visible in the shot noise. The main finding, however, was the thermal noise at zero bias. We observed its periodic suppression and emergence across the quantum dot resonance. In another device, shot noise due to the Kondo effect is found to be enhanced in the superconducting state when the Kondo temperature exceeds the value of the superconducting gap (Chapter 8). Simple design and reliable calibration of the GHz matching circuit in both reflectance and transmission pave a way for fast and non-invasive measurements of mesoscopic conductors. \n \n \n \n \n
Coupling carbon nanotube devices to microwave circuits offers a significant increase in bandwidth (BW) and signal-to-noise ratio. These facilitate fast non-invasive readouts important for quantum information processing, shot noise and correlation measurements. However, creation of a device that unites a low-disorder nanotube with a low-loss microwave resonator has so far remained a challenge, due to fabrication incompatibility of one with the other. Employing a mechanical transfer method, we successfully couple a nanotube to a gigahertz superconducting matching circuit and thereby retain pristine transport characteristics such as the control over formation of, and coupling strengths between, the quantum dots. Resonance response to changes in conductance and susceptance further enables quantitative parameter extraction. The achieved near matching is a step forward promising high-BW noise correlation measurements on high impedance devices such as quantum dot circuits. Radio-frequency circuits offer fast low-noise detection of signals in carbon nanotubes, but incompatibilities in fabrication degrade the performance of the hybrid device. Here, the authors use a deterministic mechanical transfer to couple pristine nanotubes to a gigahertz superconducting matching circuit.
Quantum point contacts (QPC) are the building blocks of quantum dot qubits and semiconducting quantum electrical metrology circuits. QPCs also make highly sensitive electrical amplifiers with the potential to operate in the quantum-limited regime. Though the inherent operational bandwidth of QPCs can eclipse the THz regime, the impedance mismatch with the external circuitry limits the operation to a few kHz regimes. Lumped-element impedance-matching circuits are successful only up to a few hundreds of MHz. QPCs are characterised by a complex impedance consisting of quantized resistance, capacitance, and inductance elements. Characterising the complex admittance at higher frequencies and understanding the coupling of QPC to other circuit elements and electromagnetic environments will provide valuable insight into its sensing and backaction properties. In this work, we couple a QPC galvanically to a superconducting stub tuner impedance matching circuit realised in a coplanar waveguide architecture to enhance the operation frequency into the GHz regime and investigate the electrical amplification and complex admittance characteristics. The device, operating at ∼ 1.96GHz exhibits a conductance sensitivity of 2.92×10−5(e2/h)/Hz and a bandwidth of 13MHz. Besides, the RF reflected power unambiguously reveals the complex admittance characteristics of the QPC, shining more light on the behaviour of quantum tunnel junctions at higher operational frequencies.
Modern experiments using high-speed, high-impedance devices such as quantum dots can suffer from strong suppression of signal power, if components are not carefully impedance-matched. To overcome this problem, the authors show that a resonant impedance-matching circuit based on superconducting transmission lines can improve the signal to noise ratio by nearly three orders of magnitude. This technology is particularly important to noise studies of electron transport in quantum systems, which can provide information that is unattainable by other means.
It can be hard to find that perfect match---even, or perhaps especially, in quantum electronics. Microwave readout of highly Ohmic quantum devices, like superconducting qubits, necessitates impedance-matching circuits to maximize power transfer. $L\phantom{\rule{0}{0ex}}C$ resonant circuits enjoy two major advantages over well-established coplanar transmission-line circuits: larger bandwidths (and thus shorter readout times), and drastically smaller on-chip footprints. With this in mind, the authors develop a compact $L\phantom{\rule{0}{0ex}}C$ resonator operating at 3 GHz to bridge a load of ~15 k$\mathrm{\ensuremath{\Omega}}$ to a typical coaxial line at 50 $\mathrm{\ensuremath{\Omega}}$---match made.
Radio frequency (RF) reflectometry is a well understood measurement technique in common use by RF engineers today. This technique has been applied at low temperature to probe the impedance of nanoscale devices for over two decades. These measurements typically employ basic resonant impedance transformers to bring the device impedance close to the characteristic impedance of the measurement apparatus. In 2013 it was discovered by Colless et al. that reflectometry could be used to measure extremely small impedance changes in a gate coupled to a quantum dot to directly probe the charge state of the dot. This has simplified measurements of certain nanoscale devices including single electron boxes (SEBs) and semiconductor qubits. Despite the maturity of RF reflectometry measurements in general, researchers working at low temperature continue to make basic mistakes in their measurements. The root cause of most of these mistakes is the failure to properly calibrate the measurement apparatus. In some cases this leads researchers to unrealistic conclusions when trying to match measurements to theory. This dissertation demonstrates that a low temperature RF calibration can be very easy and straightforward using inexpensive, off the shelf SMA terminators as calibration standards. A proper calibration allows for direct quantitative comparisons between data measured on different reflectometers or between data and simulations. The consistency between measurements and simulations then allows for straightforward design and implementation of improved impedance transformers, increasing the SNR of extremely sensitive measurements, with simple data-based simulations. A new type of single electron device, the capacitively coupled RF single electron transistor (CSET) is proposed and demonstrated which can rival the traditional RF SET in sensitivity but integrates more effectively into arrays and may offer a level of resistance to electrostatic discharge damage.
This thesis focuses on implementing radio frequency (rf) reflectometry techniques for dispersive detection of charge and spin dynamics in nanoscale devices. I have investigated three aspects of rf reflectometry using state-of-the-art silicon (Si) complementary metal-oxide-semiconductor (CMOS) nanowire field effect transistors (NWFETs). First, a high-sensitivity capacitive gate-based charge sensor is developed by optimising the external matching circuit to detect capacitive changes in the high frequency resonator. A new circuit topology is used where superconducting niobium nitride (NbN) inductor is connected in parallel with a single-gate Si NWFET resulting in resonators with loaded Q-factors in the 400-800 range. For a resonator operating at 330 MHz, I have achieved a charge sensitivity of 7.7 μe/Hz\mu e/\sqrt{\text{Hz}}μe/Hz​ and, when operating at 616 MHz, I get 1.3 μe/Hz\mu e/\sqrt{\text{Hz}}μe/Hz​. This gate-based sensor can be used for fast, accurate and scalable techniques for quantum state readout in Si CMOS based quantum computing. Second, this new circuit topology for the resonator is used with a dual-gate Si NWFET. This dual-gate device geometry provides access to a double quantum dot (DQD) system in few electron regime. The spin-state of the two-electron DQD system is detected dispersively using Pauli spin blockade between joint singlet S(2,0) and triplet T−_-−​(1,1) states in a finite magnetic field BBB. The singlet-triplet relaxation time T1T_1T1​ at B=4.5B=4.5B=4.5~T is measured to be $\sim$1 ms using standard homodyne detection technique. Third, I expand the range of applications of gate-based sensing to accurate temperature measurements. I have experimentally demonstrated a primary thermometer by embedding a single-gate Si NWFET with the rf capacitive gate-based sensor. The thermometer, termed as gate-based electron thermometer (GET), relies on cyclic electron tunneling between discrete energy levels of a quantum dot and a single electron reservoir in the NWFET. I have found that the full-width-half-maximum (FWHM) of the resonator phase response depends linearly with temperature via well known physical law by using the ratio kB/ek_\text{B}/ekB​/e between the Boltzmann constant and the electron charge. The GET is also found to be magnetic field independent like other primary thermometers such as Coulomb blockade and shot noise thermometers.