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American Journal of Physics and Applications

Average Current Through a Single-electron Transistor Under Fluctuations of an Observer’s Frame of Reference

Yun‐Sok Shin

Published Jan 1, 2019• 2 citations
Publisher

Abstract

The average current through a single-electron transistor (SET) under fluctuations of an observer’s frame of reference (OFR) is reported. To date, the average current through a SET has been studied under the assumption that an OFR remains constant throughout the performance of measurements of the current; thus, it remains an unsolved problem as to what is measured of the current when the OFR is assumed to fluctuate. In this paper, a SET comprising a source, drain, and single channel is considered, where an OFR is assumed to be matched to the electrochemical potential energy of the drain of the SET. The average current through the SET for two types of OFR fluctuation is formulated: periodic-square-wave fluctuation and periodic-sawtooth-wave fluctuation, in time representations. Under these types of fluctuation, the average current exhibits a zero-bias Coulomb peak—the amplitude of which gradually increases with the amplitude of the fluctuation type divided by temperature. The amplitude of the zero-bias Coulomb peak is greater in the case of periodic-square-wave fluctuations. Therefore, the amplitude of the zero-bias Coulomb peak together with a varying of both the energy of the channel and the temperature has the potential to reveal the distribution of fluctuations of an OFR.

DOI
10.11648/j.ajpa.20190705.14

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The average current through a single-electron transistor (SET) under fluctuations of an observer’s frame of reference (OFR) is reported. To date, the average current through a SET has been studied under the assumption that an OFR remains constant throughout the performance of measurements of the current; thus, it remains an unsolved problem as to what is measured of the current when the OFR is assumed to fluctuate. In this paper, a SET comprising a source, drain, and single channel is considered, where an OFR is assumed to be matched to the electrochemical potential energy of the drain of the SET. The average current through the SET for two types of OFR fluctuation is formulated: periodic-square-wave fluctuation and periodic-sawtooth-wave fluctuation, in time representations. Under these types of fluctuation, the average current exhibits a zero-bias Coulomb peak—the amplitude of which gradually increases with the amplitude of the fluctuation type divided by temperature. The amplitude of the zero-bias Coulomb peak is greater in the case of periodic-square-wave fluctuations. Therefore, the amplitude of the zero-bias Coulomb peak together with a varying of both the energy of the channel and the temperature has the potential to reveal the distribution of fluctuations of an OFR.
This study reports tunneling through a one-dimensional (1D) square potential barrier (SPB) under fluctuations in an observer’s frame of reference (OFR). To date, tunneling through an SPB has been studied under the assumption that the OFR remains constant throughout the tunneling measurements; therefore, the change of the tunneling probability when the OFR is assumed to fluctuate remains unanswered. In this paper, a 1D SPB is considered under fluctuations of an OFR. The average transmission probability of a particle through an SBP for two types of OFR fluctuations (periodic-square-wave and periodic-sawtooth-wave fluctuations) is formulated in time representations. Under these types of fluctuations, the average transmission probability gradually increases with a particle’s energy, which is saturated to the transmission probability in the case of the stationary OFR at a much greater energy than the amplitude of the fluctuations. The average transmission probability is much higher at the amplitude of the fluctuations in the case of periodic-square-wave fluctuations. Therefore, the average transmission probability with a particle’s energy has the potential to reveal the distribution of OFR fluctuations.
We calculate the full-frequency spectral density of voltage fluctuations in a single-electron transistor (SET), used as an electrometer biased above the Coulomb threshold so that the current through the SET is carried by sequential tunneling events. We consider both a normal-state SET and a superconducting SET. The whole spectrum, from low-frequency telegraph noise to quantum noise at frequencies comparable to the SET charging energy ${(E}_{C}/\ensuremath{\Elzxh})$ to high-frequency Nyquist noise, is described. We take the energy exchange between the SET and the measured system into account using a real-time diagrammatic Keldysh technique. The voltage fluctuations determine the backaction of the SET on the measured system, and we specifically discuss the case of superconducting charge qubit read-out and measuring the so-called Coulomb staircase of a single Cooper-pair box.
We report an experimental observation of the backaction of a single-electron transistor (SET) measuring the Coulomb staircase of a single-electron box. As current flows through the SET, the charge state of the SET island fluctuates. These fluctuations capacitively couple to the box and cause changes in the position, width, and asymmetry of the Coulomb staircase. A sequential tunneling model accurately recreates these effects, confirming this mechanism of the backaction of a SET. This is a first step toward understanding the effects of quantum measurement on solid-state qubits.
A Single-Electron Transistor (SET) is a highly sensitive charge amplifier based on the Coulomb blockade that can operate near the quantum limit: amplification can be accomplished with a back-action close to that required by the uncertainty principle. Its charge sensitivity is ultimately limited to about 1micre/sqrt(Hz) by the shot noise in the source-drain current. Despite the fact that such sensitivity has been approached by some, the shot-noise limit has yet to be reached for either the SET and its RF version, the RF-SET. A scheme is proposed and realized in this thesis for the shot noise-limited detection of a Single-Electron Transistor, based on a DC-SQUID impedance-matched with a DC-biased SET. SQUIDs can operate at the quantum limit, and a microstrip SQUID amplifier has already been demonstrated to operate at almost twice the quantum-limited noise temperature at about 500MHz by M. Muck and collaborators, with a sensitivity sufficient to resolve the SET shot noise. The device has been fabricated and characterized. A 50KOhm resistor has been used as a noise source at low temperature and the measured noise levels have been estimated. Moreover, the detection noise temperature of about 110mK was derived corresponding to a current sensitivity of about 8fA/sqrt(Hz) for the detection of a 100KOhm resistor.
We report an experimental observation of the backaction of a Single Electron Transistor (SET) measuring the Coulomb staircase of a single electron box. As current flows through the SET, the charge state of the SET island fluctuates. These fluctuations capacitively couple to the box and cause changes in the position, width, and asymmetry of the Coulomb staircase. A sequential tunnelling model accurately recreates these effects, confirming this mechanism of the backaction of an SET. This is a first step towards understanding the effects of quantum measurement on solid state qubits.