Radio-Frequency Capacitive Gate-Based Charge Sensing for Semiconductor Quantum Dots

Abstract

We present a comprehensible introduction to capacitive gate-based sensing, a technique for charge state readout of gate-defined semiconductor quantum dots. The objective of this chapter is to introduce the reader to the fundamental concepts necessary to understand the technique from a theoretical perspective and to be able to apply the method experimentally. We aimed to maintain a pedagogical tone with necessary rigour to keep it accessible and self-contained. We start by introducing the fundamentals of single-electron effects in single and double quantum dots i.e. Coulomb blockade. We use the constant interaction model and the effect of quantum confinement with a focus on explaining the electrical impedance of these systems under direct current excitation. We then explain how these properties are modified under the effect of an electrical excitation at high-frequency. More particularly, we focus on the appearance of a new component to the impedance, the so-called parametric capacitance when single-electrons tunnels back and forth between stable charge states because of the effect of the drive. We show how the parametric capacitance is composed by two physically distinct terms, the quantum capacitance and the tunneling capacitance, associated to adiabatic and non-adiabatic tunneling processes, respectively. In the second part of this chapter, we introduce the experimental technique. We show how embedding the devices in a high-Q lumpedelement LC resonator can be used to probe the parametric capacitance of the system. We first present the fabrication technique of superconducting niobium nitride (NbN) spiral inductors and how these can be used in conjunction with the devices to construct high-Q resonators at radio-frequencies (rf). We show how the resonator can be characterized and optimized for optimum sensitivity of the charge state of the quantum dot system. Finally, we present an experimental demonstration of rf capacitive gate-based sensing utilizing a NbN inductor coupled to the gate of a silicon nanowire field effect transistor. Coulomb blockade manifest in these devices at