Background

Semiconductor-based qubits that utilize the spin of individual electrons or holes (also known as spin qubits) are among the most fascinating qubit platforms, as they are, for example, compatible with existing complementary metal–oxide–semiconductor (CMOS) technology [1]. In recent years, germanium (Ge)-based spin qubits have gained prominence in research, as they effectively circumvent many challenges faced by other material systems, such as the valley states in silicon or the nuclear spins in gallium arsenide.

A common approach to read out the qubit state is radio-frequency (RF) reflectometry, which aims to increase bandwidth and reduce readout duration. Here, an LC resonator on a printed circuit board (PCB), consisting of classical surface-mount device (SMD) components or superconducting resonators, is coupled to an ohmic contact of the quantum dot spin qubit device. An RF carrier signal is fed into the circuit. A change in the chemical potential of the quantum dot alters the reflected signal, enabling readout. A major challenge is achieving impedance matching between the quantum device and a characteristic 50 Ohm transmission line that feeds the signal. Therefore, the LC resonator is equipped with a so-called varactor, placed prior to the coil, which acts as a dedicated gate-tunable matching capacitor to enable in situ impedance matching [2].

These varactors can be made of a thin metal ring structure on a quantum paraelectric substrate such as strontium titanate [3]. It is crucial to understand the impedance profile of this device as a function of the applied gate bias. This impedance can be measured with a cryogenic impedance bridge (see Fig. 1), in which two sine waves, one from the device under test and one from a known reference resistor, interfere destructively, resulting in a minimum in the output voltage. Since the resulting signal will be very small, it is amplified using a packaged high-electron-mobility transistor (HEMT) [4]. Finding the point of destructive interference, the so-called bridge point, involves optimizing and analyzing a broad parameter space. From the bridge point, we can then precisely determine the impedance and correspondingly improve qubit readout times.