Contents I. Introduction
In the last years, physicists and engineers focused on the development of new materials, devices, and technologies for quantum nanoelectronics [1], [2]. Future quantum architectures will likely require the readout, manipulation, and interaction of several ON-chip quantum bits, nanoelectronic devices, and control systems [3], [4]. As achieved in today’s integrated circuits, it is reasonable to assume that future intra- and interconnections in quantum architectures could be achieved by lithographically defined traces, removing the limit to directly address each individual quantum component from the outside. However, the current state of the art on quantum electronics requires the direct readout and manipulation of any aspect of the proposed device at cryogenic temperature, typically controlled by external measurement setups at room temperature [5], [6]. Ad hoc interface printed circuit boards (PCBs) provided with DC and RF connectors are today the standard interface between cryogenic cabling and quantum chips. Applications are reported in cryogenic filtering, microwave sample holders, and bias tee PCBs [7], [8]. Increasing the complexity of quantum chips, more sophisticated interface PCBs with a higher number of electrical connections and components are required [5]. In this framework, low-loss and low-temperature-dependent-dielectric-permittivity materials, such as those produced by the Roger Corporation (e.g., Rogers1 4003C and Rogers 4350B), have been in a leading position in the developing of microwave and millimeter-wave ad hoc cryogenic PCBs, compared to more standard substrates as the Flame Resistance n.4 (FR4) [7], [8]. Yet, manufacturing PCB prototypes with FR4 could be 10-to-50 times cheaper than the Rogers counterpart. It is known that FR4 dielectric properties are temperature dependent and are not usually provided in the cryogenic temperature range by the manufacturers [9]. Consequently, it is worth characterizing commercial FR4 laminates to estimate their dielectric properties to improve the PCB prototyping process at cryogenic temperatures. Transmission–reflection and resonance methods are generally employed to retrieve the complex permittivity of PCB laminates [10], [11], [12]. Transmission-reflection methods permit the measurement in a wider frequency range. On the other hand, resonance methods use a single frequency (or a set of frequencies), allowing the highest available accuracy for the estimation of real and imaginary permittivity. Among resonance methods, resonant cavities operate with modes resonating between metallic walls, where wall losses must be considered [10], [13]. On the other hand, dielectric resonators operate with modes resonating within the sample substrate with minimal impact of metallic losses [10]. For both the resonance methods, the material under test characterization could be performed using substrate test methods, which require no or little substrate processing. Examples are, but not limited to, Hakki–Coleman dielectric resonators [14], [15], single post dielectric resonators [16], split post dielectric resonators [16], [17], [18], [19], clamped stripline resonators [11], [20], and others [21], [22], [23]. Moreover, microwave resonant circuits can be directly manufactured on the laminate under test and used to investigate its dielectric properties, which removes limitation induced by air-gap trapping, exact estimation of the laminate thickness, and specific laminate sizes [11], [24], [25], [26]. Table S1 summarizes the resonant methods involved in the complex permittivity estimation.
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