Estimation of the FR4 Microwave Dielectric Properties at Cryogenic Temperature for Quantum-Chip-Interface PCBs Design

Ad hoc interface printed circuit boards (PCBs) are today the standard connection between cryogenic cabling and quantum chips. Besides low-loss and low-temperature-dependent-dielectric-permittivity materials, Flame Resistance n.4 (FR4) provides a low-cost solution for the fabrication of cryogenic PCBs. Here, we report on an effective way to evaluate the dielectric performance of an FR4 laminate used as a substrate for cryogenic microwave PCBs. We designed a coplanar waveguide $\lambda $ /2 open-circuit series resonator, and we fabricated the PCB using a low-cost manufacturing process, obtaining in-plane geometric features with maximum variations of 50– $100~\mu \text{m}$ compared to the PCB design. Such a geometry allows to exploit the resonance peak of the resonator to measure the variation of the complex permittivity as a function of the temperature. The resonance peak frequency was used to estimate the real permittivity, achieving a sensitivity of −470 MHz and a resolution of $1.2\times 10^{-2}$ . Similarly, the resonance peak magnitude was involved in the extrapolation of the loss tangent, reaching a sensitivity of ~−337 dB and a resolution of $1.6\times 10^{-4}$ . For the FR4 laminate used, we estimated a 9% reduction of the real permittivity and a 70% reduction of the loss tangent in the temperature range from 300 K to 4 K. The proposed approach can be immediately extended to the detection of cryogenic temperature-dependent dielectric performance of any kind of substrate.