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

Ad-hoc interface PCBs (Printed Circuit Boards) 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 fabrication of cryogenic PCBs. Here, we report on an effective way to evaluate the dielectric performance of a FR4 laminate used as 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 um 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 1.2x10^-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 1.6x^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 to 4 K. The proposed approach can be immediately extended to the detection of cryogenic temperature-dependent dielectric performance of any kind of substrate.