Heat transfer analysis of a programmable Josephson voltage standard chip operated with a mechanical cooler

We are developing a 10 V programmable Josephson voltage standard (PJVS) system using a 10 K mechanical cooler that enables liquid-helium-free operation. In our previous experiments, we identified a problem that the optimum bias operating points may be affected by the heating effect of the chip depending on the output voltage. In this study, we have observed cross-sectional and in-plane structures of the PJVS chip module in order to determine the cause of the heating effect. We perform a heat transfer analysis using the finite element method with several models based on the observation results. We confirm a temperature increase of approximately 100 mK at the chip surface, due mainly to unintended voids in the InSn solder.