Sub-kelvin thermal conductivity of substrates and on-chip routing in quantum integrated systems
The development of large-scale quantum systems increasingly relies on the close integration of heterogeneous components such as qubits, control electronics, and readout circuits, making thermal management at cryogenic temperatures a central challenge in such architectures. In this work, we present an experimental thermal study of two building blocks of such systems: the substrate and the on-chip routing. We first investigate the sub-kelvin thermal conductivity of four substrate materials: high-resistivity silicon, low-resistivity silicon, borosilicate, and sapphire. We report that high-resistivity silicon exhibits the highest thermal conductivity among the substrates studied ($5\cdot10^{-2}$~W/m$\cdot$K at 300~mK), while low-resistivity silicon, borosilicate, and sapphire show lower values ($8\cdot10^{-4}$~W/m$\cdot$K, 2$\cdot10^{-3}$~W/m$\cdot$K, and 2$\cdot10^{-3}$~W/m$\cdot$K at 300~mK, respectively). Ballistic conductance evaluation using a finite-element non-equilibrium Green's function approach further allows us to extract the phonon mean free path in each substrate and gives insights into the involved scattering mechanisms. Additionally, we employ a dedicated test vehicle to evaluate the impact of on-chip routing on the thermal conductance of the system. Our measurements with superconducting Nb routing lines reveal that the routing increases the in-plane thermal conductance of the system, but the substrate remains the dominant heat path. These results highlight the critical role of the substrate choice within quantum systems and underscore the importance of function partitioning through 3D integration approaches for more efficient thermal management in quantum architectures.