  We demonstrate highly transparent silicon-vanadium and silicon-aluminum
tunnel junctions with relatively low sub-gap leakage current and discuss how a
trade-off typically encountered between transparency and leakage affects their
refrigeration performance. We theoretically investigate cascaded
superconducting tunnel junction refrigerators with two or more refrigeration
stages. In particular, we develop an approximate method that takes into account
self-heating effects but still allows us to optimize the cascade a single stage
at a time. We design a cascade consisting of energy-efficient refrigeration
stages, which makes cooling of, e.g., quantum devices from above 1 K to below
100 mK a realistic experimental target.
