  Identifying materials and devices which offer efficient thermoelectric
effects at low temperature is a major obstacle for the development of thermal
management strategies for low-temperature electronic systems. Superconductors
cannot offer a solution since their near perfect electron-hole symmetry leads
to a negligible thermoelectric response; however, here we demonstrate
theoretically a superconducting thermoelectric transistor which offers
unparalleled figures of merit of up to $\sim 45$ and Seebeck coefficients as
large as a few mV/K at sub-Kelvin temperatures. The device is also
phase-tunable meaning its thermoelectric response for power generation can be
precisely controlled with a small magnetic field. Our concept is based on a
superconductor-normal metal-superconductor interferometer in which the normal
metal weak-link is tunnel coupled to a ferromagnetic insulator and a Zeeman
split superconductor. Upon application of an external magnetic flux, the
interferometer enables phase-coherent manipulation of thermoelectric properties
whilst offering efficiencies which approach the Carnot limit.
