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.
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