Helical spin thermoelectrics controlled by a side-coupled magnetic quantum dot in the quantum spin Hall state

We study the thermoelectric response of a device containing a pair of helical edge states contacted at the same temperature $T$ and chemical potential $mu$ and connected to an external reservoir, with different chemical potential and temperature, through a side quantum dot. Different operational modes can be induced by applying a magnetic field $B$ and a gate voltage $V_g$ at the quantum dot. At finite $B$, the quantum dot acts simultaneously as a charge and a spin filter. Charge and spin currents are induced, not only through the quantum dot, but also along the edge states. We focus on linear response and analyze the regimes, which we identify as charge heat engines or refrigerator, spin heat engine and spin refrigerator.