Band dependent magneto thermoelectric measurements on Ca$_3$Ru$_2$O$_7$

Ca$_3$Ru$_2$O$_7$ features a Neel transition at 56 K followed by a structure and metal-insulator transition at 48 K as the temperature is lowered, suggesting a complex structure-property relationship driven by electron correlated effects. However, the electronic states of Ca$_3$Ru$_2$O$_7$ are not understood. ARPES measurements revealed the presence of an electron Fermi arc, while the expected hole Fermi arc is missing. Quantum oscillations showed the existence of a small Fermi surface, but the details of the Fermi surface, including whether the hole Fermi arc is present, are not determined. We performed band-dependent thermoelectric measurement with the temperature gradient directed to a specific crystalline axis. Magneto thermopower along crystalline a and b axes, S$_a$ and S$_b$, both of which are negative in sign and nearly identical at high temperatures, were found to behave markedly differently below the metal-insulator transition, with S$_b$ changing its sign from being negative to positive. Our analysis suggests that S$_a$ and S$_b$ in Ca$_3$Ru$_2$O$_7$ are dominated by the electron and hole Fermi arcs, respectively. The implications of our data on the physics of Ca$_3$Ru$_2$O$_7$ will be discussed.