Innermost stable circular orbit of a spinning particle in Kerr spacetime

We study the stability of circular orbits of spinning test particles in Kerr spacetime. We find that some of the circular orbits become unstable in the direction perpendicular to the equatorial plane, although the orbits are still stable in the radial direction. For the large spin case $[\mathcal{S}/\mu M\lesssim O(1\left)\mathrm{}\right],$ the innermost stable circular orbit (ISCO) appears before the minimum of the effective potential in the equatorial plane disappears. This changes the radius of the ISCO and therefore the frequency of the last circular orbit.