  Using a semi-classical approach, we describe an on-chip cooling protocol for
a micro-mechanical resonator by employing a superconducting flux qubit. A
Lorentz force, generated by the passive back-action of the resonator's
displacement, can cool down the thermal motion of the mechanical resonator by
applying an appropriate microwave drive to the qubit. We show that this onchip
cooling protocol, with well-controlled cooling power and a tunable response
time of passive back-action, can be highly efficient. With feasible
experimental parameters, the effective mode temperature of a resonator could be
cooled down by several orders of magnitude.
