Linear-response conductance and magnetoresistance of ferromagnetic single-electron transistors

The current through ferromagnetic single-electron transistors (SET's) is considered. Using path integrals the linear-response conductance is formulated as a function of the tunnel conductance versus quantum conductance and the temperature versus Coulomb charging energy. The magnetoresistance of ferromagnet\char21{}normal-metal\char21{}ferromagnet (F-N-F) SET's is almost independent of the Coulomb charging energy and is only reduced when the transport dwell time is longer than the spin-flip relaxation time. In all-ferromagnetic (F-F-F) SET's with negligible spin-flip relaxation time the magnetoresistance is calculated analytically at high temperatures and numerically at low temperatures. The F-F-F magnetoresistance is enhanced by higher-order tunneling processes at low temperatures in the ``off'' state when the induced charges vanish. In contrast, in the ``on'' state near resonance the magnetoresistance ratio is a nonmonotonic function of the inverse temperature.