Modelling Electron Spin Accumulation in a Metallic Nanoparticle
A model describing spin-polarized current via discrete energy levels of a metallic nanoparticle, which has strongly asymmetric tunnel contacts to two ferromagnetic leads, is presented. In absence of spin-relaxation, the model leads to a spin-accumulation in the nanoparticle, a difference ($Δμ$) between the chemical potentials of spin-up and spin-down electrons, proportional to the current and the Julliere's tunnel magnetoresistance. Taking into account an energy dependent spin-relaxation rate $Ω(ω)$, $Δμ$ as a function of bias voltage ($V$) exhibits a crossover from linear to a much weaker dependence, when $|e|Ω(Δμ)$ equals the spin-polarized current through the nanoparticle. Assuming that the spin-relaxation takes place via electron-phonon emission and Elliot-Yafet mechanism, the model leads to a crossover from linear to $V^{1/5}$ dependence. The crossover explains recent measurements of the saturation of the spin-polarized current with $V$ in Aluminum nanoparticles, and leads to the spin-relaxation rate of $\approx 1.6 MHz$ in an Aluminum nanoparticle of diameter $6nm$, for a transition with an energy difference of one level spacing.