Kondo effect in quantum dots coupled with noncollinear ferromagnetic leads

We study the Kondo effect in a quantum dot coupled to two noncollinear ferromagnetic leads. First, we study the spin splitting δε = ε↑ - ε↓ in the quantum dot by the tunnel coupling to the ferromagnetic leads, using the Poor man's scaling method. The spin splitting takes place in an intermediate direction between the polarization directions in the two leads. δε ∞ cos(θ/2), where θ is the angle between the polarization directions; spin splitting is maximal in the parallel alignment of two ferromagnets (θ = 0) and disappears in the antiparallel alignment (θ = π). Second, we calculate the Kondo temperature T _K. The scaling calculation yields an analytical expression of T _K as a function of θ and polarization ratio p in the ferromagnets, T_K (θ,p), when δe << T_K. When δe is relevant, we evaluate T_K(δε, θ, p) using the slave-boson mean field method. The Kondo resonance is split into two by finite δε, which results in the suppression of Kondo effect.