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Spin splitting and Kondo effect in quantum dots coupled to noncollinear ferromagnetic leads

We study the Kondo effect in a quantum dot coupled to two noncollinear ferromagnetic leads. First, we study the spin splitting $δε=ε_{\downarrow}-ε_{\uparrow}$ of an energy level in the quantum dot by tunnel couplings to the ferromagnetic leads, using the Poor man&#39;s scaling method. The spin splitting takes place in an intermediate direction between magnetic moments in the two leads. $δε\propto p\sqrt{\cos^2(θ/2)+v^2\sin^2(θ/2)}$, where $p$ is the spin polarization in the leads, $θ$ is the angle between the magnetic moments, and $v$ is an asymmetric factor of tunnel barriers ($-1<v<1$). Hence the spin splitting is always maximal in the parallel alignment of two ferromagnets ($θ=0$) and minimal in the antiparallel alignment ($θ=π$). Second, we calculate the Kondo temperature $T_{\mathrm{K}}$. The scaling calculation yields an analytical expression of $T_{\mathrm{K}}$ as a function of $θ$ and $p$, $T_{\mathrm{K}}(θ, p)$, when $δε\ll T_{\mathrm{K}}$. $T_{\mathrm{K}}(θ, p)$ is a decreasing function with respect to $p\sqrt{\cos^2(θ/2)+v^2\sin^2(θ/2)}$. When $δε$ is relevant, we evaluate $T_{\mathrm{K}}(δε, θ, p)$ using the slave-boson mean-field theory. The Kondo resonance is split into two by finite $δε$, which results in the spin accumulation in the quantum dot and suppression of the Kondo effect.