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 $\delta\epsilon=\epsilon_{\downarrow}-\epsilon_{\uparrow}$ of an energy level in the quantum dot by tunnel couplings to the ferromagnetic leads, using the Poor man's scaling method. The spin splitting takes place in an intermediate direction between magnetic moments in the two leads. $\delta\epsilon \propto p\sqrt{\cos^2(\theta/2)+v^2\sin^2(\theta/2)}$, where $p$ is the spin polarization in the leads, $\theta$ 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 ($\theta=0$) and minimal in the antiparallel alignment ($\theta=\pi$). Second, we calculate the Kondo temperature $T_{\mathrm{K}}$. The scaling calculation yields an analytical expression of $T_{\mathrm{K}}$ as a function of $\theta$ and $p$, $T_{\mathrm{K}}(\theta, p)$, when $\delta\epsilon \ll T_{\mathrm{K}}$. $T_{\mathrm{K}}(\theta, p)$ is a decreasing function with respect to $p\sqrt{\cos^2(\theta/2)+v^2\sin^2(\theta/2)}$. When $\delta\epsilon$ is relevant, we evaluate $T_{\mathrm{K}}(\delta\epsilon, \theta, p)$ using the slave-boson mean-field theory. The Kondo resonance is split into two by finite $\delta\epsilon$, which results in the spin accumulation in the quantum dot and suppression of the Kondo effect.