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

Abstract

We study the Kondo effect in a quantum dot coupled to two noncollinear\nferromagnetic leads. First, we study the spin splitting\n$\\delta\\epsilon=\\epsilon_{\\downarrow}-\\epsilon_{\\uparrow}$ of an energy level\nin the quantum dot by tunnel couplings to the ferromagnetic leads, using the\nPoor man's scaling method. The spin splitting takes place in an intermediate\ndirection between magnetic moments in the two leads. $\\delta\\epsilon \\propto\np\\sqrt{\\cos^2(\\theta/2)+v^2\\sin^2(\\theta/2)}$, where $p$ is the spin\npolarization in the leads, $\\theta$ is the angle between the magnetic moments,\nand $v$ is an asymmetric factor of tunnel barriers ($-1<v<1$). Hence the spin\nsplitting is always maximal in the parallel alignment of two ferromagnets\n($\\theta=0$) and minimal in the antiparallel alignment ($\\theta=\\pi$). Second,\nwe calculate the Kondo temperature $T_{\\mathrm{K}}$. The scaling calculation\nyields an analytical expression of $T_{\\mathrm{K}}$ as a function of $\\theta$\nand $p$, $T_{\\mathrm{K}}(\\theta, p)$, when $\\delta\\epsilon \\ll T_{\\mathrm{K}}$.\n$T_{\\mathrm{K}}(\\theta, p)$ is a decreasing function with respect to\n$p\\sqrt{\\cos^2(\\theta/2)+v^2\\sin^2(\\theta/2)}$. When $\\delta\\epsilon$ is\nrelevant, we evaluate $T_{\\mathrm{K}}(\\delta\\epsilon, \\theta, p)$ using the\nslave-boson mean-field theory. The Kondo resonance is split into two by finite\n$\\delta\\epsilon$, which results in the spin accumulation in the quantum dot and\nsuppression of the Kondo effect.\n