Mean-field theory of the Kondo effect in quantum dots with an even number of electrons

We investigate the enhancement of the Kondo effect in quantum dots with an even number of electrons, using a scaling method and a mean field theory. We evaluate the Kondo temperature $T_{\mathrm{K}}$ as a function of the energy difference between spin-singlet and -triplet states in the dot, $\Delta ,$ and the Zeeman splitting, $E_{\mathrm{Z}}.$ If the Zeeman splitting is small, $E_{\mathrm{Z}}\ll T_{\mathrm{K}},$ the competition between the singlet and triplet states enhances the Kondo effect. $T_{\mathrm{K}}$ reaches its maximum around $\Delta =0\mathrm{}$ and decreases with $\Delta$ obeying a power law. If the Zeeman splitting is strong, $E_{\mathrm{Z}}\gg T_{\mathrm{K}},$ the Kondo effect originates from the degeneracy between the singlet state and one of the components of the triplet state at $-\Delta \sim E_{\mathrm{Z}}.$ We show that $T_{\mathrm{K}}$ exhibits another power-law dependence on $E_{\mathrm{Z}}.$ The mean field theory provides a unified picture to illustrate the crossover between these regimes. The enhancement of the Kondo effect can be understood in terms of the overlap between the Kondo resonant states created around the Fermi level. These resonant states provide the unitary limit of the conductance $G\sim {2e}^2/h.\mathrm{}$