Kondo effect in a nanosized particle

A magnetic impurity embedded into a metal is spin compensated into a spin singlet via the Kondo effect. The low-temperature thermodynamic properties are Fermi-liquid-like, with a small Fermi energy of the order of the Kondo temperature. In nanoscale particles, however, the conduction states have discrete energy levels and the energy spacing leads to an additional energy scale that competes with $T_K.$ We consider a small metallic sphere with a spin-$\frac{1}{2}$ impurity at its center, interacting with the metallic states via a contact Kondo exchange potential. Hence only s states couple to the impurity. For equally spaced energy levels in the host, the problem is reduced to the Bethe ansatz solution of the Kondo model in a finite box. All energy levels are obtained as a function of the Kondo coupling for three and five s states interacting with the impurity. The specific heat and the susceptibility are exponentially activated at low T due to the discreteness of the energy spectrum, until a Kondo-like Fermi-liquid state is formed at intermediate temperatures. For larger systems we obtain the equations governing the ground state and lowest nonmagnetic and magnetic excitations, which determine the leading low-T activation energies. The model also represents a quantum dot in the Coulomb blockade regime as a side branch to a short quantum wire.