2003 Toward the Controllable Quantum States  
We study inelastic electron-electron scattering mediated by the exchange interaction of electrons with magnetic impurities and find the kernel of the corresponding two-particle collision integral. In a wide region of parameters, the kernel K is proportional to the inverse square of the transferred energy, K~J 4 ͞E 2 . The exchange constant J is renormalized due to the Kondo effect. At small energy transfers, the 1͞E 2 divergence is cut off; the cutoff energy is determined by the dynamics of the
more » ... the dynamics of the impurity spins. The obtained results may provide a quantitative explanation of the experiments of Pothier et al. [Phys. Rev. Lett. 79, 3490 (1997)] on anomalously strong energy relaxation in short metallic wires. The effect of magnetic impurities on the electron properties of a metal is drastically different from that of "usual" defects which just violate the translational invariance of the crystalline lattice. The reason for the difference is that a magnetic impurity brings an additional degree of freedom-its spin. If there were no exchange interaction with the itinerant electrons, the ground state of the system would be degenerate with respect to the orientation of impurity spins. Weak exchange interaction allows an itinerant electron to flip its spin in the course of scattering on a magnetic impurity. Such scattering, accounted for even in the lowest-order (Born) approximation, yields an important effect of dephasing of the electron state. Finite dephasing time, in turn, suppresses the interference corrections to the conductivity, thus suppressing the weak localization effect [1]. The higher-order terms in the perturbation theory series for the scattering amplitude reveal one more important phenomenon. It turns out that the amplitude of scattering caused by the exchange interaction increases with lowering the temperature, as opposed to the temperature-independent scattering on a usual impurity. This increase is responsible for the nonmonotonic temperature dependence of the resistivity of a metal, the phenomenon called the Kondo effect [2] . Spin exchange between an electron and a magnetic impurity may occur in an act of elastic scattering. Accounting for these spin-exchange elastic processes is sufficient for understanding the dephasing phenomenon [1] and the Kondo effect [3] . However, such processes do not lead to any energy relaxation of electrons. In this paper we demonstrate that magnetic impurities may also mediate energy transfer between electrons. If the energy transfer E is larger than the Kondo temperature T K , then the energy relaxation occurs predominantly in two-electron collisions. We derive the kernel K of the corresponding collision integral in the kinetic equation for the distribution function. This kernel depends strongly on the transferred energy, K~J 4 ͞E 2 . The dependence of K on the energies´i of the colliding electrons (measured from the Fermi level) is relatively weak as long as j´ij ¿ T K . This dependence comes from the logarithmic in j´ij renormalization of the exchange integral J, known from the theory of Kondo effect [2] . At small energy transfers, the 1͞E 2 divergence of the kernel is cut off; the cutoff energy is determined by the dynamics of the impurity spins, which results from their interaction with the Fermi sea. The motivation for our study comes from the experiment [4, 5] where the relaxation of the electron energy distribution function in mesoscopic wires was investigated. It was found that the empirical relation K~1͞E 2 holds in a substantial interval of energies E for Au and Cu wires. The data of Refs. [4, 5] were accurate enough to rule out the direct Coulomb interaction [6] , which would yield K͑E͒~1͞E 3͞2 , as a source of relaxation. We describe the metal with magnetic impurities by means of the exchange Hamiltonian: where index l labels the magnetic impurities,Ŝ l is the spin operator of the lth impurity,Ŝ 2 l S͑S 1 1͒, and r l is its coordinate. Free electron states are labeled by the wave vector k and spin index a. The Pauli matrices are denoted by s ϵ ͑s x , s y , s z ͒. If the concentration n of the impurities is low enough, they can be considered independently. Therefore we will perform our calculations for a single impurity, omitting the impurity index l, and then will multiply the resulting expressions for the scattering rate by n. In this one-impurity problem, there is interaction only in the s channel, so we label the participating electron states with scalar index k. In the framework of the exchange Hamiltonian (1), the lowest nonvanishing order of the perturbation theory series in the exchange constant J for the inelastic scattering amplitude is the second order: 2400 0031-9007͞01͞86(11)͞2400(4)$15.00
doi:10.1142/9789812705556_0070 fatcat:wh5ak74urrblllvcjaxmid4xra