Diagrammatic analysis of the dynamics of localized moments in metals

S. E. Barnes, J. Zitkova-Wilcox

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41 Scopus citations


Using the conventional exchange Hamiltonian Hes, together with an additional model Hamiltonian which provides a source of lattice relaxation for the conduction-electron magnetization, the coupled local-moment-conduction- electron transverse dynamic susceptibility is calculated. Feynman temperature-ordered Green's functions are used and are "dressed" with self-energies which are correct to second order in interaction parameters. A pair of coupled vertex equations are constructed which includes all diagrams correct to second order in the interaction parameters. In order to obtain the desired two-particle characteristics of the response function, some new manipulative and mathematical methods are introduced. These enable the vertex equations to be reduced to a coupled pair of linear equations which determine the coupled susceptibility. At sufficiently high temperatures the Kondo "g shifts" and the weak frequency dependence of the self-energies can be ignored. These linear equations are then equivalent to the linearized version of the following Bloch equations: ddtMs=gs[Ms×(HextMe)]-(1Tse)[Ms-s0(HextMe) ]+(gsgeTes)[Me-e0(HextMs)], ddtMe=ge[Me×(HextMs)]-(1Tes+1Te1)[Me- e0(HextMs)]+(gegsTse)[Ms-s0(HextMe)]+D2[Me-e0(HextMs)], where Tse, Tes, and Te1 are the relaxation times for the local-moment-conduction-electron-spin, the conduction-electron-spin-local-moment, and the conduction-electron-spin-lattice systems. The diffusion constant is D=13vF2Ti, where Ti is the nonmagnetic-impurity scattering time, and Me and Ms are, respectively, the conduction-electron and local-moment magnetizations. The spin-orbit scattering in the presence of large-potential-impurity scattering results in relaxation effects identical to those obtained above with the model electron-lattice Hamiltonian. Hext is the external static and rf field, e0 and s0 are the conduction-electron and local-moment-unenhanced (by the s-d exchange) susceptibilities, and ge and gs are the respective g factors (and B have been set equal to unity). These Bloch equations show that the relaxation destination vectors are those appropriate to the total internal field. The presence of the g-factor ratios are consistent with the fact that Hes transfers spin, rather than magnetization, from the conduction electrons to local moments and vice versa. For temperatures kT<s, where s is the local-moment Zeeman energy, these equations fail, the self-energies become frequency dependent, and there are modifications to s0. Most important, the electronic self-energy becomes dependent on the magnitude of the electronic momentum, so that a single equation for the total electron magnetization cannot be written. A microscopic derivation of the detailed-balance condition is given.

Original languageEnglish (US)
Pages (from-to)2163-2192
Number of pages30
JournalPhysical Review B
Issue number5
StatePublished - Jan 1 1973
Externally publishedYes

ASJC Scopus subject areas

  • Condensed Matter Physics


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