New theoretical model for transition-metal impurities alloyed in copper, local magnetic moments, and the Kondo effect

Abstract

Self-consistent-field $X\alpha$ scattered-wave (SCF-$X\alpha$-SW) cluster molecular-orbital models have been constructed for transition-metal impurities (Ni, Fe, Mn, and V) alloyed in crystalline copper, in order to gain insight into the occurrence of local magnetic moments and the Kondo effect, traditionally viewed as many-body effects arising from the interaction of the transition-metal impurities with the conduction electrons of the host metal. A 19-atom cluster representing the local molecular environment, up to second-nearest neighbors, of an isolated transition-metal impurity in an otherwise perfect crystalline copper lattice yields manifolds of molecular-orbital energy levels corresponding to the copper "$d$ band," bracketed by impurity $\mathrm{dsp}$ hybrid-orbital energy levels that are bonding with respect to the bottom of the copper $d$ band and antibonding with respect to the top of the $d$ band. The latter cluster orbitals, in conjunction with Hund's rules, (i) are discrete analogs of Friedel-Anderson virtual impurity states, (ii) satisfy a "sum rule" analogous to the Friedel sum rule, (iii) have spin occupancies and polarization in accord with measured magnetic moments, (iv) are consistent with measured trends of residual electrical resistivity, and (v) describe the "renormalizing" effects of the crystalline environment on the transition-metal impurity atom. The most striking result of these theoretical studies is the discovery that local coordination chemical bonding dominates the interaction of the transition-metal impurity with its crystalline environment, yielding a large splitting (∼4-4.5 eV) between bonding and antibonding impurity $\mathrm{dsp}$-hybrid molecular-orbital energy levels and a spin splitting (∼0.2-0.6 eV) of the highest occupied antibonding $\mathrm{dsp}$-hybrid orbitals (the Friedel-Anderson-like impurity states) that is an order of magnitude smaller than values assumed in previous theoretical models (e.g., the Anderson model). The near degeneracy of the latter spin orbitals (for Fe and Mn impurities) with cluster molecular-orbital analogs of the conduction-band eigenstates of crystalline copper near the Fermi energy suggests that the "spin-compensation cloud" often associated with the Kondo effect is considerably smaller in magnetization and more spatially localized around the impurity than assumed in previous formal theories and, in conjunction with the SCF-$X\alpha$ transition-state concept, also provides a discrete orbital basis for discussing the onset of "localized spin fluctuations" around the Kondo temperature. The possible contributions of spin-orbit coupling and Jahn-Teller effects to impurity electronic structure are discussed, with detailed examples of how the former effect can lead to the quenching of the impurity magnetic moment below the Kondo temperature. The Anderson-model Hamiltonian, including the electron-electron interaction term $U$, has been reformulated to account for the large bonding-antibonding interaction and much reduced exchange splitting of the impurity-spin orbitals, and a parametrization of a "$J\overrightarrow{\mathrm{S}}·\overrightarrow{\mathrm{s}}$" antiferromagnetic exchange Hamiltonian is suggested by the cluster electronic structure. A relationship between the concept of "cluster renormalization" and the renormalization-group method of attacking the Kondo problem is suggested, and a computational procedure for determining the effects of successive surrounding shells of atoms on the impurity, analogous to that of the renormalization-group method, is demonstrated for a 43-atom cluster. Finally, the extension of cluster molecular-orbital studies to impurity-impurity interactions, to transition-metal impurities in aluminum, where permanent local magnetic moments do not occur, to Fe impurities in palladium, where "giant magnetic moments" are observed, and to rare-earth impurities in metals is discussed.