Anisotropic double exchange in a valence-delocalized [Fe2.5+Fe2.5+] cluster. Spin–orbit coupling in the Anderson–Hasegawa double exchange model

Anisotropic double exchange in valence-delocalized [Fe2.5+ Fe2.5+] cluster

“…Anisotropic DE interaction results in the ZFS of the delocalized states [6,7]. This ZFS of the anisotropic DE origin is described by the new effective ZFS Hamiltonian…”

“…Anisotropic DE interaction [6,7] originates from the combined effect of the SOC V SO at iron ions and isotropic DE between the excited states of the MV Fe ions. Strong standard isotropic DE coupling between the ground states of the MV Fe ions [2][3][4] is accompanied by the ET between the excited states of the MV Fe ions due to the SOC admixture of the excited states to the ground states [7]. The effective anisotropic DE Hamiltonian…”

“…H B includes the DE coupling between the different excited cluster states F 0 aÃb ðSÞð½ðFe 2þ aÃ Þ ex ðFe 3þ b Þ 0 Þ and F 00 abà ðSÞð½ðFe 3þ a Þ ex ðFe 2þ bÃ Þ 0 Þ fF 00 aÃb ðSÞ and F 0 abà ðSÞg of different localizations, where F 00 abà ðSÞ are formed by the excited states ( 4 T 1x;1y ) of the Fe 3+ ion and the ground ( 5 A 1 ) state of the Fe 2+ ion. The DE interaction between different excited states F 0 aÃb ðSÞ and F 00 abà ðSÞ is determined by the ET (DE) parameter t 0 (B) [7]. The one-center left spin operators S aÃn {right spin operators S 0 an } in the term B n t 0 ðS aÃn T ab S 0 an Þ (3) act on the ground state spin functions c aà ðs a ; m a Þ fj a ðs 0 a ; m 0 a Þg of the Fe 2þ aà {Fe 3þ a } ion at the a-center in the ha à bj fjab à ig localization [7].…”

“…The DE interaction between different excited states F 0 aÃb ðSÞ and F 00 abà ðSÞ is determined by the ET (DE) parameter t 0 (B) [7]. The one-center left spin operators S aÃn {right spin operators S 0 an } in the term B n t 0 ðS aÃn T ab S 0 an Þ (3) act on the ground state spin functions c aà ðs a ; m a Þ fj a ðs 0 a ; m 0 a Þg of the Fe 2þ aà {Fe 3þ a } ion at the a-center in the ha à bj fjab à ig localization [7]. The effective Hamiltonian H C…”

“…The coefficients g 1x ¼ z Fe 2þ =4D 1x , g 1y ¼ z Fe 2þ =4D 1Z describe the SOC admixture of the excited states 5 E½d yz ðxÞ; d xz ðZÞ to the ground 5 A 1 state, z Fe 2þ is the one-electron SOC constant of the Fe 2+ ion [5], D 1m ¼ Eð 5 E m Þ À E 0 ð 5 A 1 Þ. H A acts between the cluster ground states F 0 aÃb ðS; MÞ and F 0 abà ðS 0 ; M 0 Þ of different localizations. The one-center spin operators act on the ground state spin functions of the Fe 2+ ions [7]. The effective Hamiltonian H B has the form…”