Antisymmetric double exchange in trimeric mixed-valence clusters

Abstract

Abstract The theory of antisymmetric double exchange (AS DE) interaction is developed for the mixed-valence (MV) trimeric [dn–dn–dn±1] clusters of orbitally non-degenerate ions. Strong isotropic double exchange (DE) and Heisenberg exchange interactions (t-J model) form isotropic trigonal states 2S+1 Γ i (Γi=A1, A2, E) of the MV trimers. Taking into account the spin–orbit coupling results in the vector-type AS DE interaction in these clusters. For the MV trimers with si=1/2, the Hamiltonian of the AS DE coupling has the form H AS DE Z =2 i ∑ αβ K αβ Z T αβ (S β Z −S α Z ) , where KαβZ is the antisymmetric ( K → ij =− K → ji ) vector parameter of the AS DE interaction and T αβ is the isotropic transfer operator. The operator H AS DE Z describes the vector-type spin-transfer interaction induced by the spin–orbit coupling. The microscopic consideration of the AS DE coupling shows that the vector AS DE parameter K → ij is linearly proportional to the spin–orbit coupling constant λ and electron transfer parameter t. The AS DE coupling results in the new effect: the linear AS DE splittings Δ of the 2S+1 E DE terms, Δ are proportional to the cluster AS DE parameter KZ=(KabZ+KbcZ+KcaZ)/3. The vector of the AS DE interaction is directed along the trigonal Z-axis: K Z ≠0, K X =K Y =0 . For the delocalized [d9–d10–d10] ([Cu(II)Cu2(I)]) cluster, the linear AS DE fine splitting Δ=2K Z 3 of the ground 2 E DE term determines strong anisotropy of the Zeeman splitting, axial anisotropy of g-factors, and magnetic properties. The AS DE coupling mixes the 2S+1 A 1 and 2S ′ +1 A 2 , 2S+1 E and 2S ′ +1 E ′ DE terms, ΔS=0,±1. The mixing of the DE levels 2S+1 Γ i by the AS DE coupling and Dzialoshinsky–Moriya AS exchange (H DM =∑ αβ G → αβ [ S → α × S → β ]) determines the second-order AS contributions {∼[(n1KZ+n2GZ)2/(n3t+n4J)], K Z ≫G Z , G Z =(G ab Z +G bc Z +G ca Z )/3} to the cluster zero-field splitting (ZFS) parameters D S ( 2S+1 Γ i ) of the axial anisotropy ( H AN =D S ( 2S+1 Γ i )[S z 2 −S(S+1)/3] , DS≪KZ). The second-order AS DE contributions to the ZFS parameters D S ( 2S+1 Γ i ) are different for the 2S+1 A 1 , 2S+1 A 2 and 2S+1 E DE terms.