Spin-Frustrated Trinuclear Cu(II) Clusters with Mixing of 2(S = 1/2) and S = 3/2 States by Antisymmetric Exchange. 2. Orbital Origin of In-Plane Dzialoshinsky–Moriya Exchange Parameters

Abstract

The microscopic origin of the in-plane (Gx, Gy) and out-of-plane (Gz) Dzialoshinsky-Moriya (DM) exchange parameters is considered for the Cu3(II) clusters. For the systems with the d(x2-y2) ground state of the Cu ions, only Z components of the pair DM exchange parameters are active (Gz not equal to 0, G(x,y) = 0) in the cases of the orientations of the local anisotropy axes zi| (zi||Z) and perpendicular (zi perpendicular Z, xi||(- Z)) to the molecular trigonal Z axis. The dependences of the Gx, Gy, and Gz DM exchange parameters on the tilt of the local magnetic orbitals were obtained for the antiferromagnetic (AFM) clusters with the d(x2-y2) and d(z2) ground state of the Cu ions. The tilt of the local d(x2-y2) orbitals results in the change of the Gz parameter and appearance of the in-plane DM exchange interactions (Gx or/and Gy parameters). The dependence of the Gz and Gx,Gy DM exchange parameters on the tilt angle is essentially different. The in-plane DM exchange coupling (Gx,Gy parameters) can significantly exceed the out-of-plane DM coupling (Gz parameter). The nonzero Gz and Gx,Gy parameters can be positive or negative. For the {Cu3} nanomagnet with the d(x2-y2) ground state and relatively strong DM coupling, the model explains the three DM exchange parameters of the same value (|Gz| = |Gx| = |Gy|) by the small tilt of the local anisotropy axes zi of the CuO4 local groups of the trimer from the positions zi perpendicular Z. The dependence of the DM exchange parameters (Gz, Gx, Gy) on the tilt for the AFM Cu3 clusters with the d(z2) ground states of the Cu ions differs significantly from that for the AFM systems with the ground state d(x2-y2) of the individual ions. Large in-plane DM exchange parameters Gx or/and Gy result in the mixing of the 2(S = 1/2) and S = 3/2 states and zero-field splitting (ZFS) 2D(DM) of the excited S = 3/2 state. The DM exchange contribution 2D(DM) to ZFS of the excited S = 3/2 state possesses the significant dependence on the tilt of the local magnetic orbitals.