Antisymmetric exchange interactions in ${\text{Ni}}_4$ clusters

We show that in molecular ${\text{Ni}}_4$ clusters with $S_4$ symmetry, not only the local anisotropy but also the antisymmetric exchange interaction contributes to the observed fourth-order anisotropy terms in the single-spin Hamiltonian. From a group-theoretical classification of the exchange-coupled multiplets, the effective spin Hamiltonian terms are determined. We find that low-lying excited spin states possess an orbital angular momentum, which mixes into the spin ground state, leading to an effective antisymmetric exchange term. The simulation of the magnetic and spectroscopic experimental data allowed the extraction of parameter values for all interactions.

Figure 1

(Color online) ${\text{Ni}}_4$ cluster geometry and orientations of the various tensors. The thick solid lines are the isotropic exchange interactions with constants $J_A$ (blue, between ions 1 and 2, and between ions 3 and 4) and $J_B$ (red, all other interactions). The solid arrows indicate the local anisotropy coordinate systems. The dashed arrows show the antisymmetric exchange interaction vectors ${\mathbf{G}}_{\parallel }$ and ${\mathbf{G}}_{\perp }$. $\mathbf{X}$, $\mathbf{Y}$, and $\mathbf{Z}$ denote the cluster coordinate system. $S_4$ is the molecular axis of improper rotation.

Figure 2

(Color online) (a) Splitting of the $S=4$ and $S=3$ coupled spin states under the influence of symmetry lowering to $S_4$ symmetry and the action of SOC, respectively. (b) The energies of all spin multiplets as a function of spin on the basis of isotropic exchange interactions only. Curved lines indicate mixing of these multiplets by the collective action of the single-ion anisotropy (SIA, red), both components of the antisymmetric exchange (ASE, blue, solid), the axial ASE component (blue, dashed), the transverse ASE component (blue, dotted), and the simultaneous action of SIA and ASE terms (black).

Figure 3

(Color online) (a) Experimental magnetic-susceptibility temperature product (symbols) and fit (line) for (in ${\text{cm}}^{-1}$) $D=6.5$, $E=2.27$, $G_{\perp }=5$, and $G_{\parallel }=-6$. (b) Experimental magnetization as a function of magnetic field divided by temperature at various temperatures (symbols) and fits (lines) for (in ${\text{cm}}^{-1}$) $J_A=-1.2$, $J_B=7$, $D=6$, $E=2.89$, $G_{\perp }=4.8$, and $G_{\parallel }=-10$. (c) Experimental (symbols) and fitted (lines) INS spectra for the ${\text{Ni}}_4$ cluster at $T=22\text{K}$. The thick red line corresponds to the best fit including antisymmetric exchange, while the thin black line is the best fit without antisymmetric exchange. Fit parameters are given in the text.