Single-ion and exchange anisotropy effects and multiferroic behavior in high-symmetry tetramer single molecule magnets

We study single-ion and exchange anisotropy effects in equal-spin $s_1$ tetramer single molecule magnets exhibiting $T_d$, $D_{4h}$, $D_{2d}$, $C_{4h}$, $C_{4v}$, or $S_4$ ionic point group symmetry. We first write the group-invariant quadratic single-ion and symmetric anisotropic exchange Hamiltonians in the appropriate local coordinates. We then rewrite these local Hamiltonians in the molecular or laboratory representation, along with the Dzyaloshinskii-Moriay (DM) and isotropic Heisenberg, biquadratic, and three-center quartic Hamiltonians. Using our exact, compact forms for the single-ion spin matrix elements, we evaluate the eigenstate energies analytically to first order in the microscopic anisotropy interactions, corresponding to the strong exchange limit, and provide tables of simple formulas for the energies of the lowest four eigenstate manifolds of ferromagnetic (FM) and anitiferromagnetic (AFM) tetramers with arbitrary $s_1$. For AFM tetramers, we illustrate the first-order level-crossing inductions for $s_1=1/2,1,3/2$, and obtain a preliminary estimate of the microscopic parameters in a Ni$_4$ from a fit to magnetization data. Accurate analytic expressions for the thermodynamics, electron paramagnetic resonance absorption and inelastic neutron scattering cross-section are given, allowing for a determination of three of the microscopic anisotropy interactions from the second excited state manifold of FM tetramers. We also predict that tetramers with symmetries $S_4$ and $D_{2d}$ should exhibit both DM interactions and multiferroic states, and illustrate our predictions for $s_1=1/2, 1$.