Spin-flop and antiferromagnetic phases of the ferromagnetic half-twist ladder compounds Ba₃Cu₃In₄O₁₂ and Ba₃Cu₃Sc₄O₁₂

The title compounds have dominant ferromagnetic exchange interactions within one-dimensional (1D) half-twist ladders of s = 1/2 Cu²⁺ ions and antiferromagnetic (AFM) interactions between ladders, leading to ordered 3D phases at temperatures below 20 K. Here we show that a microscopic 1D model of the paramagnetic (PM) phase combined with a phenomenological model based on sublattice magnetization describes the observed temperature and field dependent magnetism. The model identifies AFM, spin-flop and PM phases whose boundaries have sharp features in the experimental magnetization M(T,H) and specific heat C_P(T,H). Exact diagonalization of the 1D model, possible for 24 spins due to special structural features of half-twist ladders, yields the magnetization and spin susceptibility of the PM phase. AFM interactions between ladders are included at the mean-field level using the field, H_AF, obtained from modelling the ordered phases. Isotropic exchange J₁ =- 135 K and g-tensor g = 2.1 within ladders, plus exchange and anisotropy fields H_AF and H_A, describe the ordered phases and are almost quantitative for the PM phase.