Unusual UUDD magnetic chain structure of the spin-1/2 tetragonally distorted spinel GeCu$_{\mathrm{2}}$O$_{\mathrm{4}}$.

GeCu$_{\mathrm{2}}$O$_{\mathrm{4}}$ exhibits a tetragonal spinel structure due to the strong Jahn-Teller distortion associated with Cu$^{\mathrm{2+}}$ ions. We show that its magnetic structure can be described as slabs composed of a pair of layers with orthogonally oriented spin 1/2 Cu chains in the basal ab plane. The spins between the two layers within a slab are collinearly aligned while the spin directions of neighboring slabs are perpendicular to each other. Interestingly, we find that spins along each chain form an unusual up-up-down-down (UUDD) pattern, suggesting a non-negligible nearest-neighbor biquadratic exchange interaction in the effective classical spin Hamiltonian. We hypothesize that spin-orbit coupling and orbital mixing of Cu$^{\mathrm{2+}}$ ions in this system is non-negligible, which calls for future calculations using perturbation theory with extended Hilbert (spin and orbital) space and calculations based on density functional theory including spin-orbit coupling and looking at the global stability of the UUDD state.