Geometric frustration on the trillium lattice in a magnetic metal–organic framework

In the dense metal-organic framework Na[Mn(HCOO)$_3$], Mn$^{2+}$ ions ($S=\\frac{5}{2}$) occupy the nodes of a `trillium' hyperkagome net. We show that this material exhibits a variety of behaviour characteristic of geometric frustration: the N\\'eel transition is suppressed well below the characteristic magnetic interaction strength; short-range magnetic order persists far above the N\\'eel temperature; and the magnetic susceptibility exhibits a pseudo-plateau at $\\frac{1}{3}$-saturation magnetisation. We demonstrate that a simple nearest-neighbour Heisenberg antiferromagnet model accounts quantitatively for each observation, and hence Na[Mn(HCOO)$_3$] is the first experimental realisation of this model on the trillium net. We develop a mapping between this trillium model and that on the two-dimensional Shastry-Sutherland lattice, and demonstrate how both link geometric frustration within the classical spin liquid regime to a strong magnetocaloric response at low fields.