Antiferromagnetic and structural phase transitions in tetragonal-like BiFeO$_3$

The recent observation that strain stabilizes a tetragonal-like (``T-like'') polymorph of BiFeO$_3$ has illustrated how epitaxial constraints can fundamentally alter the properties of this multiferroic material. We performed detailed temperature-dependent neutron and x-ray scattering experiments on epitaxial BiFeO$_3$ films on different substrates (SrTiO$_3$, LaAlO$_3$, YAlO$_3$) to study the nature of the monoclinic crystal structure and of the antiferromagnetic (AFM) order. In agreement with Monte Carlo simulations for a classical Heisenberg model, we observe a much lower N\'eel temperature in the T-like morph (T$_N$ $\simeq$ 325K) than in less-strained (``R-like") films (T$_N$ $\simeq$ 645K), and additionally a low-temperature coexistence of C-type and G-type AFM. Independent of the antiferromagnetic transition, at T $\simeq$ 375K we also observe a structural phase transition from one type of monoclinic distortion to a different one. In combination, these measurements shed light on the complexity of the phase diagram of BFO, and provide routes to explore how the material's properties can be tuned by external parameters. Research supported by the U.S. DOE, BES, MSED (H.M.C., W.S., S.L., E.D., H.M.C) and SUFD (M.D.B., G.J.M, J.L.Z, S.E.N.).