A single-phase room-temperature multiferroic with strong magnetoelectric coupling
In plain words
$\mathrm{BiFeO_3}$ is magnetic and ferroelectric at room temperature but has almost no net magnetization, while materials with strong coupling between the two orders work only at low temperature. No single material yet combines both at room temperature.
Precise statement
Identify a single-phase compound with ferroelectric polarization above about $3 \times 10^{3}\ \mathrm{statC}/\mathrm{cm}^{2}$ and net magnetization above $10\ \mathrm{emu}/\mathrm{cm}^{3}$ at $T = 300\ \mathrm{K}$, in which an electric field deterministically reverses (or rotates by $90\ \text{degrees}$) the magnetization. An answer is a material with measured $P$, $M$ and demonstrated electric-field reversal of $M$ at $300\ \mathrm{K}$ without a ferromagnetic overlayer.
What would settle it
Synthesis and measurement of a single-phase compound showing 300 K polarization and magnetization hysteresis plus electric-field-driven magnetization reversal imaged by XMCD-PEEM or magnetic force microscopy.
Status in the literature
Unverified note: it names recent work and was partly written without a live literature search; it awaits a check by an editor.
Deterministic switching at $300\,\mathrm{K}$ so far uses $\mathrm{BiFeO_3}$ coupled to a ferromagnetic layer (Heron et al., Nature 2014).
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