Multiferroic rust: magnetoelectric effect in g\"{o}thite, $\alpha$-FeOOH

By means of symmetry analysis, density functional theory calculations, and Monte Carlo simulations we show that goethite, $\alpha$-FeOOH, is a linear magnetoelectric below its N'{e}el temperature $T_{\mathrm{N}}=400$ K. The experimentally observed magnetic field induced spin-flop phase transition results in either change of direction of electric polarization or its suppression. Calculated value of magnetoelectric coefficient is 0.24 $\mu$C m${}^{-2}$ T${}^{-1}$. The abundance of goethite in nature makes it arguably the most widespread magnetoelectric material.