First-principles Study of Improper Ferroelectricity in TbMnO$_3$

Perovskite TbMnO$_3$ at room temperature forms an orthorhombically distorted lattice with the {\it Pbnm} space group. Below $\sim$27~K the magnetic moments on the Mn atoms adopt an incommensurate cycloidal wave order, and simultaneously a polarization appears along the $c$ direction. We present the results of our first-principles theoretical study of the magnetically induced polarization in TbMnO$_3$ with a {\it commensurate} cycloidal wave of Mn$^{3+}$ moments with a wave-vector close to the experimental value.\footnote{A.~Malashevich and D.~Vanderbilt, Phys.~Rev.~Lett.~\textbf{101}, 037210 (2008).} The calculations are based on density-functional theory in the local-density approximation with the on-site Coulomb correction (LDA+U). The polarization is computed using the Berry-phase technique. We show, in particular, that the spin-orbit interaction is essential for the magnetoelectric coupling. We compute both the electronic and the lattice-mediated contributions to the polarization, and find that the latter is strongly dominant. We analyze the spin-orbit induced forces and lattice displacements from both atomic and mode-decomposition viewpoints, and show that a simple model based on nearest Mn-Mn neighbor Dzyaloshinskii-Moriya interactions is not able to account fully for the results. The direction and magnitude of our computed polarization are in good agreement with experiment. If time permits, calculations on other magnetically induced improper ferroelectrics will be discussed.