Creating a three dimensional intrinsic electric dipole on rotated CrI$_3$ bilayers

Two-dimensional (2D) materials are being explored as a novel multiferroic platform. One of the most studied magnetoelectric multiferroic 2D materials are antiferromagnetically-coupled (AFM) CrI$_3$ bilayers. Neglecting magnetism, those bilayers possess a crystalline point of inversion, which is only removed by the antiparallel spin configuration among its two constituent monolayers. The resultant intrinsic electric dipole on those bilayers has a magnitude no larger than 0.04 pC/m, it points out-of-plane, and it reverts direction when the--Ising-like--cromium spins are flipped (toward opposite layers {\em versus} away from opposite layers). The combined presence of antiferromagnetism and a weak intrinsic electric dipole makes this material a two-dimensional magnetoelectric multiferroic. Here, we remove the crystalline center of inversion of the bilayer by a relative $60^{\circ}$ rotation of its constituent monolayers. This process {\em enhances} the out-of-plane intrinsic electric dipole tenfold with respect to its magnitude in the non-rotated AFM bilayer and also creates an even stronger and switchable in-plane intrinsic electric dipole. The ability to create a three-dimensional electric dipole is important, because it enhances the magnetoelectric coupling on this experimentally accessible 2D material, which is explicitly calculated here as well.