Hybrid improper dipolar density wave in NaLaCoWO6

Hybrid improper ferroelectricity (HIF) allows the generation of an electrical polarization in the AA'BB'O-6 double perovskite materials thanks to the combination of two nonpolar octahedral distortions. Nevertheless, for selected combination of the A/A' cations a nonpolar incommensurate phase is observed with average symmetry C2/m. Thanks to a detailed crystallographic description of the incommensurate phase, based on electron, neutron, and x-ray diffraction data, we show that the incommensurate modulation is related to an abrupt change of the out-of-phase tilting along the a and c axis, whereas the tilting along the b axis remains constant across the structure. By using group theory and symmetry analysis we show that we observe an incommensurate analog of HIF, which induces a hybrid improper dipolar density wave in NaLaCoWO6. The dipolar ordering is due also in this case to a trilinear invariant involving the commensurate and incommensurate octahedra tilting.