Hybrid improper dipolar density wave in NaLaCoWO$_6$

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 non-polar octahedral distortions. Nevertheless, for selected combination of the A/A' cations a non-polar 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 remain 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 NaLaCoWO$_6$. The dipolar ordering is due also in this case to a trilinear invariant involving the commensurate and incommensurate octahedra tilting's.