  Two-dimensional (2D) ferroelectrics, which is rare in nature, enable
high-density non-volatile memory with low energy consumption. Here, we propose
a theory of bilayer stacking ferroelectricity (BSF), in which, two stacked
layers of the same 2D material, with different rotation and translation,
exhibits ferroelectricity. By performing systematic group theory analysis, we
find out all the possible BSF in all the 80 layer groups (LGs) and discover the
rules about the creation and annihilation of symmetries in the bilayer. Our
general theory can not only explain all the previous findings (including
sliding ferroelectricity), but also provide new perspective. Interestingly, the
direction of the electric polarization of the bilayer could be totally
different from that of the single layer. In particular, the bilayer could
become ferroelectric after properly stacking two centrosymmetric non-polar
monolayers. By means of first-principles simulations, we demonstrate that the
ferroelectricity and thus multiferroicity can be introduced to the prototypical
2D ferromagnetic centrosymmetric material CrI3 by stacking. Furthermore, we
find that the out-of-plane electric polarization in bilayer CrI3 is interlocked
with the in-plane electric polarization, suggesting that the out-of-plane
polarization can be manipulated in a deterministic way through the application
of an in-plane electric field. The present BSF theory lays a solid foundation
for designing a large number of bilayer ferroelectrics and thus colorful
platforms for fundamental studies and applications.
