General Theory for Bilayer Stacking Ferroelectricity

Two-dimensional (2D) ferroelectrics, which are rare in nature, enable high-density nonvolatile 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, exhibit ferroelectricity. By performing systematic group theory analysis, we find all the possible BSF in all 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 a 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 nonpolar monolayers. By means of first-principles simulations, we predict that the ferroelectricity and thus multiferroicity can be introduced to the prototypical 2D ferromagnetic centrosymmetric material ${\mathrm{CrI}}_3$ by stacking. Furthermore, we find that the out-of-plane electric polarization in bilayer ${\mathrm{CrI}}_3$ 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.

Figure 1

Schematization of constructing bilayers from monolayers and illustration of the four principles. (a) The $R^{-}$ symmetry $(I,m_z,c_{2\alpha },s_{nz})$ can exist in $B$ only if $O\in R^{-}{G}_{S0}$. (b) One case of principle (b): pure translation will not break the inversion symmetry. (c) If the single layer does not have $R^{+}(E,m_{\beta },c_{nz})$, the bilayer also does not have this $R^{+}$. (d) A translation that makes two $n$-fold rotational axes coincide in both layers will preserve the $n$-fold rotation symmetry. (e) Example of the polar type transformation after bilayer stacking using different rotational operation $O$. The translational part is omitted here.

Figure 2

(a) Top and side views of bilayer ${\mathrm{CrI}}_3$ obtained using $\widehat{O}=\{m_z|0\}$. Purple and blue spheres denote I and Cr atoms, respectively. (b) The six high symmetry translation points in the “unit cell” of ${\tau }_0$ in the hexagonal crystal system, $G(0,0)$, $A(1/2,0)$, $B(0,1/2)$, $C(1/2,1/2)$, $M(1/3,2/3)$, $N(2/3,1/3)$, the minima point $D(1/3,0)$ in ${\mathrm{CrI}}_3$, and side views of bilayer ${\mathrm{CrI}}_3$ at $D$. The parallelogram with black edges is the “unit cell” of ${\tau }_0$. (c) The energy difference between FM and AFM states for different translations. (d) The polarization and energy distribution of the magnetic ground states for different translations. The global ground states around the original point are at $D_{1,2,3\dots ,6}:\pm (1/3,0),\pm (0,1/3),\pm (1/3,1/3)$. The in-plane and the out-of-plane polarization components are represented by the solid arrows and its color, respectively. The dotted arrows and dashed arrows show the small in-plane external electric field ${\overrightarrow{E}}_{\mathrm{ext}}$ and the corresponding switching pathway, respectively.