Quantum-Geometric Origin of Out-of-Plane Stacking Ferroelectricity

Stacking ferroelectricity (SFE) has been discovered in a wide range of van der Waals materials and holds promise for applications, including photovoltaics and high-density memory devices. We show that the microscopic origin of out-of-plane stacking ferroelectric polarization can be generally understood as a consequence of a nontrivial Berry phase borne out of an effective Su-Schrieffer-Heeger model description with broken sublattice symmetry, thus elucidating the quantum-geometric origin of polarization in the extremely nonperiodic bilayer limit. Our theory applies to known stacking ferroelectrics such as bilayer transition-metal dichalcogenides in $3R$ and $T_{\mathrm{d}}$ phases, as well as general $AB$-stacked honeycomb bilayers with staggered sublattice potential. Our explanatory and self-consistent framework based on the quantum-geometric perspective establishes quantitative understanding of out-of-plane SFE materials beyond symmetry principles.

Figure 1

(a) Schematic of a 1D SSH atomic chain with intracell hopping $t+\delta t$ and intercell hopping $t-\delta t$ between $A$, $B$ sublattice sites. A polar structure forms under a staggered sublattice potential ${\epsilon }_A\ne {\epsilon }_B$. (b) Winding of a $\mathit{d}$ vector defined in Eq. (1) as momentum $k$ is varied adiabatically across the 1D Brillouin zone. The Berry phase is given by $1/2$ of the solid angle $\mathrm{\Omega }$ subtended by $\mathit{d}$.

Figure 2

(a) $AB$-stacked honeycomb bilayer with direct coupling $g$ between the $1B$ and $2A$ sites. (b) Asymmetric coupling in $3R$-bilayer TMD at $+K$ occurs only between the conduction band in layer 1 and the valence band in layer 2 with the same effective angular momentum $m_z=0$. (c) Schematic band diagram at $Q$ and $Q^{\prime }$ valleys for bilayer $T_{\mathrm{d}}$ TMD. A spin-valley-dependent band splitting $2V$ leads to imbalance between spin subbands from $Q$ and $Q^{\prime }$ valleys. (d) Feedback loop for the self-consistency condition established in Eq. (6). (e)–(g) $\mathit{k}$-resolved polarization for (e) bilayer SiC, (f) $3R$-bilayer ${\mathrm{MoS}}_2$, (g) $T_{\mathrm{d}}$-bilayer ${\mathrm{WTe}}_2$. Color scales represent polarization in units of polarization quantum $e/A_{\mathrm{cell}}$, with $A_{\mathrm{cell}}$ the area of the unit cell. (h) $P_z$ of bilayer ${\mathrm{WTe}}_2$ as a function of inversion-symmetric $g_{1,-}$. The dashed gray line depicts a quadratic fit of the polarization data.