Stabilization of sliding ferroelectricity through exciton condensation

Abstract

Sliding ferroelectricity is a phenomenon that arises from the insurgence of spontaneous electronic polarization perpendicular to the layers of two-dimensional (2D) systems upon the relative sliding of the atomic layer constituents. Because of the weak van der Waals (vdW) interactions between layers, sliding and the associated symmetry breaking can occur at low energy cost in materials such as transition-metal dichalcogenides. Here we discuss theoretically the origin and quantitative understanding of the phenomenon by focusing on a prototype structure, the WTe2 bilayer, where sliding ferroelectricity was first experimentally observed. We show that excitonic effects induce relevant energy band renormalizations in the ground state, and exciton condensation contributes significantly to stabilizing ferroelectricity upon sliding beyond previous predictions. Enhanced excitonic effects in 2D and vdW sliding are general phenomena that point to sliding ferroelectricity as relevant for a broad class of important materials, where the intrinsic electric dipole can couple with other quantum phenomena and, in turn, an external electric field can control the quantum phases through ferroelectricity in unexplored ways.

Figures (6)

• Figure 1: The potential barrier for the sliding process. a) Top view, side view and Brillouin zone of bilayer WTe$_2$. In the top view the top layer is shaded for better clarity and the black rectangle marks the unit cell with the corresponding lattice parameters $a$ and $b$. Blue atoms represent W, orange ones Te. b) Top: sketch of the energy barrier of the sliding process. The two equivalent ground states have opposite out-of-plane polarization (up/down arrows). The configuration corresponding to the energy maximum is the GMS structure. The top layer displacement from the GMS structure to the ground state structure is largely magnified in this sketch. Bottom: DFT-PBE bands corresponding to the ground state structure (left) and to the GMS structure (right). The bands are plotted along the $\Gamma X$ path and the zero of the energy axis is the Fermi energy, $E_F$. c) Effects of the formation of the excitonic insulator phase (red curves). The sketch shows that tightly bound excitons can modify the band structure (bottom), and possibly open a gap, hence lowering the energy of the system and thus changing the energy barrier for the sliding process (top). The red solid bands represent qualitatively how the DFT bands change after including excitonic effects; the dashed bands, shown as reference, reproduce the DFT-PBE bands of panel b) around $E_F$ for the displaced and GMS structures (left and right, respectively).
• Figure 1: Model BSE and EI gap equation with four bands. Exciton wavefunction (left) and EI bands reconstruction (right) in the four band model case. Here the exciton wave function is decomposed in the transition components $\Psi_x^{vc\boldsymbol{k}}$, with $v=1,2$ and $c=3,4$ the indices of the two valence and two conduction bands in ascending order of energy. The range of the color scale is $[-0.1,0.1]$, with positive [negative] values in red [blue]. Analogously to Fig. 3 of the main text, the maximum amplitude is reached at the two locations in the Brillouin zone region that correspond to the two energy gaps along the $\Gamma$X cut between bands $2$ and $3$, and the corrections to the bands in the EI phase given by $\Delta_{ij}(\boldsymbol{k})$ (see main text) are larger where the amplitudes $\Psi_x^{vc\boldsymbol{k}}$ are larger. The results shown here refer to the GMS structure (black bands as in the main text). Notice that at difference with the calculations discussed in the main text, here the Brillouin zone sampling is reduced along the $k_y$ direction: This impacts significantly the exciton binding energy, thus a lower screening parameter $\alpha_{\mathrm{2D}} = 1.9$ Å has been used to still obtain a gap $>0$ in the EI phase.
• Figure 2: Effect of layer sliding on band structure. DFT-PBE bands for different displacements of the top layer along the sliding direction. The plot shows the cut along $\Gamma X$ (thus $k_y=0$); $k_x$ is in units of the corresponding reciprocal lattice vector (rlu). The bottom layer (dyed in black) is fixed and $0.00$ displacement of the top layer corresponds to the GMS structure. Images on the right show the displacements of the top layer (coloured) in real scale, with the dashed line indicating the fixed position of the bottom layer; negative values of the displacements are indicated in Å, as in the key; positive displacements -- not shown -- are analogous and give the same bands.
• Figure 2: Energy of the EI phase upon variation of the screening parameter. Gap (left) and energy gain (right) in the EI phase for increasing screening of the e-h interaction. See main text for the definition of $E_{\mathrm{EI}}$ and $\alpha_{\mathrm{2D}}$. The connecting lines are only a guide to the eye. The plots show that although both $E_{\mathrm{gap}}$ and $E_{\mathrm{EI}}$ vary considerably with $\alpha_{\mathrm{2D}}$, the variation with respect to the top layer displacement ($y$ as in the main text) at fixed $\alpha_{\mathrm{2D}}$ is about the same for all values of $\alpha_{\mathrm{2D}}$ (i.e. the lines are parallel). As a consequence, the renormalized switching barrier is the same for all these values of $\alpha_{\mathrm{2D}}$.
• Figure 3: Excitons and excitonic insulator. a) Exciton wave function in $\bm{k}$ space. The contour plot represents the probability amplitude $\Psi_x^{\bm{k}}$ of exciting an e-h pair by transferring an electron from the valence to the conduction band state at fixed momentum $(k_x,k_y)$. The range of the color scale is $[0,0.11]$. b) Condensation of excitons leads to the reconstruction of the bands (EI) with the opening of a gap. The plot shows the cut for $k_y = 0$. For both panels the layer displacement is $\bar{d}$ and $\alpha_{\mathrm{2D}} = 4.3$ Å.