Negative piezoelectric response of van der Waals layered bismuth tellurohalides

The polarization and piezoelectric response of the $\mathrm{BiTe}X$ $(X=\mathrm{Cl},\mathrm{Br},and\mathrm{I})$ layered tellurohalides are computed from first principles. The results confirm a mixed ionic-covalent character of the bonding, and demonstrate that the internal structure within each triple layer is only weakly affected by the external stress, while the changes in the charge distribution with stress produce a substantial negative piezoelectric response. This suggests a mechanism for negative piezoelectric response that should remain robust even in the ultrathin film form in this class of materials.

Figure 1

Layered structure of $\mathrm{BiTe}X$ for (a) $X=\mathrm{I}$ or Br in the $P3m1$ structure, and (b) $X=\mathrm{Cl}$ in the doubled-cell $P{6}_{3}mc$ structure. Horizontal solid lines indicate $c$ lattice constants. Distances $\alpha$ and $\beta$ denote Bi-$X$ and Bi-Te intralayer distances, while $\gamma$ is the separation between triple layers. (c) Wannier functions constructed from $p$ bands, having Bi-$X$ (lower) and Bi-Te (upper) bond-orbital character.

Figure 2

(a), (b) Schematic view of two simple models of the polarization in $\mathrm{BiTe}X$. (a) Fully ionic model suggests a negative dipole moment. (b) Model with a covalent Bi-Te unit suggests a positive dipole moment. (c) Wannier function perspective. Solid vertical lines indicate Wannier center locations, with ${\delta }_{X,\mathrm{Te}}$ denoting their displacements relative to the neighboring anion layers (dotted vertical lines).

Figure 3

(a)–(d) Calculated polarization vs strain $\eta =c/c_0-1$ of $\mathrm{BiTe}X$ at fixed in-plane lattice constant for relaxed-ion, frozen-ion, and frozen-triple-layer cases. Solid lines are fits whose slopes at equilibrium ($\eta =0$, dashed vertical line) correspond to $e_{33}$. (e)–(h) Changes of relaxed structural parameters under uniaxial strain. Here $\alpha$ and $\beta$ are the Bi-$X$ and Bi-Te internal atomic interlayer distances (see Fig. 1); $\gamma$ is inter-triple-layer spacing.

Figure 4

Calculated polarization of $\mathrm{BiTe}X$ vs uniaxial stress at relaxed in-plane lattice constant. Solid lines are fits whose slopes at equilibrium (${\sigma }_{33}=0$, dashed vertical line) correspond to $d_{33}$.