Phase competition and negative piezoelectricity in interlayer-sliding ferroelectric $\mathrm{Zr}{\mathrm{I}}_2$

The so-called interlayer-sliding ferroelectricity was recently proposed as an unconventional route to pursuit electric polarity in van der Waals multilayers, which was already experimentally confirmed in a $\mathrm{W}{\mathrm{Te}}_2$ bilayer even though it is metallic. Very recently, another van der Waals system, i.e., the $\mathrm{Zr}{\mathrm{I}}_2$ bilayer, was predicted to exhibit the interlayer-sliding ferroelectricity with both in-plane and out-of-plane polarizations [Zhang et al., Phys. Rev. B 103, 165420 (2021)]. Here the $\mathrm{Zr}{\mathrm{I}}_2$ bulk is studied, which owns two competitive phases ($\alpha$ vs $\beta$), both of which are derived from the common parent $s$ phase. The $\beta -\mathrm{Zr}{\mathrm{I}}_2$ owns a considerable out-of-plane polarization ($0.39\mu \mathrm{C}/{\mathrm{cm}}^2$), while its in-plane component is fully compensated. Their proximate energies provide the opportunity to tune the ground state phase by moderate hydrostatic pressure and uniaxial strain. Furthermore, the negative longitudinal piezoelectricity in $\beta -\mathrm{Zr}{\mathrm{I}}_2$ is dominantly contributed by the enhanced dipole of $\mathrm{Zr}{\mathrm{I}}_2$ layers as a unique characteristic of interlayer-sliding ferroelectricity, which is different from many other layered ferroelectrics with negative longitudinal piezoelectricity like $\mathrm{Cu}\mathrm{In}{\mathrm{P}}_2{\mathrm{S}}_6$.

Figure 1

(a)–(c) Schematic structures (side view) of various phases of $\mathrm{Zr}{\mathrm{I}}_2$ bulk. Each unit cell (u.c.) contains two vdW layers and four Zr's. (d) The corresponding phonon spectra. No imaginary frequency exists for the $\alpha$ and $\beta$ phases, while weak imaginary frequencies exist from $\mathrm{\Gamma }$ to $Z$ for the $s-\mathrm{Zr}{\mathrm{I}}_2$, implying its dynamic instability. The corresponding Brillouin zones are shown in Fig. S1 of SM [28]. (e) The evolution tree of these three phases. Starting from the high symmetric space group of parent phase, the space groups of $\alpha$ and $\beta$ phases can be obtained by reducing four symmetric operations, respectively.

Figure 2

Density of states and their atomic projects of (a) $\alpha -\mathrm{Zr}{\mathrm{I}}_2$ and (b) $\beta -\mathrm{Zr}{\mathrm{I}}_2$, calculated using the pure GGA method.

Figure 3

Origin of interlayer-sliding ferroelectricity in $\beta -\mathrm{Zr}{\mathrm{I}}_2$. (a) The planar-averaged differential charge density ($\mathrm{\Delta }\rho$) along the $c$ axis (characterized by the height $z$), with $s-\mathrm{Zr}{\mathrm{I}}_2$ as the reference. Inset: the opposite polarization. Here the positive value of $\mathrm{\Delta }\rho$ represents more electrons while its negative value means more holes. The red and cyan spheres represent the charge centers of positive and negative regions (distinguished by colors), respectively. The arrows denote the dipoles. (b) The corresponding structure of (a) for comparison. The in-plane component of polarization (indicated by arrows) can exist in bilayer but is canceled in bulk for its antiferroelectric ordering between neighboring bilayers.

Figure 4

(a) Two possible switching paths of polarization ($P$) in $\beta -\mathrm{Zr}{\mathrm{I}}_2$. The $s-\mathrm{Zr}{\mathrm{I}}_2$ is the intermediate state of path I, while the $\alpha -\mathrm{Zr}{\mathrm{I}}_2$ is the intermediate state of path II. Both of these paths are achieved via the interlayer sliding of vdW layers, while the path II also includes the monoclinic angle distortion of cell. The interlayer sliding is normalized to its optimized value. (b) The corresponding energy barriers of these two switching paths for $\beta -\mathrm{Zr}{\mathrm{I}}_2$.

Figure 5

The evolution of $\mathrm{Zr}{\mathrm{I}}_2$ under the uniaxial strain applied along the $c$ axis, as a function of the lattice constant of the $c$ axis. (a) The compressive strain prefers the ferroelectric $\beta -\mathrm{Zr}{\mathrm{I}}_2$, while the pulling stress can lead to the $\alpha -\mathrm{Zr}{\mathrm{I}}_2$. The corresponding pressures of strain are also shown (right axis). (b) The negative piezoelectricity, namely the out-of-plane polarization is enhanced by pressure along the $c$ axis. Both the net polarization (left axis) and the electric dipole of a u.c. (right axis) are enhanced by pressure. The open circles denote the metastable $\beta -\mathrm{Zr}{\mathrm{I}}_2$.

Figure 6

The effects of hydrostatic pressure to $\beta -\mathrm{Zr}{\mathrm{I}}_2$. (a) The lattice constants (left axis) and the volume (right axis). (b) The polarization (left axis) and electric dipole of a u.c. (right axis). (c) The enthalpy difference between $\alpha -\mathrm{Zr}{\mathrm{I}}_2$ and $\beta -\mathrm{Zr}{\mathrm{I}}_2$ as a function of pressure, where the enthalpy of $\beta -\mathrm{Zr}{\mathrm{I}}_2$ is taken as the reference. $\alpha -\mathrm{Zr}{\mathrm{I}}_2$ becomes more stable above $\sim 1.3$ GPa. (d) Phonon spectrum for $\beta -\mathrm{Zr}{\mathrm{I}}_2$ at 3 GPa, which remains dynamically stable.