Polarization Vortices in Germanium Telluride Nanoplatelets: A Theoretical Study

Using first-principles calculations based on density functional theory, we study the properties of germanium telluride crystalline nanoplatelets and nanoparticles. Above a diameter of 2.7 nm, we predict the appearance of polarization vortices giving rise to an unusual ferrotoroidic ground state with a spontaneous and reversible toroidal moment of polarization. We highlight the crucial role of inhomogeneous strain in stabilizing polarization vortices. Combined with the phase-change properties of germanium telluride, the ferrotoroidic properties could be of practical interest for ternary logic applications.

Figure 1

(a) Atomic structure of a $n\times n\times 2$ GeTe nanoplatelet for $n=8$. The structure consists in two consecutive atomic layers of Ge (dark balls) and Te (light balls) atoms along $z$ so that a Ge–Te pair can be affected to any given position $i$ in the $xy$ plane. (b) Top view of the relaxation pattern of a $16\times 16\times 2$ nanoplatelet with respect to initial bulk atomic positions, under the constraint of $D_{2d}$ symmetry. (c) Cohesive energy per Ge–Te pair, $E_C$, as a function of size for $n\times n\times 2$ nanoplatelets in the $D_{2d}$ symmetry. The dashed and dotted horizontal lines correspond to $E_C$ for a 2D infinite slab and for 3D crystal, respectively.

Figure 2

Top view of (a) the individual ferrotoroidic polar distortions, $\Delta {\mathbf{r}}_i^{\mathrm{Ge}}$ (light arrows) and $\Delta {\mathbf{r}}_i^{\mathrm{Te}}$ (dark arrows), for the $16\times 16\times 2$ nanoplatelet. For the same nanoplatelet, decomposition of $\Delta {\mathbf{r}}_i^{\mathrm{Ge}}$ and $\Delta {\mathbf{r}}_i^{\mathrm{Te}}$ into (b) the displacement ${\eta }_i$ of the center of mass of each vertical Ge–Te pair and (c) the structural distortion ${\xi }_i^{\mathrm{Ge}}$ (light arrows) and ${\xi }_i^{\mathrm{Te}}$ (dark arrows).

Figure 3

(a) Evolution of the spontaneous toroidal moment of polarization, $\mathbf{G}$ (square data points) in terms of the number of atoms in $n\times n\times 2$ nanoplatelets. The values of $\mathbf{G}$ for platelets in which from 1 to 3 pairs of Ge–Te atoms are cut at each corner (rounder cross section) are also reported (triangular data points). A top view of the local dipole moments ${\mathbf{p}}_i$ (arrows) is shown in the inset for $n=8$ and $n=16$. (b) The variation of internal-energy double-well depth, $\Delta E_{\mathrm{DW}}$ in terms of the number of atoms in $n\times n\times 2$ nanoplatelets. The dependence of double-well on $\mathbf{G}$ for $16\times 16\times 2$ is given as the inset. (c) Evolution of $\mathbf{G}$ for $10\times 10\times m$ ($m=2$ to 10) (square data points) and $12\times 12\times m$ ($m=2$ to 8) (triangular data points) nanoparticles.