Symmetry breaking, strain solitons, and mechanical edge modes in monolayer antimony

Two-dimensional materials exhibit a variety of mechanical instabilities accompanied by spontaneous symmetry breaking. Here, we develop a continuum description of the buckling instability of antimonene sheets. Regions of oppositely directed buckling constitute domains separated by domain walls that are solitons in our model. Perturbations about equilibrium propagate as waves with a gapped dispersion in the bulk, but there is a gapless mode with linear dispersion that propagates along the domain walls in a manner reminiscent of the electronic modes of topological insulators. We establish that monolayer antimonene is a mechanical topological insulator by demonstrating a mapping between our continuum model and an underlying Dirac equation of the symmetry class BDI, which is known to be a topological insulator in one dimension and a weak topological insulator in two dimensions. Monolayer antimony can be produced by exfoliation as well as epitaxy, and the effects predicted in this paper should be accessible to standard experimental tools such as scanning probe microscopy and Raman spectroscopy. We surmise that the effects studied here (namely, low-scale symmetry breaking, strain solitons, and gapless edge modes) are not limited to antimonene but are common features of two-dimensional materials.

Figure 1

The atoms of antimony form a honeycomb pattern when the sheet is viewed from above (upper diagram). The A sublattice atoms are shown dark and the B sublattice atoms are light (blue and green in color, respectively). The buckling of the sheet wherein the B sublattice is displaced upward relative to the A sublattice is shown in a perspective view in the lower diagram.

Figure 2

The domain-wall profile ${\zeta }_W\left(x\right)$ plotted in natural units [solid blue curve; see Eq. (6)]. Also shown are the zero mode function ${\psi }_{\mathrm{zero}}\left(x\right)$ [dotted black curve; see Eq. (14)]; the potential $U\left(x\right)$, of which the zero mode is the ground state [dashed orange curve; see Eq. (13)]; and the potential $W\left(x\right)$, the superpartner of $U\left(x\right)$ [dot-dashed red curve; see Eq. (17)].

Figure 3

The zero mode has a linear dispersion, $\omega =\left|k\right|$ in natural units, where $k$ is the wave vector along the domain wall (solid blue curve). Also shown is the dispersion ${\omega }_{\mathrm{ex}}\left(k\right)$ for the excited mode that is localized along the wall [dashed red curve; see Eq. (18)] and the bulk modes [dotted black curve; see Eq. (21)]. As shown in Eq. (21) the bulk mode frequency depends on $p$, the component of the wave vector normal to the domain wall, as well as $k$, the component along the wall, but the dependence is isotropic. The plot is for the case $p=0$. Note that in natural units $c=1$ and ${\mathrm{\Omega }}^2=2$ in Eq. (21).