Realization of strain induced multiple topological phases in Cu$_2$SnS$_3$: An $ab$-$initio$ study

Abstract

The search of multiple topological phases (TPs) and their transitions by tuning different parameters through chemical substitutions, electric field, magnetic field, strain and Floquet engineering, etc has garnered a widespread attention in recent time. In spite of great effort, the observations of multiple TPs in a single material and multiple TP transitions in the presence of one parameter remain elusive. Here we demonstrate the presence of multiple TPs and their transitions with uniaxial compressive strain (UCS) in orthorhombic Cu$_2$SnS$_3$ by using $state$-$of$-$the$-$art$ $ab$-$initio$ calculations. In the absence of spin-orbit coupling (SOC), the Cu$_2$SnS$_3$ exhibits a single (type-II) nodal-ring and in the presence of SOC, it hosts Weyl phase with seven Weyl points (three at $Γ$ and four at general positions) along with nodal arcs. On the application of UCS, it remains type-II nodal-ring $<5.5$\%, which further evolves into type-III nodal-ring for $5.5\% \leq$ UCS $<5.6$\%. Interestingly, at 5.6\% of UCS, it shows Weyl phase with four Weyl nodes even in the absence of SOC. All the above-mentioned seven Weyl points persist below $5$\% of UCS. For 5\% $\leq$ UCS $<5.6$\%, four Weyl points (at general positions) disappear and nodal-arcs remain intact in all the studied range of UCS. The TPs observed in the absence of SOC appears to arise due to the presence of strain driven topological flat band, which is typically reported to be seen in kagome and Lieb lattices.

Figures (4)

• Figure 1: (Color online) (a) The crystal structure of Cu$_2$SnS$_3$. The blue, gray, and yellow balls denote Cu, Sn, and S atoms, respectively. Here a, b, and c correspond to the $x$-, $y$-, and $z$-axes, respectively. (b) The conventional Brillouin zone of Cu$_2$SnS$_3$.
• Figure 2: (Color online) Evolution of bulk band structure of Cu$_2$SnS$_3$ along high-symmetry points X-$\Gamma$-A in the first BZ under uniaxial strains in different magnitudes when SOC is excluded. The negative sign in the figures stands uniaxial compressive strain (UCS) along the $a$ axis. The red, green and blue curves represent the band number 1, 2 and 3, respectively. Zero energy represents the Fermi level. (a) and (c) plots show the energy dispersions at 0% and 5.5% UCS, respectively. The subplot of (a) and (c) show the energy-gap bwtween band 2 and 3 along X-$\Gamma$ direction. We show the obtained topological nodal line in (a) and (c) at different magnitudes of strain in the first BZ apart from the X-$\Gamma$-A direction. (b)((d)) 3D representation of the energy (E) dispersions at 0 (-5.5)% strain around type-II (type-III) nodal line in the $k_x$-$k_z$ plane.
• Figure 3: (Color online) Evolution of bulk band structure of Cu$_2$SnS$_3$ along high-symmetry direction X-$\Gamma$-A in the first BZ under uniaxial strains in different magnitudes when SOC is included. The negative sign in the figures stands UCS along the $a$ axis. The black, red, green blue, magenta and brown curves represent the band number 1, $1^\prime$, 2, $2^\prime$, 3 and $3^\prime$, respectively. (a) and (b) plots show the energy dispersions at 0% and 5.5% UCS. We show the zoom-up of the touching of different combinations of bands. The chiralities (C) for each band are also indicated in the plots. The coordinates of the Weyl point between bands $2^\prime$ and 3 are ($\pm\frac{2\pi}{a}$(0.226), $\pm\frac{2\pi}{b}$(0.004), 0.000), and the energy dispersion around the obtained Weyl point is shown in subplot (a). We also show the nodal arc in the subplot of Fig. (a) and (b).
• Figure 4: (Color online) Surface density of states for the (001) surface (a) band 1-$1^\prime$, (b) band 2-$2^\prime$, and (c) band 3-$3^\prime$ of Cu$_2$SnS$_3$ at 0% strain along two dimensional high-symmetric directions. (d)-(f) plot show non-trivial topological Fermi arcs of Cu$_2$SnS$_3$ of (001) surface for band 1-$1^\prime$ at energy -185 meV, band 2-$2^\prime$ at energy -104.5 meV and band 3-$3^\prime$ at energy 35.5 meV for the top surface of the unit cell.