Strain-induced indirect-to-direct band-gap transition in bulk ${\mathrm{SnS}}_2$

While ${\mathrm{SnS}}_2$ is an earth-abundant large-band-gap semiconductor material, the indirect nature of the band gap limits its applications in light harvesting or detection devices. Here, using density functional theory in combination with the many-body perturbation theory, we report indirect-to-direct band-gap transition in bulk ${\mathrm{SnS}}_2$ under moderate, $2.98\%$ uniform biaxial tensile (BT) strain. Further enhancement of the BT strain up to $9.75\%$ leads to a semiconductor-to-metal transition. The strain-induced weakening of the interaction of the in-plane orbitals modifies the dispersion as well as the character of the valence- and the conduction-band edges, leading to the transition. A quasiparticle direct band gap of 2.17 eV at the $\mathrm{\Gamma }$ point is obtained at $2.98\%$ BT strain. By solving the Bethe-Salpeter equation to include excitonic effects on top of the partially self-consistent ${\mathrm{GW}}_0$ calculation, we study the dielectric functions, optical oscillator strength, and exciton binding energy as a function of the applied strain. At $2.98\%$ BT strain, our calculations show the relatively high exciton binding energy of 170 meV, implying strongly coupled excitons in ${\mathrm{SnS}}_2$. The effect of strain on vibrational properties, including Raman spectra, is also investigated. The Raman shift of both in-plane ($E_{2g}^1$) and out-of plane ($A_{1g}$) modes decreases with the applied BT strain, which can be probed experimentally. Furthermore, ${\mathrm{SnS}}_2$ remains dynamically stable up to $9.75\%$ BT strain, at which it becomes metallic. A strong coupling between the applied strain and the electronic and optical properties of ${\mathrm{SnS}}_2$ can significantly broaden the applications of this material in strain-detection and optoelectronic devices.

Figure 1

(a, b) Unit cell and corresponding Brillouin zone of bulk ${\mathrm{SnS}}_2$. (c–e) Electronic band structure and angular-momentum-resolved density of states at $0.0\%, 2.98\%$, and $9.75\%$ BT strain, respectively. Dashed red lines represent the Fermi level. Red circles represent the VBM and CBM.

Figure 2

Orbital-resolved band structure at $0.0\%$ and $2.98\%$ BT strain. Red circles represent the corresponding VBM and CBM in the band structures. The dashed red line represents the Fermi level.

Figure 3

Band-gap variations along the symmetric directions $\mathrm{\Gamma }-\mathrm{\Gamma }, \mathrm{\Gamma }-\mathrm{M}$, and $\mathrm{\Gamma }-\mathrm{L}$ as a function of the BT strain.

Figure 4

Atom projected band structures at $2.98\%$ BT strain showing variations in the surface atomic states with an increasing number of layers (2L to 5L) of ${\mathrm{SnS}}_2$. Red and yellow spheres represent the energy states of the surface and the inner atoms. The dashed red line represents the Fermi level.

Figure 5

(a) Dispersion of the CBM and VBM of bilayer ${\mathrm{SnS}}_2$ at $2.98\%$ BT strain. Blue spheres represent the conduction-band edges (CBEs) at the $\mathrm{\Gamma }$- and $M$-symmetric points. Valence-band edges at $\mathrm{\Gamma }$ and $\delta \mathrm{\Gamma }$ are represented by red spheres. (b) The energy difference ($\mathrm{\Delta }E$) of the CBE at $\mathrm{\Gamma }$ ($E_{\mathrm{CBE}}^{\mathrm{\Gamma }}$) and the CBM at $M$ (${\mathrm{E}}_{\mathrm{CBM}}^M$) as a function of the inverse of the layer thickness $d^{-1}$ is plotted by the blue line. $d$ is the distance between the outermost sulfur atoms in 2L-${\mathrm{SnS}}_2$, as shown in the inset. Similarly, the energy difference between the valence-band edge (VBE) at $\mathrm{\Gamma }$ ($E_{\text{VBE}}^{\mathrm{\Gamma }}$) and the VBM at $\delta \mathrm{\Gamma }$ ($E_{\text{VBM}}^{\delta \mathrm{\Gamma }}$) as a function of $d^{-1}$ is plotted by the red line. $\delta \mathrm{\Gamma }$ is the $\mathbf{k}$ point, at which the VBM of bilayer ${\mathrm{SnS}}_2$ lies.

Figure 6

(a–c) Phonon dispersion corresponding to the strain: $0.0\%, 2.98\%$, and $9.75\%$. (d) Raman shift observed in the in-plane ($E_{2g}^1$) and out-of-plane ($A_{1g}$) modes with the BT strain. The modes of vibrations of Sn (blue) and S (yellow) atoms are shown.

Figure 7

Imaginary part of the dielectric constant $\left[{\varepsilon }_2\left(\omega \right)\right]$ shown by red lines at (a) $0.0\%$ and (b) $2.98\%$ BT strain. Vertical blue lines represent the oscillator strength. Dashed green lines indicate a ${\mathrm{GW}}_0$ band gap of 2.82 eV (at $\mathrm{\Gamma }$, without including the phonon-assisted effect) and 2.17 eV at $0.0\%$ and $2.98\%$ strain, respectively. Dashed black lines represent the optical gaps, which are 2.66 and 2.0 eV at $0.0\%$ and $2.98\%$ BT strain, respectively. The difference between the position of the green and that of the black lines gives the excitonic binding energy.

Figure 8

Reflectivity $R(\omega$) and absorption coefficient $\alpha (\omega$) as a function of the photon energy (eV), where red and blue lines correspond to $0.0\%$ and $2.98\%$ BT strain.