^{1}, Brad D. Malone

^{1}and Efthimios Kaxiras

^{1,2,a)}

### Abstract

SnS is a metal monochalcogenide suitable for use as absorber material in thin film photovoltaic cells. Its structure is an orthorhombic crystal of weakly coupled layers, each layer consisting of strongly bonded Sn-S units. We use first-principles calculations to study model single-layer, double-layer, and bulk structures of SnS in order to elucidate its electronic structure. We find that the optoelectronic properties of the material can vary significantly with respect to the number of layers and the separation between them: the calculated band gap is wider for fewer layers (2.72 eV, 1.57 eV, and 1.07 eV for single-layer, double-layer, and bulk SnS, respectively) and increases with tensile strain along the layer stacking direction (by ∼55 meV/1% strain).

The authors wish to thank Prasert Sinsermsuksakul and Roy G. Gordon for the helpful discussions. Computations were performed on the Odyssey cluster, supported by the FAS Science Division Research Computing Group at Harvard University, the Extreme Science and Engineering Discovery Environment (XSEDE), which is supported by National Science Foundation grant number OCI-1053575, and the SEAS HPC cluster, supported by the Academic Computing Group.

I. INTRODUCTION

II. MODELS AND METHODS

III. RESULTS AND DISCUSSION

IV. CONCLUSIONS

### Key Topics

- Band gap
- 48.0
- II-VI semiconductors
- 47.0
- Materials properties
- 15.0
- Valence bands
- 12.0
- Solar cells
- 11.0

## Figures

Structural model of orthorhombic tin sulfide (SnS). Gray and yellow balls represent Sn and S, respectively. (a) Unit cell, repeated twice in the bc-plane, and (b) corresponding Brillouin zone.

Structural model of orthorhombic tin sulfide (SnS). Gray and yellow balls represent Sn and S, respectively. (a) Unit cell, repeated twice in the bc-plane, and (b) corresponding Brillouin zone.

Band structure for model (a) single-layer, (b) double-layer and (c) bulk SnS, and [(d)–(f)] corresponding Brillouin zone and constant energy surfaces close to the edge of the conduction (purple) and valence (yellow) band. In [(a)–(c)] the band structure is shown along the path that traces the edge of the Brillouin zone in a clock-wise fashion. At each k-point the energy eigenvalues are color-coded with respect to the relative contribution of the s- (red), p- (blue), and d- (green) levels of Sn (top panel), and s- and p-levels of S (bottom panel), and the symbol size is proportional to the relative total contribution of the element. The shaded horizontal strip delimits the band gap. The lowest-energy level of each band structure is shifted to −17 eV (level is not shown).

Band structure for model (a) single-layer, (b) double-layer and (c) bulk SnS, and [(d)–(f)] corresponding Brillouin zone and constant energy surfaces close to the edge of the conduction (purple) and valence (yellow) band. In [(a)–(c)] the band structure is shown along the path that traces the edge of the Brillouin zone in a clock-wise fashion. At each k-point the energy eigenvalues are color-coded with respect to the relative contribution of the s- (red), p- (blue), and d- (green) levels of Sn (top panel), and s- and p-levels of S (bottom panel), and the symbol size is proportional to the relative total contribution of the element. The shaded horizontal strip delimits the band gap. The lowest-energy level of each band structure is shifted to −17 eV (level is not shown).

Maximally-localized Wannier functions constructed from the occupied valence states in bulk SnS, describing (a) Sn lone-pair states of which there are 4, one for each Sn atoms, (b) bonding states of which are 12, three for each Sn-S pair, and (c) states localized around the S atoms, of which there are 4. (d) Projection of the Wannier functions onto the Bloch band structure. At each k-point, the energy eigenvalues are color-coded with respect to the relative contribution of the Sn lone-pair states (red), and the bonding and S-states states (blue).

Maximally-localized Wannier functions constructed from the occupied valence states in bulk SnS, describing (a) Sn lone-pair states of which there are 4, one for each Sn atoms, (b) bonding states of which are 12, three for each Sn-S pair, and (c) states localized around the S atoms, of which there are 4. (d) Projection of the Wannier functions onto the Bloch band structure. At each k-point, the energy eigenvalues are color-coded with respect to the relative contribution of the Sn lone-pair states (red), and the bonding and S-states states (blue).

Minimum band gap Eg for (a) direct and (b) indirect transitions and associated band extrema in bulk (filled circles) and double-layer (empty circles) SnS with respect to strain, , applied along the layer stacking direction. Negative and positive values of correspond to compressive and tensile strain, respectively, ( = 0 for the unstrained SnS). The horizontal dashed line marks the band gap of single-layer SnS.

Minimum band gap Eg for (a) direct and (b) indirect transitions and associated band extrema in bulk (filled circles) and double-layer (empty circles) SnS with respect to strain, , applied along the layer stacking direction. Negative and positive values of correspond to compressive and tensile strain, respectively, ( = 0 for the unstrained SnS). The horizontal dashed line marks the band gap of single-layer SnS.

Absorption coefficient, , as a function of photon energy, , for the model single-layer, double-layer, and bulk structures of SnS. Solid, dashed, and dotted lines correspond to unstrained structures, and structures under 5% compressive and 5% tensile strain along the layer stacking direction (a-axis in Fig. 1 ).

Absorption coefficient, , as a function of photon energy, , for the model single-layer, double-layer, and bulk structures of SnS. Solid, dashed, and dotted lines correspond to unstrained structures, and structures under 5% compressive and 5% tensile strain along the layer stacking direction (a-axis in Fig. 1 ).

Imaginary part of the dielectric function, , as a function of photon energy for the model bulk (top) and double-layer (bottom) structure of SnS with light polarized in the direction of each structural axis. Solid, dashed, and dotted lines correspond to unstrained structures, and structures under 5% compressive and 5% tensile strain along the layer stacking direction (a-axis in Fig. 1 ).

Imaginary part of the dielectric function, , as a function of photon energy for the model bulk (top) and double-layer (bottom) structure of SnS with light polarized in the direction of each structural axis. Solid, dashed, and dotted lines correspond to unstrained structures, and structures under 5% compressive and 5% tensile strain along the layer stacking direction (a-axis in Fig. 1 ).

## Tables

Kohn-Sham band gap (EKS), derivative discontinuity ( ), indirect ( ), and direct ( ) band gaps, all in eV, and the electron ( ) and hole ( ) effective masses in units of bare electron mass at k-points involved in indirect transitions along the b- and c-direction (see also Fig. 2 ).

Kohn-Sham band gap (EKS), derivative discontinuity ( ), indirect ( ), and direct ( ) band gaps, all in eV, and the electron ( ) and hole ( ) effective masses in units of bare electron mass at k-points involved in indirect transitions along the b- and c-direction (see also Fig. 2 ).

Static dielectric constants for the model single-layer, double-layer, and bulk structures of SnS, obtained within the random phase approximation and the bootstrap approximation.

Static dielectric constants for the model single-layer, double-layer, and bulk structures of SnS, obtained within the random phase approximation and the bootstrap approximation.

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