Tuning Excitonic Properties of Monochalcogenides via Design of Janus Structures

Two-dimensional (2D) Janus structures offer a unique range of properties as a result of their symmetry breaking, resulting from the distinct chemical composition on each side of the monolayers. Here, we report a theoretical investigation of 2D Janus Q′A′AQ P3m1 monochalcogenides from group IV (A and A′ = Ge and Sn; Q, Q′ = S and Se) and 2D non-Janus QAAQ P3̅m1 counterparts. Our theoretical framework is based on density functional theory calculations combined with maximally localized Wannier functions and tight-binding parametrization to evaluate the excitonic properties. The phonon band structures exhibit exclusively real (nonimaginary) branches for all materials. Particularly, SeGeSnS has greater energetic stability than its non-Janus counterparts, representing an outstanding energetic stability among the investigated materials. However, SGeSnS and SGeSnSe have higher formation energies than the already synthesized MoSSe, making them more challenging to grow than the other investigated structures. The electronic structure analysis demonstrates that materials with Janus structures exhibit band gaps wider than those of their non-Janus counterparts, with the absolute value of the band gap predominantly determined by the core rather than the surface composition. Moreover, exciton binding energies range from 0.20 to 0.37 eV, reducing band gap values in the range of 21% to 32%. Thus, excitonic effects influence the optoelectronic properties more than the point-inversion symmetry breaking inherent in the Janus structures; however, both features are necessary to enhance the interaction between the materials and sunlight. We also found anisotropic behavior of the absorption coefficient, which was attributed to the inherent structural asymmetry of the Janus materials.


S1 Introduction
This Supplementary Information provides essential technical details that complement and support the reproduction of the published simulations.It encompasses specific simulation parameters, optimized geometries, total energy values, relevant published results, local density of states, spin-orbit coupling tests, electronic band structures, data utilized for calculating the band alignment, methodology details for calculating the Wannier functions, and visual plots illustrating the excitonic properties.These supplementary materials are intended to facilitate a comprehensive understanding of the research findings and aid fellow researchers in replicating and extending the reported results.

S2 Selected PAW Projectors: Technical Details
Table S1: Key specifications of selected PAW projectors, including Species, PAW-PBE projector name (Title), electronic valence configuration (Valence), number of valence electrons (Z val ), and maximum recommended cutoff energy (ENMAX).

Species Title
Valence Z val ENMAX (eV)

S5 Energetic and Structural Reference Values
This section presents information of the reference systems used in the evaluation of the energetic properties, such as (i) total energy of free atoms used for calculating the cohesive energy; (ii) total energies of non-Janus structures with P 3m1 space group used for calculating formation energies; and (iii) total energies of phosphorene-like (ph-like) Janus structures for evaluate relative energies.

S5.1 Free Atoms
The free-atom calculations adopt an orthorhombic box of 20 Å×21 Å×22 Å to avoid undesirable symmetry constraints, we adopt a small Gaussian smearing parameter value (σ ) to guarantees the absence of partial occupation of the atomic orbitals.Table S2 presents the total energies of free atoms that compose the investigated Janus systems.We also present the Gaussian smearing parameter (σ ), the occupation of the highest-occupied-molecular-orbital, HOMO, (OCC HOMO ), and the HOMO eigenvalue.

S5.2 Monolayers
Table S3 presents a comparison between our calculated lattice parameters and values reported in the literature, with the aim of validating our simulations.We conclude that our simulations agree with recent published data.The Atomic Simulation Environment (ASE) tool classifies the space group for each compound by adopting symprec=1 × 10 −3 Å.
The discrepancies are shorter than 0.01 Å, demonstrating that averages of non-Janus lattice patameters can be accuratelly used to determine lattice parameters for Janus monolayers.Table S4: Janus lattice parameter estimate from non-Janus systems.a QAA ′ Q ′ 0 is the optimized lattice parameter for Janus monolayers; and a QAAQ 0 is the lattice parameter for non-Janus monolayers, both obtained from stress-tensor calculations with ENCUT=520 eV and a k-mesh of 9×9×1.

S6.2 Spin-Orbit Coupling Effects
Figure S13 contrasts the electronic band structures of P3m1 Janus monolayers for plain PBE and PBE with spin-orbit coupling (PBE+SOC) approaches.Here, the spin-orbit coupling SOC is included using the fully relativistic scheme within the second-order approach in the framework of non-collinear spin density functional theory as implemented in the VASP code. 3 We found a band gap reduction of0.03 eV for SeSnSnSe, which is the system with the heaviest atoms.The highest band gap change is 0.08 eV, occurring for the SeGeSnS Janus monolayer.However, despite these small changes in the band gap, there are splittings around 0.4 eV occurring around 2 eV above the VBM for various compounds.S6.3 PBE .vs.HSE06 electronic band structures

S6.4 Workfunction
The workfunction is the photon energy necessary for remove one electron from a surface calculated as the energy difference between the vacuum energy and the Fermi level.The vacuum energy is determined from a plateau of the Hartree potential in the vacuum region (between periodic images of monolayers).Due to the non-centrosymmetric character of Janus monolayers, the two monolayer surfaces could present different workfunction vaues due to distinct vacuum energies.Figure S15 depicts the average Hartree potential in a direction perpendicular to the monolayer surfaces.Table S5 present the Fermi energy and the vacuum energies employed in the calculation of the workfunctions.
-10.0 Fermi level Table S5: Energies used for determining the Band offset and workfunctions calculated with the HSE06 exchange-correlation energy functional, namely, energies for the valence band maximum (E VBM ) and conduction band minimum (E CBM ).Vacuum levels for the ), their respective workfunctions, Φ Q and Φ Q ′ , and the workfunction variation

S7 Wannierization
For the Wannierization procedure, with HSE06 functional, we use the s and p orbital projections for Ge, S and Se, for Sn we use s,p and d projections, the MLWF-TB Hamiltonian basis set is complemented with random projections in order to obtain the same number of bands of the DFT simulation, this procedure was done with the flag use_ws_distance=.false.and guiding_centres=.true. .

S6. 1
Figure S2 shows the density of states of all systems showing the contribution of each element, while Figures S3-S12 shows the contribution of each orbital in the density of states.All density of states were calculated with the PBE exchange-correlation energy functional.

Figure S2 :
Figure S2: Density of states calculated with the PBE exchange-correlation energy functional.The Fermi level is set to zero.

Figure S3 :
Figure S3: SGeGeS non-Janus: orbital resolved density of states calculated with the PBE exchange-correlation energy functional.The Fermi level is set to zero.

Figure S4 :
Figure S4: SeGeGeSe non-Janus: orbital resolved density of states calculated with the PBE exchange-correlation energy functional.The Fermi level is set to zero.

Figure S5 :
Figure S5: SSnSnS non-Janus: orbital resolved density of states calculated with the PBE exchange-correlation energy functional.The Fermi level is set to zero.

Figure S6 :
Figure S6: SeSnSnSe non-Janus: orbital resolved density of states calculated with the PBE exchange-correlation energy functional.The Fermi level is set to zero.

Figure S7 :
Figure S7: SGeGeSe external-Janus: orbital resolved density of states calculated with the PBE exchange-correlation energy functional.The Fermi level is set to zero.

Figure S8 :
Figure S8: SSnSnSe external-Janus: orbital resolved density of states calculated with the PBE exchange-correlation energy functional.The Fermi level is set to zero.

Figure S9 :
Figure S9: SGeSnS internal-Janus: orbital resolved density of states calculated with the PBE exchange-correlation energy functional.The Fermi level is set to zero.

Figure S10 :
Figure S10: SeGeSnSe internal-Janus: orbital resolved density of states calculated with the PBE exchange-correlation energy functional.The Fermi level is set to zero.

Figure S11 :
Figure S11: SGeSnSe full-Janus: orbital resolved density of states calculated with the PBE exchange-correlation energy functional.The Fermi level is set to zero.

Figure S12 :
Figure S12: SeGeSnS full-Janus: orbital resolved density of states calculated with the PBE exchange-correlation energy functional.The Fermi level is set to zero.

Figure S13 :
Figure S13: Electronic band structures calculated with spin-orbit-coupling (SOC) in red lines, and without SOC in dashed black lines.These calculations adopts the PBE exchange-correlation energy functional, and the Fermi energy is set to zero in all graphs.

Figure S14 :
Figure S14: Contrast of the electronic band structures calculated with PBE and HSE06 exchange-correlation energy functionals.The Fermi level is set to zero.

Figure S15 :Figure S16 :
Figure S15: Internal-and external-Janus QAA ′ Q ′ compositions: average Hartree potential in the direction perpendicular to monolayer surfaces.The dashed red lines indicate the Fermi level and the blue dashed lines indicate the atomic layers for each specie.In each panel it is also shown workfunctions at the Q (Φ Q ) and Q ′ (Φ Q ′ ) surfaces in units of eV.

Figure S23 :
Figure S23: Absorption Coefficient, considering linear light polarization at x and ŷ at BSE (solid lines) and IPA (dashed lines) levels for non-Janus structures.

Figure S24 :
Figure S24: Absorption Coefficient, considering linear light polarization at x and ŷ at BSE (solid lines) and IPA (dashed lines) levels for external/internal-Janus structures.

Figure S25 :
Figure S25: Absorption Coefficient, considering linear light polarization at x and ŷ at BSE (solid lines) and IPA (dashed lines) levels for full-Janus structures.

Table S3 :
Lattice parameter and total energy for free-standing monolayers, namely, Composition; classification as non-Janus, internal-Janus, external-Janus, or full-Janus; space group; lattice parameters (a 0 ); and total energy (E tot ).The P 3m1 spacial group contains inversion center symmetry, whereas P3m1 not.
Table S4 contrast values for simulated lattice parameters of 2D Janus monolayers with chemical formula QAA ′ Q ′ with average lattice parameters for monolayers with chemical formula QAAQ and Q

Table S6 :
5arameters used for BSE simulations: k-points density, R k and their correspondent k-mesh, n v , number of valence bands, n c , number of conduction bands, in order to get all optical transitions in the solar emission spectrum range.i.e 0.5 eV to 4.0 eV and dielectric function smearing η.All simulations were done using a Coulomb truncated 2D potential (V2DT),4implemented in WanTiBEXOS package.5

Table S7 :
MLWF-TB+BSE calculated excitonic properties: fundamental band gap, E g , direct band gap, E d g , exciton ground state, Ex gs , direct exciton ground state, Ex d gs , and exciton binding energy, Ex b , obtained from E g − Ex gs .All direct excitons ground states are bright.