## Introduction

Atomically thin two-dimensional (2D) materials are nowadays the subject of intense fundamental and technological research1,2,3,4,5,6. Due to the reduced dimensionality and the weak dielectric screening, their opto-electronic response, even at room temperature, is dominated by strongly bound electron-hole pairs, called excitons7. Among a plethora of semiconducting 2D materials, the family of group VI transition metal dichalcogenides (TMDs) is most attractive for very high absorbance and optical bandgaps ranging from the visible to the near-infrared region of the electromagnetic spectrum8,9. TMDs thus are promising candidates for next generation opto-electronic and photonic devices, such as photovoltaic solar cells10,11, photodetectors12,13, and light-emitting diodes14. However, the opto-electronic performance of TMDs is affected by spin and momentum-forbidden dark excitons. The former originate from the strong spin-orbit coupling at the K and K′ points15,16, the latter are mainly due to the presence of lateral valleys at the Λ (or Q) point, midway between the Γ and K points of the Brillouin zone17. Their energy ordering relative to the bright direct K-K excitons has a profound impact on the photoluminescence efficiency. A spin-forbidden dark exciton as lowest-energy state, as in WS2, and finite momentum excitons energetically close to the bright exciton, as in MoS2, introduce additional non-radiative decay channels that hamper the photoluminescence quantum yield17.

The 2D semiconducting material MoSi2N4 recently has been synthesized by chemical vapor deposition18. It has a layered structure with P$$\overline{6}$$m2 space group, in which an inner MoN2 layer is coordinated to two outer SiN layers, compare Fig. 1. It exhibits an indirect band gap, a high strength (66 GPa), and remarkable ambient stability. Theoretical calculations based on the deformation potential approach predict high intrinsic electron and hole mobilities of 270 cm2 V−1s−1 and 1200 cm2 V−1s−1, respectively18,19, which result in a high on/off ratio of 4000 at 77 K in a back-gated field-effect transistor. The carrier type and bandgap of MoSi2N4 can be easily manipulated by doping20. Interestingly, other 2D materials with the general formula MoSi2X4 and the same structure as MoSi2N4 can be obtained by elemental substitution and exhibit versatile properties21. The indirect bandgap becomes direct after replacing the N atoms by P or As atoms22,23. The absence of inversion symmetry in the structure of MoSi2X4 allows for valley polarization by optical pumping with circularly polarized light22,24, making the materials suitable for application in spintronics and valleytronics21. Several members of the family exhibit high piezoelectric coefficients and are predicted to show high lattice thermal conductivity, comparable with Cu25.

Despite the appeal of the materials, theoretical investigations in the literature have focused on the electronic and optical properties22,26,27 mainly within the limited density functional theory (DFT) approach. It is known that DFT fails at properly predicting both the electronic bandgap of semiconductors and the light absorption spectra. In particular, DFT intrinsically misses the excitonic features that are essential in low-dimensional materials. For this reason, in this work, we study MoSi2X4 (X = N, P, or As) employing state-of-the-art many-body perturbation theory techniques, to disclose the excitonic nature of the optical properties, which inevitably must be taken into account to provide thorough insights into the basic properties of these materials and reliable guidance to future experimental works.

## Results and discussion

### Structure and electronic properties

Figure 1 shows the relaxed atomic structures of MoSi2N4, MoSi2P4, and MoSi2As4. Their optimized lattice constants are 2.91, 3.47, and 3.62 Å, respectively, in agreement with previous reports18,27. The PBE and G0W0 band structures are reported in Fig. 2. We confirm that MoSi2N4 is an indirect bandgap semiconductor both at the PBE and G0W0 levels, with the valence band maximum (VBM) located at the Γ point and the conduction band minimum (CBM) located at the K point. Both the valence and conduction band edges are split due to spin-orbit coupling and lack of inversion symmetry. The valence band edge at the K point shows a splitting of 158 (141) meV at the G0W0 (PBE) level, while the conduction band edge shows a much smaller splitting of 10 (3) meV. The G0W0 bandgap is 2.79 eV, i.e., 1.02 eV larger than the PBE bandgap. The direct gap at the K point is 2.96 eV, i.e., 0.18 eV larger than the indirect bandgap. In MoSi2P4 and MoSi2As4 the bandgaps turn direct with the VBM shifting to the K point. The splittings at the VBM and CBM are much larger than for MoSi2N4, specifically, 158 (139) and 14 (4) meV for MoSi2P4 and 203 (181) and 28 (16) meV for MoSi2As4 at the G0W0 (PBE) level due to the heavier masses of P and As. The G0W0 bandgap is 1.17 eV for MoSi2P4 and 1.01 eV for MoSi2As4. As a comparison, the HSE06 hybrid functional27 underestimates the direct bandgap with respect to the G0W0 bandgap by about 0.35 eV for MoSi2P4 and MoSi2As4 and by 0.65 eV for MoSi2N4. This fact has sizeable implications on the optical properties of the materials. It is worth mentioning that the G0W0 bandgaps of MoSi2X4 are inversely proportional to the atomic number of the X element, as explained by the decreasing electronegativity.

Figure 2 also displays a first significant difference from traditional TMDs. The absence of lateral valleys (e.g., the Λ valleys in TMDs) along the Γ-K high-symmetry direction. Due to the absence of Λ valleys close in energy to the K valleys, excitons have fewer non-radiative recombination channels in MoSi2X4 than in TMDs, hinting at a less suppressed quantum yield and high efficiencies in optoelectronic devices. It recently has been proven that non-radiative recombination channels can be efficiently suppressed in TMDs by electrostatic doping28 and strain29. We anticipate that the same approaches can be applied to further enhance the quantum yield of MoSi2X4 (X = N, P, or As).

### Optical properties

In the upper panels of Fig. 3 we report the absorption spectra computed at the BSE level. The absorption onset falls in the visible region for MoSi2N4, at 2.31 eV, and in the near infrared region at 0.77 and 0.71 eV for MoSi2P4 and MoSi2As4, respectively. The overlap of the absorption spectra with the incident AM1.5G solar flux Φs(ω) (shaded gray region in Fig. 3), while limited for MoSi2N4, is larger for MoSi2P4 and MoSi2As4. To quantify the theoretical efficiency of the absorption of solar light we compute the maximum short-circuit current density as $${J}_{{{{\rm{sc}}}}}^{\max }=e\int \,d\omega A(\omega ){\Phi }_{s}(\omega ),$$ where A(ω) is the absorbance11. The value of $${J}_{{{{\rm{sc}}}}}^{\max }$$ is only 0.7 mA/cm2 in the case of MoSi2N4, while MoSi2P4 and MoSi2As4 exhibit large values of 3.6 and 4.5 mA/cm2, respectively, because of the lower absorption onset. These values are comparable with well-studied TMDs30, such as MoS2 (3.9 mA/cm2) and MoSe2 (4.6 mA/cm2), indicating great potential of MoSi2X4 in photovoltaic applications.

### Exciton fine structure

Selected absorption peaks A, B, $$\widetilde{{{{\rm{B}}}}}$$, and C are marked in the upper panels of Fig. 3, with the electronic transitions taking place at the k-points correspondingly highlighted in the lower panels. Similarly to TMDs, peaks A and B are due to well-defined excitons with contributions from transitions at the K and K’ points, reflecting the band splitting induced by the spin-orbit coupling. As shown in more detail for MoSi2P4 in Fig. 4a–c, excitons A and B originate from the allowed transitions between same-spin bands, namely VBM → CBM for exciton A and VBM−1 → CBM+1 for exciton B. The binding energy of exciton A is 0.68 eV in MoSi2N4 and 0.44 eV in both MoSi2P4 and MoSi2As4, i.e., smaller than the binding energy of the lowest-energy bright exciton of a MoS2 monolayer, estimated to be larger than 0.70 eV at the same level of theory31,32. Excitons A and B in the case of MoSi2N4 are located at 2.31 and 2.45 eV, respectively, in agreement with the experimental values (2.21 and 2.35 eV)18. The 0.1 eV blueshift of the computed exciton energies with respect to the experimental values can be attributed to three main effects that were not included in the calculations: the presence of a substrate, which has been shown to reduce the optical gap of single-layer TMDs by up to 0.1 eV33,34, temperature effects, which already at 0 K cause a redshift of 75 meV in single-layer MoS2 due to zero-point renormalization35, and tensile strain, which has a strong impact on the band structure of MoSi2N4 already at small magnitudes (see Supplementary Fig. 4).

Spin-forbidden transitions between opposite-spin bands at the K and K′ points give rise to the dark exciton AD with zero dipole moment for in-plane light polarization. Above peaks A and B, it is possible to identify in Fig. 3 several peaks with lower intensities. As a representative, we select a peak $$\widetilde{{{{\rm{B}}}}}$$, which is due to VBM−1 → CBM+1 transitions (as peak B) taking place near the K and K’ points (Fig. 4). These transitions are slightly more energetic and less intense, as they occur only in the vicinity of the high-symmetry K and K′ points.

Finally, we find in Fig. 3 an intense absorption peak at higher energy, well above the G0W0 direct bandgap, which originates from direct transitions (VBM → CBM) within the Brillouin zone (lower panels of Fig. 3). We mark it as C to suggest a possible analogy with the intense peaks that are found in TMDs and are mainly given by similar transitions at k-points intermediate between Γ and K. In the case of the TMDs these transitions take place in regions of the Brillouin zone where the valence and conduction bands are almost parallel, i.e., ‘nested’36. In such conditions, photo-generated electrons and holes relax with opposite momenta to their closest band extrema, i.e., to the Λ valley and Γ hill, respectively. Hence, photo-generated carriers remain separated in momentum space, and the quantum yield drops significantly37. This does not happen for MoSi2P4 and MoSi2As4 according to Fig. 4, since no lateral valley is interposed between the transitions and the band extrema at the K and K’ points. In Fig. 5 we show a schematic of the electron-hole relaxation mechanisms in (a) MoS2 and (b) MoSi2P4. In MoS2 the excited electrons from peak C quickly relax via intraband scattering to the Λ valley between the Γ and K points (dashed red arrow). Simultaneously, the excited holes relax to the Γ hill (blue dashed arrow). Once relaxed, the charge carriers can follow phonon-mediated radiative recombination (dot-dashed green arrow), or non-radiative recombination, or phonon scattering to the K and K′ points (dotted red and blue arrows). Only a relatively small fraction of the photo-excited carriers reaches the K and K′ points and recombines radiatively, with moderate yield37. In the cases of MoSi2P4 and MoSi2As4, see Fig. 5b, the photo-generated electrons and holes are statistically favored to quickly relax to the closest K or K’ points, via interband scattering (red and blue dashed arrows). It is not possible to fully exclude relaxation paths leading holes to the Γ hill, however they are disfavored by the longer interband scattering chain.

Band nesting in TMDs results in a large enhancement of the joint density of states (jDOS) and, in turn, in an increased absorbance. In MoSi2P4 and MoSi2As4 the large intensity of peak C can be similarly traced back to the jDOS, see Fig. 6a. However, this enhancement is not related to band nesting, see Fig. 4, but rather to the reduced curvature of the valence and conduction bands in the region where the electronic transitions contributing to peak C occur. In MoSi2N4 the intense peak C is mainly due to contributions from electronic transitions in the vicinity of K/K’, as well as at Γ (see the lower panels of Figs. 3 and 4d). The band structure of MoSi2N4 shows flat conduction and valence bands around Γ, which enhance the jDOS and, consequently, the intensity, see Fig. 6b. This feature is absent in MoSi2P4 and MoSi2As4.

As mentioned, the energetic ordering of excitons AD and A is crucial for the optoelectronic performance. We have verified that in our calculations the bright-dark splitting $$\Delta \equiv {E}_{{{{\rm{A}}}}}-{E}_{{{{{\rm{A}}}}}_{{{{\rm{D}}}}}}$$ is converged within 2 meV (see Supplementary Fig. 2c). It is important to notice that, while in the case of MoSi2N4 the dark exciton AD is the lowest-energy state, with a bright-dark splitting Δ = 9 meV, the ordering is reversed in the cases of MoSi2P4 and of MoSi2As4, where we have Δ = −3 and −13 meV, respectively (lower panels of Fig. 3). This is the result of interplay between the splitting of the conduction band due to the spin-orbit coupling and the exchange interaction between the electron and hole in the exciton32,38. Indeed, in all three studied materials the band splitting provides a negative contribution to Δ, since the opposite-spin bands involved in exciton AD are the VBM and CBM+1, implying that the electronic transitions contributing to exciton A (VBM → CBM) are less energetic than those contributing to exciton AD. On the contrary, the exchange interaction always contributes positively to Δ, lowering the energy of the dark exciton with respect to the bright exciton. The contribution of the exchange interaction to Δ is found to be between 12 meV (MoSi2P4) and 18 meV (MoSi2N4), being large enough to overcome the band splitting due to the spin-orbit coupling only in the case of MoSi2N4 (positive Δ). The negative Δ implies that MoSi2P4 and MoSi2As4 will show an enhanced quantum yield at low temperatures, unlike TMDs such as WS2, WSe239, and MoS240,41. In the case of MoSi2As4, due to the large Δ = −13 meV, the quantum yield is expected to be only moderately suppressed even at room temperature, unlike MoSe2 with Δ = −1 meV16,42. The energetic ordering of the excitons and bright-dark splitting represent the second main specificity of the materials under consideration with respect to traditional TMDs.

The lower panels of Fig. 3 also give the intrinsic radiative lifetimes of excitons A and B at 0 K, $${\tau }_{{{{\rm{A}}}},{{{\rm{B}}}}}^{0}$$, which all fall in the range of tenths of ps, in line with the values obtained for 2D TMDs43. As shown in Refs. 43,44,45, in a 2D material the finite temperature radiative lifetime 〈τX〉(T) of exciton X is proportional to $${k}_{{{{\rm{B}}}}}T{M}_{{{{\rm{X}}}}}^{* }/{E}_{{{{\rm{X}}}}}^{2}$$, where $${M}_{{{{\rm{X}}}}}^{* }$$ and EX are the effective mass and energy of the exciton at zero momentum, respectively. We report in Table 1 the effective masses of the electrons, holes, and excitons, together with the finite temperature exciton radiative lifetimes calculated at T = 4 and 300 K.

The values obtained for MoSi2N4 are slightly larger than those of MoS2, MoSe2, and WSe246. Conversely, the values obtained for MoSi2P4 and MoSi2As4 are about two orders of magnitude larger. This is primarily due to the combination of low energy (EA,B < 1 eV) and high effective mass of the excitons, which enhances the temperature-dependent factor in 〈τ〉(T). Long radiative lifetimes are often associated with exciton localization due to defects or strain, as observed for TMD monolayers and bilayers47,48, or with interlayer excitons in 2D van der Waals heterostructures of TMDs49,50,51.

We plot in Fig. 7 the square moduli of the wavefunctions of excitons A, B, $$\widetilde{{{{\rm{B}}}}}$$, and C of MoSi2P4. Corresponding results for MoSi2N4 and MoSi2As4 are reported in Supplementary Figs. 5 and 6, respectively. The hole (white diamond) is kept fixed where the valence band edge has the highest probability density. In all cases, excitons A and B both have a circular shape around the hole. The Bohr radii of excitons A and B are estimated to be, respectively, aA = 1.79 nm and aB = 1.76 nm in MoSi2P4, 0.98 nm and 0.96 nm in MoSi2N4, and 1.65 nm and 1.64 nm in MoSi2As4, in line with the values found in TMDs52,53,54. Exciton $$\widetilde{{{{\rm{B}}}}}$$ is more delocalized, and exciton C spreads along the zigzag direction. The side views in Fig. 7 show that excitons A, B, and $$\widetilde{{{{\rm{B}}}}}$$ are concentrated on the inner MoSi2 layer, whereas exciton C extends to the outer SiP layers. Exciton C shows a similar delocalization in MoSi2As4, while it does not extend to the outer SiN layers in MoSi2N4.

Exciton-exciton annihilation (strong exciton Auger recombination) is a key technological bottleneck to obtain high quantum yield in TMDs, and the less localized C exciton may enhance the annihilation rate by increasing the annihilation radius55,56,57. Lee et al.58 have employed ultra-flat Au substrates with atomically controlled h-BN spacers to suppress the exciton-exciton interaction and improve the quantum yield of TMDs by an order of magnitude. In MoSi2X4 layers, due to the contribution of the excitons in the inner MoX2 layer, the outer SiX layers can act as the h-BN spacers. Thus, chemical vapor deposition of MoSi2X4 on a metallic substrate may directly lead to highly efficient optoelectronic devices.

We provide a state-of-the-art study of the electronic and optical properties of 2D MoSi2X4 based on many-body perturbation theory. The G0W0 approximation overcomes the drawbacks of DFT, being superior to hybrid-functionals, to achieve reliable electronic band structures and bandgaps. The bandgap of MoSi2N4 is indirect, while the bandgaps of MoSi2P4 and MoSi2As4 are direct at the K point with a bandgap value of around 1 eV. Unlike traditional TMDs, they have no lateral valleys close in energy to the CBM. By solving the Bethe-Salpeter equation, we obtain absorption spectra displaying excitonic features. The absorption onset falls into the visible region for MoSi2N4, and into the near-infrared region for MoSi2P4 and MoSi2As4, resulting, together with a strong light-carrier coupling, in a significant absorption of the solar flux and in maximum short-circuit current densities comparable to those of TMDs. The analysis of the exciton fine-structure reveals a crucial interplay between the band splitting induced by the spin-orbit coupling and the exchange interaction within the exciton, which results in a bright lowest-energy exciton for both MoSi2P4 and MoSi2As4. The negative bright-dark splitting is particularly large (−13 meV) for MoSi2As4. Our detailed analysis suggests that the materials under investigation provide remarkable quantum yields, especially MoSi2As4. Absence of lateral valleys, negative bright-dark splitting, and long exciton radiative lifetime give rise to high performance optoelectronic materials.

## Methods

We perform DFT calculations within the plane-wave expansion using the Quantum Espresso package59. The gradient-corrected Perdew–Burke–Ernzerhof (PBE) functional60 is employed together with fully relativistic norm-conserving pseudopotentials61 that account for the spin-orbit coupling. We consider as semicore states the Mo 4s and 4p orbitals, as well as the As 3d orbitals. Convergence is achieved with an 80 Ry kinetic energy cutoff and a uniform 24 × 24 × 1 Monkhorst-Pack k-point mesh62. A 15 Å vacuum region along the c-direction, the direction perpendicular to the layer plane, is introduced to ensure decoupling of periodic replicas. The structure relaxation is assumed to have reached convergence when the maximum component of the residual ionic forces is smaller than 10−8 Ry/Bohr.

Building up on the obtained DFT results, we perform many-body perturbation theory calculations within the G0W0 approximation using the Yambo code63,64. A box cutoff along the c-direction is applied to the bare Coulomb potential. The inverse dielectric matrix, $${\varepsilon }_{{{{\bf{G}}}}{{{\bf{G}}}}^{\prime} }^{-1}$$, is obtained within the plasmon-pole approximation64. We adopt the quasiparticle bandgap value at the K point as convergence parameter and fix a convergence threshold of 50 meV. Following this criterion, we employ a 36 × 36 × 1 k-point mesh, 300 bands and a 10 Ry cutoff for the G vectors in $${\varepsilon }_{{{{\bf{G}}}}{{{\bf{G}}}}^{\prime} }^{-1}$$ and 240 bands for the computation of the correlation self-energy. In order to speed up the convergence with respect to empty states we adopted the technique described in Ref. 65 as implemented in the Yambo code64.

The quasiparticle band structures are then used to build and solve the Bethe-Salpeter equation on a 36 × 36 × 1 k-point mesh, including eight valence and ten conduction bands in the excitonic Hamiltonian (compare the Supplementary Information).