Excitons and light-emission in semiconducting MoSi₂X₄ two-dimensional materials

Abstract

Semiconducting two-dimensional materials with chemical formula MoSi₂X₄ (X = N, P, or As) are studied by means of atomistic ground- and excited-state first-principles simulations. Full-fledged quasi-particle bandstructures within the G₀W₀ approach substantially correct the electronic bandgaps previously obtained with hybrid-functional density functional theory and highlight the absence of lateral valleys close in energy to the conduction band minimum. By solving the Bethe–Salpeter equation, we show that the optical properties are dominated by strongly bound excitons with the absorbance and maximum short-circuit current densities of MoSi₂P₄ and MoSi₂As₄ comparable to those of transition metal dichalcogenides. Due to the presence of the outer SiX layers, the exciton binding energies are smaller than those generally found for transition metal dichalcogenides. Long radiative lifetimes of bright excitons, over 10 ns at room temperature for MoSi₂As₄, and the absence of band-nesting are very promising for application in efficient ultra-thin optoelectronic devices.

Introduction

Atomically thin two-dimensional (2D) materials are nowadays the subject of intense fundamental and technological research^1,2,3,4,5,6. 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 spectrum^8,9. TMDs thus are promising candidates for next generation opto-electronic and photonic devices, such as photovoltaic solar cells^10,11, photodetectors^12,13, 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′ points^15,16, 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 cm²V⁻¹s⁻¹ and 1200 cm²V⁻¹s⁻¹, respectively^18,19, 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 atoms^22,23. The absence of inversion symmetry in the structure of MoSi₂X₄ allows for valley polarization by optical pumping with circularly polarized light^22,24, 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.

Fig. 1: Atomic structure.

Top a and side b views. In b the MoX₂ and SiX layers are marked. The violet, blue, and green spheres are Mo, Si, and X atoms, respectively.

Despite the appeal of the materials, theoretical investigations in the literature have focused on the electronic and optical properties^22,26,27 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 reports^18,27. 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.

Fig. 2: Band structures.

The energies refer to the VBM.

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 Φ_s(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.

Fig. 3: Absorption spectra and exciton radiative lifetimes.

Upper panels: Absorption spectra computed at the BSE level. The dashed vertical lines mark the G₀W₀ bandgaps. Lower panels: Intrinsic radiative lifetimes of excitons A and B at 0 K, and exciton weights in the Brillouin zone. A_D denotes the dark exciton.

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 theory^31,32. 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 eV^33,34, 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).

Fig. 4: Weighted band structures.

G₀W₀ bandstructure of a MoSi₂P₄ and d MoSi₂N₄ with weights of the single-particle transitions giving rise to excitons A, B, $\widetilde{{{{\rm{B}}}}}$, and C (Fig. 3). In b and c we report details of the CBM(+1) and VBM(−1), respectively, around the K point.

Spin-forbidden transitions between opposite-spin bands at the K and K′ points give rise to the dark exciton A_D 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.

Fig. 5: Relaxation.

Relaxation processes of the excited charge carriers from peak C in a MoS₂ and b MoSi₂P₄.

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.

Fig. 6: jDOS and absorbance.

jDOS per unit cell (red line) and absorbance (blue line) of a MoSi₂P₄ and b MoSi₂N₄. The jDOS is obtained from the PBE eigenvalues and only comprises transitions involving the VBM−1, VBM, CBM, and CBM+1. The absorbance is shifted to match the jDOS, as reported in each panel.

As mentioned, the energetic ordering of excitons A_D 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 A_D 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 exciton^32,38. Indeed, in all three studied materials the band splitting provides a negative contribution to Δ, since the opposite-spin bands involved in exciton A_D 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 A_D. 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 MoS₂^40,41. 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 meV^16,42. 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 〈τ_X〉(T) of exciton X is proportional to ${k}_{{{{\rm{B}}}}}T{M}_{{{{\rm{X}}}}}^{* }/{E}_{{{{\rm{X}}}}}^{2}$, where ${M}_{{{{\rm{X}}}}}^{* }$ and E_X 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.

Table 1 Effective exciton masses and finite temperature radiative lifetimes.

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 (E_A,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 bilayers^47,48, or with interlayer excitons in 2D van der Waals heterostructures of TMDs^49,50,51.

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, a_A = 1.79 nm and a_B = 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 TMDs^52,53,54. 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.

Fig. 7: Wavefunctions.

Top (upper panels) and side (lower panels) views of the square moduli of the wavefunctions of excitons A, B, $\widetilde{{{{\rm{B}}}}}$, and C in MoSi₂P₄. The white diamond marks the highest probability density of the hole.

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 radius^55,56,57. 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 code^63,64. 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).