Strain-free MoS2/ZrGe2N4 van der Waals Heterostructure: Tunable Electronic Properties with Type-II Band Alignment

Vertically stacked van der Waals heterostructures (vdW-HS) amplify the scope of 2D materials for emerging technological applications, such as nanodevices and solar cells. Here, we present a first-principles study on the formation energy and electronic properties of the heterobilayer (HBL) MoS2/ZrGe2N4, which forms a strain-free vdW-HS thanks to the identical lattice parameters of its constituents. This system has an indirect band gap with type-II band alignment, with the highest occupied and lowest unoccupied states localized on MoS2 and ZrGe2N4, respectively. Biaxial strain, which generally reduces the band gap regardless of compression or expansion, is applied to tune the electronic properties of the HBL. A small amount of tensile strain (>1%) leads to an indirect-to-direct transition, thereby shifting the band edges at the center of the Brillouin zone and leading to optical absorption in the visible region. These results suggest the potential application of HBL MoS2/ZrGe2N4 in optoelectronic devices.


INTRODUCTION
−7 Various classes of two-dimensional (2D) sheets have been realized experimentally 8−16 and many more have been theoretically predicted. 17−37 On top of this, individual layers with different lattice parameters can give rise to complex patterns and, in some cases, to moirésuperlattices. 38−41 Although vdW-HS carry fascinating physics, treating them from first principles can be computationally prohibitive due to the enormous size of the supercells that are often required to simulate them strain-free.Some strategies have been proposed to reduce these numerical efforts.For example, the so-called "two-step approach" allows one to predict the electronic structure of vdW-HS by performing two separate calculations adopting the lattice parameters of each constituent alone. 32,37his method offers reasonable accuracy but is applicable only to systems with a type-II level alignment; also, it neglects the impact of residual strain on the structural and electronic properties of the heterostructure.The increasing number of available 2D materials continuously enhances the amount of vdW-HS that can be explored computationally, making the above-mentioned limitations particularly restraining.
The recent discovery of MoSi 2 N 4 , 42 a new 2D semiconductor with P6̅ m2 space group and a thickness of seven atomic planes, has stimulated the development of a new class of 2D materials with chemical formula MA 2 Z 4 , 42−48 where M is a transition metal (from group IVB, VB, and VIB), while A and Z are semimetallic and nonmetallic species of group IVA and VA, respectively.−53 However, all of these systems suffer from lattice mismatch and require very large supercells to be simulated strain-free.
Interestingly, a computationally predicted member of the MA 2 Z 4 family, ZrGe 2 N 4 , 42 has the same lattice parameter as MoS 2 .ZrGe 2 N 4 has a direct band gap of 0.85 eV at Γ and is an excellent thermoelectric material due to its low thermal conductivity. 54Due to these intriguing characteristics and the lattice matching with MoS 2 , 21 the heterobilayer (HBL) formed by ZrGe 2 N 4 and MoS 2 represents an ideal platform to study from first principles a strain-free vdW-HS.In particular, it allows for a systematic assessment of the effects of strain, distributed equally on both layers, on the electronic properties of this interface.Hence, we chose this material combination to investigate the effects of strain on the HBL without spurious contributions arising from a lattice mismatch.It should be noticed that both constituting materials exhibit the 1T and 2H phase.However, 1T-MoS 2 is metastable and shows instabilities at room temperature. 55For ZrGe 2 N 4 , both phases are predicted to be stable, 56 but more studies focus on the 1T phase due to its better thermoelectric performance. 54ased on this evidence, we focus herein on the strain-free 2H-MoS 2 /1T-ZrGe 2 N 4 HBL studying its formation energy and electronic structure.After the characterization of the pristine system, which has an indirect band gap and a type-II level alignment, we simulate it under both tensile and compressive biaxial strain with a focus on the interplay between strained lattices and charge redistribution between the two layered materials.We find that values of tensile strain >1% cause an indirect-to-direct band gap transition preserving the type-II level alignment with optical absorption peaks predicted in the visible range, suggesting intriguing perspectives for the MoS 2 /ZrGe 2 N 4 vdW-HS as a suitable candidate for optoelectronic devices and solar cell applications.

COMPUTATIONAL METHODS
The results presented in this work are obtained from density functional theory (DFT) 57 using the Vienna ab initio simulation package (VASP) 58 implementing the projector augmented wave method. 59The HBL is modeled in an unit cell containing a total of 10 atoms (three from MoS 2 and seven from ZrGe 2 N 4 ) and a vacuum layer of 30 Å in the nonperiodic direction to avoid spurious interactions between periodic images.In the structural optimization step and in the evaluation of the charge-density distribution, the exchange correlational potential is treated at the level of the generalized gradient approximation proposed by Perdew, Burke, and Ernzerhof (PBE) 60 and supplemented by Grimme's DFT-D3 correction 61 to account for dispersive interactions.Spin−orbit coupling (SOC) is included in all calculations except for the postprocessing runs to visualize the wave function distribution in real space: in those cases, we checked that SOC did not induce any perceivable change in the plots.An 18 × 18 × 1 kpoint mesh and an energy cutoff of 520 eV are adopted for volume and structural relaxation with convergence thresholds of 1 × 10 −8 eV for the energy and 10 meV Å −1 for the interatomic forces.The electronic structure is subsequently computed with the range-separated hybrid functional by Heyd Scuseria, and Ernzerh (HSE06). 62Due to higher computa- tional costs, in these runs, the k-point mesh is halved after checking the convergence of the electronic structure, see Figure S2.Biaxial strain defined as = a a a 0 0 , where a(a 0 ) corresponds to the lattice constant of the strained (unstrained) heterostructure, is applied adopting positive (negative) values for tensile (compressive) strain up to ±4%.Crystal structures and wave-function plots are visualized using VESTA. 63

Structural Properties.
In this study, we consider the vdW-HS formed by monolayer ZrGe 2 N 4 in the 1T-phase, where the N atoms form a distorted octahedron with Zr atoms in the middle (see Figures 1 and S1), and a single MoS 2 sheet in the 2H phase, where Mo is surrounded by six S atoms forming a centrosymmetric trigonal prism. 54,64Both monolayers are initially optimized and the resulting lattice parameters (a = 3.17 Å for both) are in good agreement with earlier works performed at the same level of theory. 65ptimized MoS 2 has bond length d Mo−S = 2.41 Å, while in ZrGe 2 N 4 , three different bond lengths are relevant: d Zr−N = 2.18 Å, d Ge−N (in-plane) = 1.91 Å, and d N−Ge (out of plane) = 1.87 Å; they all match with earlier reports. 54,56,66We checked the dynamical stability of computationally predicted ZrGe 2 N 4 by calculating its phonon spectrum (see Figure S7), which does not feature any imaginary frequency.The thermal stability of this compound in the 1T phase was previously demonstrated with molecular dynamics simulations up to 2000 K. 54 We build the MoS 2 /ZrGe 2 N 4 HBL considering three stacking arrangements, labeled as AA, AB, and AC, see Figure 1a−h.Due to the different types of atoms included in the vdW-HS, several configurations emerge for each stacking order.AA structures are obtained by placing Mo atoms on top of Ge atoms (labeled as AA Mo/Ge ) and Zr atoms (AA Mo/Zr ).In the AB stacking, S atoms are on top of Ge, while N is at the center of the hexagon formed by MoS 2 .In the AC and AC′ stackings, Ge atoms are on top of the Mo−S bonds with their projection closer to Mo.The interlayer distance d between the S and N atoms is in the range of 2.97−3.38Å, see Table S1, and it is shortest in the AB-stacked structures.
We assess the relative stability of the considered vdW-HS in terms of their total energies computed from DFT, see Figure 1i, since all materials have the same number and types of atoms.The AB Mo/Zr HBL (Figure 1d), is the most stable structure: for visualization purposes, its energy is set to zero in Figure 1i and marked by a gray arrow.The HBL with AC stacking (Figure 1g) is energetically very close to the AB Mo/Zr one with an energy difference of 1 meV only.The AC′ (Figure 1h) and AB S/Ge (Figure 1c) HBL exhibit energies that are only 2 meV higher than the minimum.The remaining AB configurations, AB Mo/Ge and AB Mo/N , are less stable by about 5 meV, while larger energies (>60 meV) are found for AA Mo/Ge and AA Mo/Zr .The small differences in the formation energies of the considered stackings, except for the AA configurations, indicate that this structural parameter does not play a crucial role in the formation of the HBL.This assumption is supported by the fact that all the bond lengths in each constituent are identical regardless of the stacking, see Table S1.In the following, we continue with the analysis of the electronic properties, focusing on the most stable structure AB Mo/Zr .We confirmed its dynamical stability by calculating its phonon dispersion (see Figure S7) which does not show any imaginary frequency.
A deeper analysis of the structural properties of the considered vdW-HS shows that for all stackings the Mo−S bond in MoS 2 (d Mo−S ) is slightly reduced in the HBL compared to the isolated sheet, see Table S1, as a consequence of the vdW interactions occurring at the interface with ZrGe 2 N 4 .Interestingly, this effect is not present in ZrGe 2 N 4 where the interatomic distances remain unchanged compared with the isolated material.By stretching the HBL in the AB Mo/Zr configuration, the interlayer distance increases to 3.01 Å with 2% strain but decreases to 2.82 Å by enhancing strain to 4%, see Table S2.Conversely, upon compression, the interlayer distance first decreases by 0.04 Å with −1% strain, it is equal to the value in the unstrained system under −2% strain (2.97 Å), and then increases to 3.02 Å with larger strain.A similar trend was also observed for the other two configurations energetically close to AB Mo/Zr , see Table S2.Consistent with intuition, compressive strain reduces bond lengths while tensile strain increases them; see Table S3.However, in the ZrGe 2 N 4 monolayer, the length of the Ge−N bond at the interface changes more significantly compared to the inner bonds of the same kind, whereas in MoS 2 , both Mo− S bonds change equally, see Table S3.
3.2.Electronic Properties of the Strain-free Heterostructure.To set a proper reference point for the analysis of the electronic properties of the MoS 2 /ZrGe 2 N 4 HBL, it is instructive to start by examining its constituents.According to our HSE06 calculations, monolayer MoS 2 features a direct band of 2.16 eV at K (Figure 2a) in agreement with the existing literature, 32,67 while monolayer ZrGe 2 N 4 has an indirect band gap of 2.34 eV between Γ and M. To the best of our knowledge, there is no report of the band gap of monolayer ZrGe 2 N 4 computed with HSE06 but our result obtained with PBE (see Figure S3) agrees well with corresponding values from the literature. 54The direct band gap of ZrGe 2 N 4 is at Γ, and it is 250 meV larger than the indirect one.SOC gives rise to a 210 meV splitting at the top of the valence band (VB) of MoS 2 . 31,32In contrast, no SOC splitting appears at any of the frontier states of ZrGe 2 N 4 .
The character of the electronic states can be inferred from the projected density of states (PDOS).In the case of MoS 2 , the highest occupied state exhibits hybridization between Mo and S atoms, while the lowest unoccupied state has a predominant Mo character, see Figure 2a.In the considered energy range, the contribution of Mo is always larger than that of the S atoms in the unoccupied region.In the valence, states with S character dominate below −1.0 eV, while equal contribution from Mo and S is found at lower energies. 68In ZrGe 2 N 4 , the highest occupied state originates solely from N atoms, whereas the lowest unoccupied state is a hybrid state with contributions from all elements with a predominance of Zr, see Figure 2b.The valence states of ZrGe 2 N 4 have mainly N character.
The MoS 2 /ZrGe 2 N 4 HBL in the AB Mo/Zr stacking has an indirect band gap with the valence-band maximum (VBM) at K and the conduction band minimum (CBM) at M, see Figure 2c.A comparison with Figure 2a reveals that the top of the valence region is inherited from MoS 2 , with the VBM at K and a spin-orbit splitting of the highest occupied band of 159 meV slightly reduced compared to the isolated TMDC monolayer, see Figure 2a.Interestingly, the valence state at Γ becomes energetically closer to the VBM (77 meV) in the HBL compared to the isolated monolayer, as seen in TMDC heterostructures. 32,37On the other hand, the bottom of the conduction region is dominated by the features of ZrGe 2 N 4 with the CBM at M as in the isolated system, while the lowest unoccupied state at the zone center remains 250 meV above CBM.The second unoccupied band originates from MoS 2 where the valley at K remains clearly visible.These characteristics appear with even more clarity from the inspection of the PDOS.The type-II level alignment is evident, and so is the energy separation between the frontier states localized at opposite ends of the HBL.Deeper valence states and higher conduction states exhibit hybridization between the two constituents of the vdW-HS.As an example, a ZrGe 2 N 4 -related bulge appears immediately below the highest occupied states, Figure 2c.

Effects of Biaxial Strain.
We continue our analysis by considering the MoS 2 /ZrGe 2 N 4 HBL under strain.Due to the matching lattice constants of its building blocks, this interface is ideally suited to explore the effects of biaxial extensions and compressions equally distributed on both layers.−71 In this study, we consider values of strain up to ±4% with intermediate steps at ±1 and ±2%.We chose this range as it is mostly explored in experiments on TMDCs. 72−76 Larger values of strain in flat TMDC monolayers have been studied from first principles 34 to provide a point of reference for such extreme cases.
−79 However, the nature of the fundamental gap changes depending on the applied deformation.Under compressive strain (Figure 3a−c), the gap remains indirect with the VBM at K and the CBM at M. Direct comparison with the band structure of the unstrained HBL (cyan lines) reveals a significant downshift of the CBM which is responsible for the reduction of the gap upon increasing strain.Notably, in the valence region, the highest occupied states at Γ are found at lower energy compared with the VBM as strain is enhanced.Under tensile strain, the situation is more intricate.With 1% deformation, the fundamental band gap (blue arrow) is still indirect, but the direct band gap at Γ is only 88 meV larger (see Figure 3d).Further deformation (strain > 1%) is sufficient to downshift the lowest conduction state at Γ and to raise its counterpart in the valence region, giving rise to a direct band gap (Figure 3e).This trend is amplified, especially in the unoccupied region, upon 4% strain, under which the gap remains direct at Γ and shrinks further compared with the unstrained HBL (Figure 3f).We checked that these trends are independent of the stacking order, see Figure S6.As shown in Figure 3, strain primarily induces an indirect-to-direct band gap transition and shifts the valleys.No visible effect is produced on the band dispersion and thus on the corresponding effective masses (see Table S4).In Figure 4, we summarize the band-gap values of the HBL as a function of strain, distinguishing between fundamental and direct band gaps.At a glance, we notice that a large amount of tensile strain (≥2%) leads to an indirect-to-direct band gap transition but with band gap sizes smaller than in the unstrained case.Overall, the application of strain leads to a reduction of the fundamental gap, except for 1% strain.The direct band gap follows a completely different trend, being maximized under moderate compressive strain (−1%) and decreasing with tensile deformations.This behavior is generally reflected in the optical spectra computed in the independentparticle approximation on top of the HSE06 electronic structure, see Figure S8.
To gain a deeper understanding of the consequences of strain on the electronic structure of the MoS 2 /ZrGe 2 N 4 HBL and in particular on the reason for the indirect-to-direct band gap transition as a function of strain, we analyze the character of its frontier states by plotting in real space the square modulus of the corresponding wave functions (WFs), see Figure 5.In the unstrained configuration, the VBM at K is entirely localized on MoS 2 , while the CBM at M is solely on ZrGe 2 N 4 and in particular around the Zr atoms, see Figure 5c.The distribution on the lowest conduction state at Γ remains localized on ZrGe 2 N 4 , although the character of the state changes and the probability density is maximized around the Ge−N bonds in the outermost layer.In contrast, in the valence region, the highest state at Γ is partially distributed also on ZrGe 2 N 4 and in particular on the outermost layer of N atoms in close proximity with the outer distribution of the S p-orbitals in MoS 2 .
Under compressive strain, the WF probability remains localized on MoS 2 in the valence region and on ZrGe 2 N 4 in the conduction regardless of the applied amount (see Figure 5a,b).Visually, the two distributions are identical except for a small contribution from the S atoms in the highest occupied state at K. On the other hand, under tensile strain, a delocalization of the WF distribution is evident.This is especially remarkable in the conduction region, where the probability extends to MoS 2 .The larger contribution of MoS 2 (ZrGe 2 N 4 ) to the lowest unoccupied (highest occupied) state at the Γ-point is mirrored by an increase in the interlayer distance, see Table S2.This increased level of hybridization is also associated with the downshift of the lowest unoccupied valley at Γ, leading to an indirect-to-direct band gap transition.
3.4.Summary and Conclusions.In summary, we presented a comprehensive analysis of the electronic properties of HBL formed by monolayer MoS 2 on top of the 1T-phase of  ZrGe 2 N 4 .These materials have identical lattice parameters offering the opportunity to build a strain-free HBL.Among the considered stackings, the one in which the Mo atoms lie on top of the Zr atom is energetically most favorable, although differences in the total energies with most of the other configurations are on the order of a few meV.This HBL is characterized by an indirect band gap with the VBM and CBM at the high symmetry points K and M, respectively, and a type-II level alignment with the VBM (CBM) localized on MoS 2 (ZrGe 2 N 4 ).We analyzed the effects of biaxial strain on the electronic properties of HBL considering deformations up to ±4% of the in-plane lattice parameter at equilibrium.The size of the band gap reduces with increasing amounts of both tensile and compressive strain.Under tensile strain >1%, an indirect-to-direct transition occurs, shifting the CBM and VBM to the center of the Brillouin zone.The real-space analysis of the WF distribution confirms the localization of the highest occupied and lowest unoccupied states on the MoS 2 and ZrGe 2 N 4 monolayers, respectively.Yet, the VBM at Γ also includes a small contribution from the N atoms of ZrGe 2 N 4 facing MoS 2 .Increasing tensile strain increases the amount of MoS 2 character in the CBM as well as the contribution of the N atoms of ZrGe 2 N 4 in the highest occupied states.Contrarily, compression only increases the concentration of each layer in the frontier states of HBL.
In conclusion, the vdW-HS MoS 2 /ZrGe 2 N 4 subject to ∼2% of tensile strain with its direct band gap and type-II band alignment offers favorable perspectives for enhanced photoluminescent efficiency 80 and ultrafast charge separation, 35,81−83 thus covering a broad range of possible optoelectronic applications.Moreover, the direct band gap at Γ promises high solar efficiency in analogy with group III−V semiconductors (e.g., GaAs and InAs).Most importantly, the insight offered by this study suggests the potential of using strain to customize the electronic properties of vdW-HS in a controlled way.We finally emphasize the predictive power of ab initio simulations in discovering new material combinations such as MoS 2 /ZrGe 2 N 4 with favorable structural characteristics such as absence of lattice mismatch.Further studies in this direction may extend the spectrum of available vdW-HS, starting, for example, by considering both polymorphs of MoS 2 and ZrGe 2 N 4 (1T and 2H phases) or using data-driven methods to identify other strain-free material combinations.

Figure 1 .
Figure 1.Side and top views of the HBL MoS 2 /ZrGe 2 N 4 in the (a,b) AA, (c−f) AB, and (g,h) AC stacking configurations, with their point of reference for the stacking indicated by the dashed red lines and their primitive unit cells marked in black.(i) Relative stability of the considered configurations compared to the most stable structure AB Mo/Ge (gray arrow) whose energyis set to zero.The inset shows the Brillouin zone of the investigated HBL with the relevant high-symmetry points and the path connecting them highlighted in bold.

Figure 2 .
Figure 2. Electronic band structure and density of states of (a) monolayer MoS 2 , (b) monolayer ZrGe 2 N 4 , and (c) the heterostructure MoS 2 / ZrGe 2 N 4 with stacking AB Mo/Zr , respectively, calculated with the HSE06 functional.The Fermi level is set to zero in all panels and marked by a horizontal dashed line.The blue arrows mark the fundamental gap.

Figure 3 .
Figure 3. Electronic band structures of vdW AB Mo/Zr HBL (black lines) under different values of compressive strain (a−c) and tensile strain (d−f) calculated using the HSE06 functional.The band structure of the unstrained HBL is shown for comparison (cyan lines) in each panel.The fundamental gap is marked by a blue arrow.The Fermi energy (E f ) is set to zero at the VBM.

Figure 4 .
Figure 4. Fundamental (dotted lines) and direct (solid lines) band gaps of the MoS 2 /ZrGe2N 4 HBL, calculated with the HSE06 functional including SOC, as a function of strain.