Background {#Sec1}
==========

Ever since the discovery of graphene \[[@CR1], [@CR2]\], interest in two-dimensional (2D) layered materials has been growing steadily.  Many graphene-like 2D materials such as transition-metal dichalcogenides \[[@CR3]\], monolayer honeycomb structures of group V elements and III-V binary compounds \[[@CR4]--[@CR8]\], and post transition metal chalcogenides (PTMCs)\[[@CR9]\] have gained a lot of interest due to their exceptional physical properties and promising applications.  Among these diverse 2D materials, GaTe monolayer, as a member of PTMCs \[[@CR9]\], has successfully been fabricated by molecular beam epitaxy \[[@CR10]\].  Theoretical calculations showed that GaTe monolayer is an indirect-bandgap semiconductor and its bandgap can be modulated by applying strains \[[@CR11]\].  Besides, monolayer C~2~N, a new 2D layered material with uniform pore and nitrogen atom distributions, was also successfully synthesized via a bottom-up wet-chemical reaction and found to be a direct-gap semiconductor \[[@CR12]\].  Many studies demonstrated that its bandgap, band edge positions, and optical properties can be engineered by varying their stacking order, layer number, external electric field or strain and alloying/substituting with other elements \[[@CR13]--[@CR16]\].  It should be noted that the tunable direct bandgap and porous nature of C~2~N is expected to exhibit desirable properties for electronics, optoelectronics, and energy conversion as well as photocatalytic water splitting, etc \[[@CR15]\].  However, a significant challenge still remains for the use of C~2~N in photocatalysis and photovoltaic cells: The photogenerated electron-hole pairs stay in the same regions spatially, which can lead to a high rate of recombination of photogenerated carriers, thus reducing the solar energy conversion

In parallel with the efforts on single 2D materials, the van der Waals (vdW) heterostructures fabricated by stacking different 2D semiconductor materials have opened up new avenues for creating new materials and designing new devices \[[@CR17]--[@CR23]\].  This sort of heterostructure can be generally classified as three types: the type I (straddling gap), type II (staggered gap), and type III (broken gap) according to the relative positions of the valence band maximum (VBM) and conduction band minimum (CBM) of the respective semiconductors \[[@CR18], [@CR24], [@CR25]\].  For the type I heterostructures, the energies of the VBM and CBM of one material straddle those of the other material and all the photogenerated electrons and holes are accumulated in the same layer, which induces the ultrafast recombination of the excited carriers and thus can be utilized in optoelectronic devices, such as light-emitting diode.  In the case of type II heterostructures, both the CBM and VBM of one material are lower or higher in energy than the ones of the other material.  As a result, photogenerated electrons and holes are confined separately in the two materials, respectively, thereby inhibiting the rate of recombination.  Therefore, they can be used as building blocks for photovoltaic devices \[[@CR18], [@CR24]\].  As for the type III heterostructures, the VBM level of one material is higher than the CBM level of the other, which is desirable for tunneling field effect transistors \[[@CR25], [@CR26]\].  Very recently, many GaTe-based heterostructures have been extensively studied both theoretically and experimentally.  The GaTe/InSe heterostructure has been fabricated experimentally and presents the type II band alignment \[[@CR27], [@CR28]\].  Quasi-2D GaTe/GaSe heterostructure was created by transferring exfoliated few-layer GaSe onto bulk GaTe sheets and found to form type I band alignment at the interface \[[@CR29]\].  The GaTe/SnI heterostructure was verified to be a large-gap quantum spin Hall insulator and exhibits a noticeable Rashba splitting that can be modulated by changing the interlayer distance of heterosheets \[[@CR30]\].  In addition, construction of semiconductor/C~2~N heterostructures, such as g-C~3~N~4~/C~2~N \[[@CR31]\], MoS~2~/C~2~N \[[@CR32]\], and CdS/C~2~N \[[@CR33]\], demonstrated an enormous potential for promoting the photocatalytic performance of C~2~N due to the efficient separation of the electron-hole pairs, thereby restraining the recombination of photogenerated carriers. 

In this work, we construct the GaTe/C~2~N vdW heterostructure and perform first-principles density functional theory (DFT) calculations to investigate its structural parameters and electronic, optical properties.  The results show that the heterostructure possesses intrinsic type II band alignment and better visible-UV light absorption than the constituent layers.  Moreover, we predict the strain dependences of the bandgap, band alignments, and band edge positions of the GaTe/C~2~N heterostructure, which are essential in the design of new multi-functional nano-devices. 

Methods {#Sec2}
=======

In our research, we perform first-principles calculations by using the Vienna ab initio simulation package (VASP) \[[@CR34]\].  A plane-wave basis set with a kinetic energy cutoff of 500 eV and Perdew-Burke-Ernzerhofer (PBE) projected augmented wave pseudopotential \[[@CR35]\] are adopted to expand the wave functions and to describe the electron-ion potential, respectively.  The computationally more expensive hybrid Heyd-Scuseria-Ernzerhof (HSE06) functional method \[[@CR36]\] is adopted to correct the underestimated bandgaps obtained by DFT/PBE calculations.  The weak vdW interaction between the two monolayers is described by the DFT-D2 correction of Grimme \[[@CR37]\].  A vacuum space in the *z*-direction more than 25 Å is used to avoid interactions between adjacent heterobilayers.  A 21×21×1 (11×11×1) *k*-mesh for the PBE (HSE06) calculations is utilized to sample the Brillouin zone.  The atomic positions are fully relaxed until energy and forces are converged to 10^−5^ eV and 0.01 eV/Å, respectively. 

Results and Discussion {#Sec3}
======================

Let us start from the investigations of the pristine GaTe and C~2~N monolayers.  The optimized configurations of the two monolayers are shown in Fig.  [1](#Fig1){ref-type="fig"}[a](#Fig1){ref-type="fig"}, [b](#Fig1){ref-type="fig"}, respectively.  Their structural parameters are listed in Table [1](#Tab1){ref-type="table"}.  For GaTe monolayer, the optimized lattice constant and Ga-Te bond length are 4.14 and 2.41Å respectively.  In the case of the C~2~N monolayer, the optimized lattice constant, C-N, and C-C(1)/C-C(2) distances are 8.26, 1.34, and 1.47/1.43Å, respectively.  Furthermore, their band structures are also investigated by the PBE/HSE06 calculations and presented in Additional file [1](#MOESM1){ref-type="media"}: Figure S1a and b, respectively.  Apparently, the GaTe monolayer is a semiconductor with an indirect bandgap of 1.43/2.13 eV while C~2~N monolayer is a direct bandgap semiconductor with a value of 1.65/2.44 eV. Meanwhile, we find that aside from a rigid shift, the band structures of C~2~N monolayer calculated with PBE and HSE06 differ significantly, especially for the valence bands.  However, the CBMs and VBMs calculated using PBE and HSE06 are all at *Γ* points, indicating the band dispersions given by the two functionals are relatively consistent though there is some difference in accuracy.  All the results are in good agreement with those of previous reports \[[@CR11], [@CR38]\] and suggest the reliability of our calculation method.  As is well known, bandgaps of semiconductors are generally underestimated by the PBE functional because of the lacking of the derivative discontinuity in the energy functional.  Our subsequent presentation for the electronic and optical properties will be based on the HSE06 results.  Fig.  1Top and side views of (**a**) GaTe and (**b**) C~2~N monolayers.  Top views of (**c**--**e**) *α*-, *β*-, and *γ*-stacking GaTe/C~2~N heterostructures, in which the corresponding base vectors of the heterostructures are labeledTable 1The calculated ground-state properties of the monolayers and their heterostructures: lattice parameters of primitive unit cell *a* and *b*, bond lengths, interlayer distances (*d*~il~), interlayer binding energy *E*~b~ (per unit area), and bandgaps calculated within vdW-DFT/PBE $\documentclass[12pt]{minimal}
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                \begin{document}$\left (E_{\mathrm {g}}^{\text {HSE}}\right)$\end{document}$*a*=*bd* ~Ga−Te~*d* ~Ga−Ga~*d* ~C−C(1)/C(2)~*d* ~C−N~*d* ~il~*E* ~b~$\documentclass[12pt]{minimal}
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                \begin{document}$E_{\mathrm {g}}^{\text {HSE}}$\end{document}$(Å)(Å)(Å)(Å)(Å)(Å)(meV/Å^2^)(eV)(eV)GaTe4.142.712.471.47/1.431.432.13C~2~N8.321.46/1.421.341.652.44*α*-stacking8.262.692.431.46/1.421.333.70− 15.060.701.42*β*-stacking8.262.692.431.46/1.421.333.71− 14.970.681.42*γ*-stacking8.262.702.421.46/1.421.333.50− 15.800.601.38

The GaTe/C~2~N heterobilayer is constructed by combining a 2×2 supercell of GaTe sheet and a 1×1 unit cell of C~2~N layer, with the only 0.48% lattice mismatch.  In order to find the stable configuration of the heterostructure, we shift the GaTe monolayer in different directions.  As a result, three energetically favorable stacking types with high symmetry named as *α*-, *β*-, and *γ*-stacking are obtained, as illustrated in Fig.  [1](#Fig1){ref-type="fig"}[c](#Fig1){ref-type="fig"}--[e](#Fig1){ref-type="fig"}.  In the *α*-stacking, the hexagonal C~4~N~2~ rings are right over the hexagonal GaTe rings.  As for the *β*- and *γ*-stacking, they can be obtained by moving GaTe layer in the *α*-stacking about 1.21 and 2.42 Å along the ***a*** + ***b*** direction, respectively.  To compare the relative stabilities of the three stacking configurations, we calculate their interface binding energies, $\documentclass[12pt]{minimal}
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                \begin{document}$\phantom {\dot {i}\!}E_{\mathrm {b}} = (E_{\mathrm {GaTe/C_{2}N}}-E_{\text {GaTe}}-E_{\mathrm {C_{2}N}})/S$\end{document}$, where $\documentclass[12pt]{minimal}
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                \begin{document}$\phantom {\dot {i}\!}E_{\mathrm {GaTe/C_{2}N}}$\end{document}$, *E*~GaTe~, and $\documentclass[12pt]{minimal}
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                \begin{document}$E_{\mathrm {C_{2}N}}$\end{document}$ represent total energies of the GaTe/C~2~N heterostructure, free-standing GaTe and C~2~N monolayers, respectively, and *S* is the surface area of the 2D supercell.  As shown in Table [1](#Tab1){ref-type="table"}, the binding energies of GaTe/C~2~N heterostructures with *α*-, *β*-, and *γ*-stacking configurations are − 15.06 meV, − 14.97 meV, and − 15.80 meV/Å^2^, respectively.  The three binding energies are very close to each other though the *γ*-stacking is energetically more favorable, which is consistent with its smallest interlayer distance.  We further confirm the dynamic and thermal stabilities of these heterostructures with different stacking forms by calculating their phonon spectra and performing ab initio molecular dynamics (MD) simulations and show the results in Additional file [1](#MOESM1){ref-type="media"}: Figure S2.  All phonon modes have positive frequencies except for the transverse acoustic mode near the *Γ* point due to the phonon softening, confirming the dynamic stability \[[@CR5]\].  In the MD simulations, the total energies of the systems oscillate in certain energy ranges, and no geometric reconstructions and broken bonds are found to occur in the heterostructures, indicating that these systems are thermally stable at room temperature \[[@CR39]\].  We note that during MD simulation the *γ*-stacking configuration possesses the least energy undulation (less than 7 meV/atom), indicating its more prominent thermal stability.  The very close binding energies of the three stacking configurations implies that their electronic structures may also be very similar.  To confirm this, we calculate the band structures for the three configurations (see Additional file [1](#MOESM1){ref-type="media"}: Figure S3).  One can see that the three band structures are indeed almost identical.  Although the *γ*-stacking configuration is the most stable one, the three configurations may still be populated with some probabilities at room temperature because of their similar formation energies.  However, because their electronic structures are also very close to each other, we can choose only one configuration to present our work.  Here, we choose the most stable *γ*-stacking configuration in the following analyses and discussions. 

We now go to the electronic properties of the GaTe/C~2~N vdW heterostructure.  As shown in Fig.  [2](#Fig2){ref-type="fig"}[a](#Fig2){ref-type="fig"}, the bandgap of GaTe/C~2~N heterostructure is calculated to be about 1.38 eV. In comparison with that of its components, its bandgap is reduced due to the GaTe-C~2~N interaction and the resulting band alignment.  Also, the electronic structure of C~2~N monolayer is well-preserved.  Nevertheless, the projected band structure of GaTe in the heterostructure have considerable changes comparing to the monolayer, which can be attributed to the fact that the interlayer vdW and electrostatic interactions can result in the overlap of electronic states in the bands of the heterostructure.  Similar behavior is also found in MoS~2~/PbI~2~ vdW heterostructure \[[@CR40]\].  Furthermore, we find that its VBM and CBM are mainly localized on GaTe and C~2~N sublayers, respectively.  From the calculated total and partial density of states (PDOS) in Fig.  [2](#Fig2){ref-type="fig"}[a](#Fig2){ref-type="fig"} (right panel), it can be seen that the CBM mainly originates from the *p* states of N and C atoms, whereas the VBM is mainly dominated by the *p* states of Te and Ga atoms.  The band decomposed charge densities of the CBM and VBM in Fig.  [2](#Fig2){ref-type="fig"}[c](#Fig2){ref-type="fig"}, [d](#Fig2){ref-type="fig"} reveal that the lowest-energy electrons and holes are distributed in the C~2~N layer and GaTe layer, respectively, consistent with the detailed PDOS results above.  The band alignment of the GaTe/C~2~N heterostructure including both VB offset (VBO) and CB offset (CBO) is illustrated in Fig.  [2](#Fig2){ref-type="fig"}[b](#Fig2){ref-type="fig"}, which is according to the analysis of Fig.  [2](#Fig2){ref-type="fig"}[a](#Fig2){ref-type="fig"}.  Clearly, the VB and CB of the GaTe layer are higher in energy than the corresponding bands of the C~2~N layer, and the VBO and CBO between the GaTe and C~2~N layers are about 1.03 and 0.72 eV, respectively.  When the heterostructure is illuminated with light, the electrons with energy obtained from the sunlight leap into the CB from the VB.  And then these photogenerated electrons on the CB of the GaTe sheet can be easily shifted to that of the C~2~N layer due to the observed CBO.  Conversely, the photogenerated holes on the VB of the C~2~N sheet transfer to that of the GaTe layer because of the VBO.  The above results suggest that a type II band alignment is formed at the interface between GaTe and C~2~N layers, which is a prerequisite to separate the electrons and holes efficiently.  In addition, the calculated plane-averaged charge density difference of the heterostructure, shown in Additional file [1](#MOESM1){ref-type="media"}: Figure S4, indicates that some electrons transfer from the C~2~N layer to the GaTe layer.  It means that an intrinsic built-in electric field (*E*~in~) is induced with its direction pointing from C~2~N layer to GaTe layer.  Also note that the *E*~in~ acts in opposite (same) direction to the transfers of photogenerated electrons (holes) and thus inhibits the recombination of photogenerated electron-hole pairs.  As a result, under the combined effect of intrinsic *E*~in~ and band offset, the photogenerated carries can be effectively separated on different surfaces, which can improve the energy conversion efficiency and finally enhance the performance of optoelectronic devices.  Fig.  2**a** The projected band structure of the GaTe/C~2~N heterostructure with *γ*-stacking configuration and the corresponding total and partial density of states. ** b** Schematic representation of type II band alignments for the carrier transfer and separation in the GaTe/C~2~N heterostructure, referring to the vacuum level.  The redox potentials (red dashed line) of water splitting at pH =0 are shown for comparison.  Band decomposed charge densities of the **c** VBM and **d** CBM of the heterostructure

Besides, we notice that the CBM of the heterostructure locates more positive than the reduction potential (− 4.44 eV vs vacuum level) of hydrogen evolution, whereas its VBM almost overlaps with the oxidation potential (− 5.67 eV vs vacuum level) of oxygen evolution.  Hence, it has only limited photocatalytic capacity to split water by producing hydrogen at pH =0.  Nevertheless, changing the inter-layer spacing and pH value can ignite the potential application of the heterostructure as a visible light photocatalyst (see later discussion in details). 

Actually, a promising photoelectric nano-device should absorb as much visible-UV light as possible.  Thus, we further explore the optical absorptions of the GaTe/C~2~N heterostructure and its components.  The computational details have been fully described in our previous works \[[@CR22], [@CR23]\].  As displayed in Fig.  [3](#Fig3){ref-type="fig"}, the GaTe/C~2~N heterostructure exhibits stronger visible-UV light absorption and a wider absorption range compared with its components, especially in the energy range of 2.20 to 4.71 eV. This stems from the new optical transitions induced by the charge transfer and interlayer coupling in the heterostructure \[[@CR41]\].  Fig.  3The calculated optical absorption spectra *A*(*ω*) of the GaTe/C~2~N heterostructure and its components using hybrid HSE06 functional. * A*(*ω*) of the heterostructures with vertical strains of 0.5 Å and 1.5 Å and in-plane strains of +6% and -6%.  And the solar spectrum are also shown for comparison

It is widely known that strains, including inter-layer (normal) and in-plane strains, provide an effective way to tune the electronic properties and thus enhance the performance of materials \[[@CR42]\].  Here, we first explore the normal strain effect in GaTe/C~2~N vdW heterostructure.  The normal strain is evaluated by *Δd*=*d*−*d*~0~, where *d* and *d*~0~ are the actual and equilibrium distances, respectively, between GaTe and C~2~N sublayers.  Thus, if *Δd*\>0, the system is under a normal tensile strain, and vice versa.  The change in the interaction between the GaTe and C~2~N layers should be reflected by the intensity of charge transfer between them.  The calculated plane-averaged charge density differences of the GaTe/C~2~N heterostructures with different interlayer distances are shown in Additional file [1](#MOESM1){ref-type="media"}: Figure S5.  The results show that as the distance between the GaTe and C~2~N sheets decreases, the charge transfer obviously intensifies as a result of the enhanced interlayer interaction.  Thus, the electronic behavior of the GaTe/C~2~N heterostructure is expected to be well tuned by normal strain. 

The calculated bandgap and binding energy of the heterostructure as functions of the applied strain are shown in Fig.  [4](#Fig4){ref-type="fig"}[a](#Fig4){ref-type="fig"}, and the evolutions of the CBM and VBM of the heterostructure under normal strain are shown in Fig.  [4](#Fig4){ref-type="fig"}[b](#Fig4){ref-type="fig"}.  It is clearly shown that an increasing normal compressive strain reduces the bandgap due to the enhanced inter-layer interaction.  In contrast, an increasing normal tensile strain first increases slowly the bandgap and then reach nearly a convergence at *Δd*≃0.8Å, which can arise from the greater reduction of the inter-layer interaction \[[@CR32]\].  We find the equilibrium structure at *Δd* = 0 has the lowest binding energy, which is consistent with the result shown in Table [1](#Tab1){ref-type="table"}.  Meanwhile, we notice that the type II band alignments and enhanced visible-UV light absorption are preserved, being nearly irrespective of the inter-layer distance (see Fig.  [3](#Fig3){ref-type="fig"} and Additional file [1](#MOESM1){ref-type="media"}: Figure S6).  More interestingly, the large tensile normal strains (*Δd*≃0.3 Å) shift the VBM below the O~2~/H~2~O oxidation potential, making the system suitable for water splitting at pH = 0.  During the photocatalytic water splitting, the hydrogen and oxygen production processes will occur separately in the C~2~N layer and GaTe layer, respectively.  We note that under such situation, the VBM over-potential is so small that it may not be sufficient for O~2~ production \[[@CR43]\], but such bias potentials can be tuned by changing the pH value of the medium \[[@CR44]\].  In other words, the photocatalytic properties for water splitting can be further modulated by controlling the pH to match the redox potential of water.  As illustrated in Fig.  [4](#Fig4){ref-type="fig"}[b](#Fig4){ref-type="fig"}, in the acidic environment of pH = 2, the band edges of the heterostructure perfectly straddle the water redox potential, showing the heterostructure is well suitable for H~2~/O~2~ production from water, especially for large vertical strains applied.  Fig.  4Normal strain effects on **a** the bandgap and biding energy, and **b** the band-edge positions of GaTe/C~2~N vdW heterostructure.  The redox potentials of water splitting at pH 0 (red dashed line) and pH 2 (blue dashed line) are shown for comparison

To further reveal the mechanism of the photocatalytic hydrogen generation on GaTe/C~2~N heterostructure, we simulate the water adsorption and decomposition on the surface of the C~2~N layer, where hydrogen is produced during the photocatalytic water splitting.  Since the formation of hydrogen molecules starts from the decomposition of absorbed water, we firstly investigate the absorption energy of H, OH, and H~2~O on the C~2~N surface at the DFT/PBE level.  The corresponding adsorption energies are −1.03, −0.51, and −0.56 eV, respectively, as illustrated in Fig.  [5](#Fig5){ref-type="fig"}[a](#Fig5){ref-type="fig"}.  The negative values indicate that the absorptions are energetically favorable \[[@CR45]\].  Subsequently, the calculated reaction energy of water decomposition is about 1.48 eV (from − 0.56 to 0.92 eV).  This means that water decomposition is an endothermic reaction on this surface.  Furthermore, as the generated hydrogen atoms are adsorbed on C~2~N surface, the remotely separated hydrogen adatom will be energetically favorable to migrate close to form hydrogen molecules \[[@CR46]\].  As displayed in Fig.  [5](#Fig5){ref-type="fig"}[b](#Fig5){ref-type="fig"}, the reaction energy required for removing one H~2~ from C~2~N is relatively small (0.04 eV), which indicates that the adsorbed H~2~ is easy to be released and is beneficial for photocatalytic hydrogen gas production.  Fig.  5**a** Adsorption configurations of H, OH, H~2~O and decomposition mechanism of H~2~O on C~2~N surface in GaTe/C~2~N vdW heterostructure. ** b** Interaction between two hydrogen adatoms, formation and releasing of hydrogen molecular on C~2~N surface in GaTe/C~2~N vdW heterostructure

Finally, we turn to exploring the effect of in-plane biaxial strains, which is simulated by changing the crystal lattice parameter and calculated by *ε*=(*a*−*a*~0~)/*a*~0~, where *a* and *a*~0~ are the lattice constants of the strained and pristine structures, respectively.  To guarantee the in-layer biaxial strains considered are within the range of elastic response, we first examine the strain energy per atom, *E*~s~= (*E*~strained~−*E*~unstrained~)/*n*, with *n* being the number of atoms in the unit cell.  The calculated strain-energy curve (see Fig.  [6](#Fig6){ref-type="fig"}[a](#Fig6){ref-type="fig"} (right *y*-axis)) shows a characteristic of the quadratic function, indicating that all the strains considered are within the elastic limit and, therefore, are fully reversible.  The bandgap evolution under various biaxial strains is given in Fig.  [6](#Fig6){ref-type="fig"}[a](#Fig6){ref-type="fig"}.  One can see that the bandgap reaches its maximum value (∼1.45 eV) under the strain of about − 2%.  At *ε* = − 12% the system undergos a semiconductor to metal transition, implying tunable conductive and transport properties of this heterostructure.  Meanwhile, an interesting indirect-direct-indirect (Ind-D-Ind) bandgap transitions are found at *ε*≃− 3% and − 8%, respectively.  These transitions are derived from the strain-induced band-energy shifts at different k-points (see the Additional file [1](#MOESM1){ref-type="media"} for detail: Figure S7).  The Ind-D transition and the changes in electronic structure due to strain may enhance the optical absorption \[[@CR47]\].  In Fig.  [3](#Fig3){ref-type="fig"}, we compare the optical absorptions of the GaTe/C~2~N heterostructures under strains of ± 6%, where their bandgaps are nearly the same.  The results show that biaxial strains red-shift the optical spectra in the range of visible light, being consistent with the decreased bandgap discussed above.  Interestingly, a − 6% strain leads to a significantly enhanced optical absorption in the region of \[1.60--2.65 eV\].  Furthermore, it is also found that strain can change the band alignment.  As shown in Fig.  [6](#Fig6){ref-type="fig"}[b](#Fig6){ref-type="fig"} and Additional file [1](#MOESM1){ref-type="media"}: Figure S7, for *ε*≥+ 6%, the CBM of the GaTe sublayer shifts downwards and becomes the CBM of the heterostructure.  As a result, the energies of the CBM and VBM in the GaTe sublayer are straddled by those in the C~2~N sublayer, leading to a transition from the type II to the type I. Here, we note that the CBM and VBM of the GaTe sublayer approach each other under large tensile strains and form a very small bandgap while those of the C~2~N sublayer have only a minor change.  This behavior can be understood by first considering the strain effects on the electronic structures of the two isolated monolayers.  Previous calculations showed that the bandgap of GaTe monolayer is much more sensitive to large tensile strains than that of C~2~N monolayer: Under large tensile strains, the former will become very small while the latter remains \[[@CR11], [@CR16]\].  This may be due to the buckling structure of GaTe, which is affected more significantly by in-plane strains.  Since the overall interlayer interactions in the heterostructure is weak, mainly the vdW and the electrostatic interactions which have only minor effects on the bandgap, the behaviors of the two monolayers under large tensile strains are preserved in the GaTe/C~2~N heterostructure.  In addition, for *ε*≥−12%, both the CBM and VBM of the GaTe sublayer become higher than those of the C~2~N sublayer, and thus, the type III band alignment is formed.  However, when the compressive strain is further increased to be larger than − 13%, this type III band alignment is broken, where the C~2~N sublayer will become metallic.  In a word, the strain can engineer effectively the type and value of the bandgap and band alignment of the GaTe/C~2~N heterostructure.  This will be useful to design multi-functional high-performance electronic and optoelectronic devices.  Fig.  6**a** In-plane biaxial strain effects on the bandgap and strain energy of the GaTe/C~2~N heterostructure. The mistyrose, blue, and green regions represent the metal (M), Ind and D bandgap ranges, respectively. ** b** The evolutions of the band-edge positions of the sublayers in heterostructure as a function of the in-plane biaxial strain.  The I, II, and III regions correspond to type-I, -II, and -III band alignments, respectively

Conclusions {#Sec4}
===========

In summary, by performing first-principles hybrid DFT calculations, we have investigated systematically the strain-dependent structural, electronic, and optical properties of the GaTe/C~2~N heterostructure.  It is predicted to be an indirect-gap semiconductor showing improved optical absorptions in the visible-UV range compared to its components.  The type II band alignment and intrinsic built-in electric field inhibit the energy-wasted recombination of the photogenerated carriers and thus enhance the performance of optoelectronic devices.  In particular, large normal tensile strains can make the system suitable for water splitting at certain pH. By studying the absorption and decomposition behaviors of a water molecule on the C~2~N sublayer in the heterostructure, we find that the absorption of H~2~O and the formation of H~2~ on the C~2~N surface are all energetically favorable, which is beneficial for photocatalytically producing hydrogen gas.  In-plane compressive strains will induce the Ind-D-Ind and semiconductor-metal transitions, whereas in-plane tensile strains will induce the type II to type I or type III transition.  These results demonstrate that the GaTe/C~2~N heterostructure has great potential in applications of multi-functional optoelectronic devices. 

Additional file
===============

 {#Sec5}

Additional file 1**Figure S1**.  Band structures of the GaTe and C~2~N monolayers; **Figure S2**.  Phonon spectrum and MD simulations of GaTe/C~2~N heterostructures; **Figure S3**.  Band structures of the of GaTe/C~2~N heterostructures; **Figure S4**.  The plane-averaged charge density difference of the heterostructure; **Figure S5**.  The effect of normal strain on the interlayer interaction **Figure S6**.  Projected band structures of the heterostructures under the various normal strains; **Figure S7**.  Projected band structures of the heterostructures under the various in-plane strains. ( PDF 3733 kb)

2D

:   Two-dimensional

CBM

:   Conduction band minimum

CBO

:   Conduction band offset

DFT

:   Density functional theory

HSE06

:   Hybrid Heyd-Scuseria-Ernzerhof

PBE

:   Perdew-Burke-Ernzerhofer

PDOS

:   Partial density of states

PTMCs

:   Post transition metal chalcogenides

VBM

:   Valence band maximum

VBO

:   Valence band offset

vdW

:   van der Waals

The authors would like to acknowledge the amenity provided by the High-performance Grid Computing Platform of Henan Polytechnic University for the density functional theory calculations. 

Funding {#d29e1832}
=======

This work was supported by the National Natural Science Foundation of China (Nos.  11174220, 11374226 and 11804081), the Key Scientific Research Project of the Henan Institutions of Higher Learning (Nos.  16A140009 and 18A140018), the Key Research Project for the Universities of Henan Province (No.  19A140009), and the Doctoral Foundation of Henan Polytechnic University (Nos.  B2015-46 and B2018-38). 

Availability of data and materials {#d29e1837}
==================================

The datasets supporting the conclusions of this article are included within the article. 

BJW and SHK conceived the idea and supervised the project.  XHL carried out the simulation.  BJW and XHL wrote the manuscript, and SHK revised the manuscript.  XLC, WYY, YYZ, FYL, ZXF, and YSZ contributed to the data analysis.  All the authors discussed the results and approved the final manuscript. 

Competing interests
===================

The authors declare that they have no competing interests. 

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