Strain induced piezoelectric effect in black phosphorus and MoS2 van der Waals heterostructure

The structural, electronic, transport and optical properties of black phosphorus/MoS2 (BP/MoS2) van der Waals (vdw) heterostructure are investigated by using first principles calculations. The band gap of BP/MoS2 bilayer decreases with the applied normal compressive strain and a semiconductor-to-metal transition is observed when the applied strain is more than 0.85 Å. BP/MoS2 bilayer also exhibits modulation of its carrier effective mass and carrier concentration by the applied compressive strain, suggesting that mobility engineering and good piezoelectric effect can be realized in BP/MoS2 heterostructure. Because the type-II band alignment can facilitate the separation of photo-excited electrons and holes, and it can benefit from the great absorption coefficient in ultra-violet region, the BP/MoS2 shows great potential to be a very efficient ultra-violet photodetector.

In this work, we investigate the effect of normal compressive strain on the structural, electronic, transport and optical properties of semiconducting BP/MoS 2 vdW heterostructure. The band gap of BP/MoS 2 decreases with the applied strain and a semiconductor-to-metal transition is observed when the applied compressive strain is more than 0.85 Å. The compressive strain also exerts influence on the charge carrier effective mass and concentration, suggesting that mobility engineering and good piezoelectric effect can be realized in BP/MoS 2 heterostructure. The calculated optical property of BP/MoS 2 shows that BP/MoS 2 may be a very efficient ultra-violet photodetector because the type-II band alignment can facilitate the separation of photo-excited electrons and holes, and it can benefit from the great absorption coefficient in ultra-violet region.

Results and Discussion
Top and side views of the relaxed atomic structure of BP/MoS 2 bilayer are shown in Fig. 1. To minimize the lattice mismatch between the stacking sheets, a rectangular unit cell of MoS 2 is constructed. The supercell of this system is composed by 1 × 4 unit cells of BP and 1 × 5 unit cells of MoS 2 , which is the same with our previous work 27 . The optimised lattice constants of monolayer MoS 2 are a M = 3.19 Å, b M = 5.53 Å and the calculated lattice constants of monolayer BP are a P = 3.30 Å, b P = 4.62 Å. To determine the stable structure of BP/MoS 2 bilayer, the total energy as a function of uniaxial strains along X (zigzag) and Y (armchair) directions is depicted in Fig. 1(c). Both curves show their minimas under zero strain, at which point, the lattice constants of the supercell employed here are a = 3.26 Å, b = 22.18 Å.
Tuning the band gaps of 2D materials has been a challenge in band gap engineering. Many techniques, such as in-plane strains 28-31 and a strong external electric field 32,33 , are promising but suffer from lack of practical applicability. Therefore, in this work, the effect of applied normal compressive strain on the electronic and optical properties of BP/MoS 2 is studied. The strain is calculated as ε = d 0 − d, where d 0 and d are the equilibrium and instantaneous distances between the top phosphorus atom layer and the molybdenum atom layer. In Fig. 2, we show the projected band structure of BP/MoS 2 bilayer under different compressive strain. Figure 1(d) presents the Brillouin zone of rectangular cells with high-symmetry points labeled. In Fig. 2, the bands dominated by BP and MoS 2 are plotted by blue red dots, respectively. It is clear that the conduction band minimum (CBM) and the valence band maximum (VBM) are dominated by MoS 2 and BP, respectively, regardless of the applied compressive strain. More importantly, BP/MoS 2 vdW heterostructure has a type-II band alignment and thus the lowest energy electron-hole pairs are spatially separated with electrons and holes locating in MoS 2 and BP layer, respectively. These results indicate that BP/MoS 2 vdW heterostructure may be suitable for optoelectronics and solar energy conversion.
Evolution of band edges and total energy of BP/MoS 2 bilayer as a function of the applied compressive strain is concluded in Fig. 3(a). It can be proved that the interlayer distance d 0 at equilibrium state is 6.92 Å. The vdW interaction exerts little influence on the band edges of BP/MoS 2 . The BP/MoS 2 bilayer shows a finite indirect band gap up to 0.45 eV. Upon applying compressive strain, both the VBM and CBM move towards the Fermi level, resulting in a decreasing band gap with the applied strain. The BP/MoS 2 bilayer experiences a semiconductor-to-metal transition when the applied compressive strain is larger than 0.85 Å, which may lead to tunable conductivity and transport properties.
The difference between the integrated charge density of BP/MoS 2 bilayer under different compressive strain and that of the isolated monolayers as a function of the perpendicular distance is shown in Fig. 3 are the charge density at the (x, y, z) point in BP/MoS 2 supercell, BP and MoS 2 monolayer supercell, respectively. It is found that there is a small amount of charge exchange between BP layer to MoS 2 layer. Furthermore, the applied compressive strain can facilitate electrons transferring from BP to MoS 2 layer and holes transferring from MoS 2 to BP layer. More charge transfer between BP layer and MoS 2 layer suggests an increased carrier concentration and a stronger interlayer interaction. When under compressive strain, the quasi-fermi level of MoS 2 moves upward and the quasi-fermi level of BP layer moves downward, as shown in Fig. 3(c,d). As a result, the band gap of BP/MoS 2 bilayer decreases with the applied compressive strain.  For a three-dimensional material, when it is compressed along z direction, it usually will expand in the planar x and y directions. This effect is also taken into consideration in this work. Figure 4(a,b) gives the variation of the total energy with planar lattice constant, a and b, in BP/MoS 2 bilayer under a normal compressive stran of 0.8 Å. It is found that the total energy reaches its minima when a = 3.28 Å, b = 22.40 Å, respectively. Both a and b are a little larger than that of BP/MoS 2 bilayer without strains. It can be concluded that for the BP/MoS 2 heterostructure, when exerting compressive strain along z direction, it also will expand in the planar x and y direction, just as a three-dimensional material.
Monolayer MoS 2 has been reported to have a intrinsic spin-orbit gap. Whether the spin-orbit coupling (SOC) effect will exert obvious influence on the band gap of BP/MoS 2 heterostructure is unknown. In Fig. 4(c,d), the band structures of BP and MoS 2 monolayer, including SOC effect, are shown. BP and MoS 2 monolayer show a spin-orbit gap of 22 meV and 147 meV at K point in the valence band, which are in good agreement with previous works 15,34 . As shown in Fig. 4(e), though the SOC gap of MoS 2 can be comparable to the band gap of the BP/MoS 2 heterostructure under compressive strain, the SOC exerts little influence on the band gap of BP/MoS 2 because the VBM in the band structure of BP/MoS 2 bilayer is dominated by BP whose SOC gap is very small rather than by MoS 2 . So the SOC effect is not taken into consideration in following results.
With the applied normal compressive strain exerting influence on the band structure of BP/MoS 2 bilayer, a change in the effective masses of electrons and holes can be expected. Figure 5 displays the changes of electron and hole effective masses with the compressive strain. The effective mass is calculated using = / ( ∂ /∂ ) ħ m E k 2 2 2 , and the k points closely approach the VBM and CBM. In the band structure of BP/MoS 2 bilayer, it is clear that the CBM is located at a point between Y and Γ . The curve labeled CBM[Y] shows the effective mass of electron at CBM along Y direction. The electron effective mass (m e ) at CBM in the Γ direction is much smaller than that in the Y direction and it decreases gradually with applied compressive strain. The hole effective mass (m h ) at Γ is quite sensitive to the compressive strain. m h in the X direction increases drastically in the case of compressive strain larger than 0.4 Å, while that in the Y direction decreases gradually with the compressive strain. It also can be seen that the minimum of both m e and m h is much smaller than 1, suggesting that BP/MoS 2 as a type-II heterostructure may possesses great transport properties such as high mobility and conductivity. BP/MoS 2 vdW heterostructure may have great potential for applications in nanoelectronics and optoelectronics. Very recently, it is predicted that the BP/TMD system is a more efficient solar cell than the graphene/ TMD systems [35][36][37] because the former can benefit from the absorption of wider range of wavelength in the solar spectrum, and the type-II heterojunction alignment can allow more efficient hole-electron separation. In Fig. 6, therefore, the real (ε 1 ) and imaginary (ε 2 ) parts of the frequency-dependent dielectric functions are calculated by GGA-PBE and the frequency-dependent reflectivity R and absorption coefficient I are computed. Different with hexagonal lattice, the dielectric tensor has three independent components, namely, ε X , ε Y and ε ⊥ . Strong anisotropy is observed in the imaginary parts of dielectric functions ε X 2 , ε Y 2 and ε ⊥ 2 . The variations of reflectivity and absorption coefficients as the function of frequency also exhibit similar trend, as illustrated in middle and bottom planets. From the absorption spectrum (the bottom planet), we can find that BP/MoS 2 bilayer may have good application in ultraviolet light (above 3.27 eV) detecting. Furthermore, the effect of applied normal compressive strain on the optical properties of BP/MoS 2 bilayer is also studied. The applied compressive strain shows little influence on the optical properties in the X and Y directions, while exerts much more remarkable influence on the three optical parameters in perpendicular direction (ε ⊥ 2 , R ⊥ and I ⊥ ). It is because the compressive strain just enhances the interlayer interaction while exerts no influence on the in-plane interaction.

Conclusion
In summary, we have provided total energy and band structure calculations for the p-type BP/n-type MoS 2 vdW heterostructure and investigated its structural, electronic, transport and optical properties  by using first principles calculations. The decreased band gap by applied compressive strain indicates the great application potential of BP/MoS 2 vdW heterostructures in future nanoelectronics such as field effect transistors. A semiconductor-to-metal transition can be observed in BP/MoS 2 bilayer under the compressive strain. The reduced carrier effective mass and improved carrier concentration by the applied compressive strain suggest that mobility engineering and good piezoelectric effect can be realized in BP/MoS 2 bilayer. Furthermore, because the type-II heterojunction alignment can facilitate the separation of photo-excited electrons and holes, and it can benefit from the great absorption coefficient in ultra-violet region the BP/MoS 2 heterostructure may be a very efficient ultra-violet photodetector. According to our results, the BP/MoS 2 vdw p-n heterojuntion will present abundant opportunities for application in future nano-and optoelectronics such as photovoltaic cell, photodetector and logical device.

Methods
The first-principles calculations are performed by VASP code within plane-wave basis sets and Perdew-Burke-Ernzerhof (PBE) projector augmented wave pseudopotentials [38][39][40] . The semi-empirical DFT-D2 method method of Grimme is utilized for the dispersion correction in the interlayer interaction 41 . The plane-wave cutoff energy is set to be 500 eV and a vacuum larger than 12 Å is used to simulate the isolated sheet. The first Brillouin zone is sampled with a (15 × 15 × 1) Monkhorst-Pack grid for relaxation of BP and MoS 2 unitcells 42 . A (5 × 20 × 1) Monkhorst-Pack grid is used for relaxation of supercells. All the structures are fully relaxed with a force tolerance of 0.01 eV/Å.