Elastic Anisotropy and Optic Isotropy in Black Phosphorene/Transition-Metal Trisulfide van der Waals Heterostructures.

Anisotropic two-dimensional materials with direction-dependent mechanical and optical properties have attracted significant attention in recent years. In this work, based on density functional theory calculations, unexpected elastic anisotropy and optical isotropy in van der Waals (vdW) heterostructures have been theoretically proposed by assembling the well-known anisotropic black phosphorene (BP) and transition-metal trisulfides MS3 (M = Ti, Hf) together. It is interesting to see that the BP/MS3 vdW heterostructures show anisotropic flexibility in different directions according to the elastic constants, Young's modulus, and Poisson's ratio. We have further unraveled their physical origin of the type-II band structure nature with their conduction band minimum and valence band maximum separated in different layers. In particular, our results on the optical response functions including the excitonic effects of the BP/MS3 vdW heterostructures suggest their unexpected optical isotropies together with the enhancements of the solar energy conversion efficiency.

Introduction

Since the successful synthesis of graphene, an artificial two-dimensional (2D) atomically thin honeycomb carbon allotrope,[1] the family of 2D materials has been burgeoning to hundreds of members in the past decade.[2−11] Among them, low-symmetry 2D materials such as black phosphorene (BP),[12−14] tellurene,[15−17] monochalcogenides,[18−22] dichalcogenides,[23−25] and trichalcogenides[26−28] have attracted global attention because of their unique asymmetric 2D crystal structures together with their intraplane mechanical, electrical, and optical anisotropies.[23,29] For instance, the angle-dependent optical conductivity of BP may allow the construction of plasmonic devices where the surface plasmon polariton frequency will have a strong directional dependence on the wave vector.[30] Strong anisotropy quasi-one-dimensional charge density wave states have been observed in TiS3, a typical transition-metal trisulfide.[31] It is interesting to note that very different transport properties can be achieved in the field-effect transistors made from the same anisotropic material as well.[32] Nevertheless, the small species and amount of 2D systems with asymmetric crystal structures limit the effective application of their anisotropic properties.[33,34] Therefore, the development and investigation of novel anisotropic 2D materials is a never-ending task and of considerable interest and importance.

Combing two different 2D materials together via the van der Waals (vdW) interaction to obtain the so-called vdW heterostructure is an effective way to extend the number of 2D material family.[35−38] It is worth noting that the formation of a vdW heterostructure can not only combine the advantages of two 2D affiliations together but also introduce unexpected novel properties and phenomena.[39−42] For instance, flexible and semitransparent devices have been fabricated by combining different 2D semiconductor materials with fine-tuning of the emission spectra and enhancing electroluminescence.[43] By tuning the interfacial distance or applying an external electric filed, the band alignment between graphene and GaSe can be effectively modulated in the GaSe/graphene vdW heterostructure.[44−46] ZrS3 and HfS3 monolayers can form a stable type-II vdW heterostructure and show high solar power conversion efficiency up to 18%.[47] Superior electrical conductivity, omnidirectional flexibility, and high Li capacity can be achieved at the same time in the BP–TiC2 vdW heterostructure.[48] It is noted that the single-layer structures of BP and MX3 (X = Ti, Hf) sharing the same asymmetric orthogonal 2D crystal structure with reasonable lattice mismatch, which are good counterparts for establishing high-quality heterostructures.[27,49] Hence, it is very interesting to raise the question that whether BP and MX3 monolayers can form vdW heterostructures and show attractive novel physical and chemical properties.[50] Therefore, in this work, BP/MS3 vdW heterostructures were theoretically explored based on advanced vdW-corrected density functional theory (DFT) calculations in order to unravel their interesting mechanical properties, electronic structures, and optical properties. The present results reveal unexpected elastic anisotropy and optical isotropy in the BP/MS3 vdW heterostructures. We assume that these remarkable 2D materials will find their applications in nanoscaled flexible optical and photoelectrical devices.

Results and Discussions

BP, TiS3, and HfS3 monolayers share a similar orthogonal 2D lattice, and the corresponding structure sketches are illustrated in Figure a,b. As listed in Table , the optB88-vdW-optimized 2D lattice parameters of BP, TiS3, and HfS3 are a(BP) = 4.506 Å, b(BP) = 3.304 Å; a(TiS3) = 4.974 Å, b(TiS3) = 3.384 Å; and a(HfS3) = 5.094 Å, b(HfS3) = 3.576 Å, respectively, all of which are in excellent agreement with the previous literature.[26,27] Herein, the lattice mismatches between TiS3 (HfS3) and BP monolayers are Δa = 10.4% and Δb = 2.4% (Δa = 13.0%, Δb = 8.2%), which are larger than the well-known vdW heterostructures.[51,52] However, because the BP monolayer can sustain very large tensile strain and retain good lattice stability,[49,53] the construction of a BP–TiS3 (BP–HfS3) heterostructure is still expected. As there are many possible structural configurations between two 2D orthogonal lattices, a global total energy minimum search by combining the TiS3 (HfS3) and BP monolayers together is necessary. We first placed the original TiS3 (HfS3) and BP monolayers together in the same 2D orthogonal lattice and then we shifted the BP monolayer along the directions of lattice a and b independently by the step size of 0.1. After all, we have achieved a total of 100 stacking configurations for the BP–TiS3 (BP–HfS3) heterostructure. Herein, all the configurations were under full structural optimization with convergence criteria in terms of both energy and force. Figure c,d represents the cohesive energy mapping and the most energy favorable stacking configurations for the BP–TiS3 and BP–HfS3 heterostructures, respectively, after structural optimization. It is interesting to show that BP–TiS3 and BP–HfS3 heterostructures show a very different cohesive energy mapping feature and stable stacking configuration. The optB88-vdW-optimized 2D lattice parameters of BP–TiS3 and BP–HfS3 heterostructures with the most energy favorable stacking configuration are a(BP–TiS3) = 4.885 Å, b(BP–TiS3) = 3.340 Å, and a(BP–HfS3) = 4.996 Å, b(BP–HfS3) = 3.464 Å, respectively. We found that the BP monolayers sustain tensile strains in the heterostructures. On the contrary, the TiS3 and HfS3 monolayers withstand compression strains in the heterostructures. In addition, it is worth noting that the deformation of BP is much larger than that of TiS3 and HfS3 in the heterostructure, which could enhance the stability of the heterostructure due to the good tensile flexibility of BP.[49,53] Furthermore, the binding energy between BP and TiS3 (HfS3) monolayers in the heterostructures is calculated to be 24.44 (22.46) meV/Å2, which is close to the typical vdW binding energy of around 20 meV/Å2 by the advanced DFT calculations.[38] Hence, the BP–TiS3 (BP–HfS3) heterostructures belong to the novel growing family of vdW heterostructures.[54] The calculated P–P and M–S bond length and the corresponding changes from the monolayers to heterostructures of the BP–TiS3 (BP–HfS3) vdW heterostructure with the most stable stacking configuration are also listed in Table . For both cases, the P–P bond length is larger than that in the BP monolayer with positive bond length change values, whereas the M–S bond length is smaller than that in the MS3 monolayer with negative bond length change values, which represent the results that the BP monolayers sustain tensile strain but the MS3 monolayers withstand compression strain in the heterostructures. It is worth noting that all the bond length changes are smaller than 0.06 Å, showing the very small structure rearrangement and good structure stability of the monolayers in the vdW heterostructures.

Figure 1

Structure sketch of (a) BP and (b) MS3. The cohesive mapping and most stable structure configuration sketch of (c) BP–TiS3 heterostructure and (d) BP–HfS3 heterostructure. Herein, the small black balls are P atoms; the small yellow balls are S atoms; and the large red and blue balls are Ti and Hf atoms, respectively.

Table 1

Lattice Constants (Å), P–P and M–S Bond Length (Å) Range (from LM–Ss to LM–Sl) for the BP and MS3 Monolayers and the BP/MS3 Heterostructures, and the Corresponding Changes (Å) of the Bond Length from the Monolayers to Heterostructures

system	a	b	LP–Ps	ΔLP–Ps	LP–Pl	ΔLP–Ps	LM–Ss	ΔLM–Ss	LM–Sl	ΔLM–Sl
BP	4.506	3.304	2.226		2.260
TiS3	4.974	3.384					2.456		2.641
HfS3	5.094	3.576					2.580		2.699
BP–TiS3	4.885	3.340	2.243	0.017	2.277	0.017	2.453	–0.003	2.607	–0.034
BP–HfS3	4.996	3.464	2.278	0.052	2.287	0.027	2.568	–0.012	2.661	–0.038

The elastic constants were calculated by a step-by-step stress–strain method[55,56] to explore the mechanical stability and biaxial/omnidirectional stretchability of the BP/MS3 vdW heterostructures. For the 2D orthogonal system, there are only four independent elastic constants, C11, C22, C12, and C44, as summarized in Table . The elastic coefficient matrix can be represented as

Table 2

Estimated Elastic Constants C11, C22, C12, and C44 (N/m) for the BP and MS3 Monolayers and the BP/MS3 Heterostructures

system	C11	C22	C12	C44
BP	28.4	115.9	23.3	30.7
TiS3	93.4	137.3	14.1	23.6
HfS3	89.0	125.3	11.3	23.3
BP–TiS3	164.6	264.5	34.1	48.7
BP–HfS3	145.4	200.9	19.1	38.9

Herein, the elastic constants C11 and C22 describe the response stiffness of the 2D crystal when applied uniaxial tensile strains along the x and y direction, respectively. The elastic constant C12 implies the ability of the material to resist biaxial tensile strain. The elastic constant C44 expresses the deformation resistance of the in-plane shear strain. For all the cases, the calculated elastic constants satisfy the so-called Born’s mechanical stability criteria, C11, C22, C44 > 0 and Δ = C11C22 – C122 > 0, indicating that all the monolayers and vdW heterostructures are mechanically stable. From Table , three general roles can be concluded for the 2D monolayers and vdW heterostructures herein: first, the strong mechanical anisotropic behaviors can be seen from the elastic constants table. The calculated elastic constants C22 of both the monolayers and vdW heterostructures are significantly larger than that of C11, indicating that all these 2D materials are stiffer against strain in the zigzag (y) direction than in the armchair (x) direction. Second, the MS3 monolayers present higher rigidity than the BP monolayer with larger elastic constants. Third, the in-plane elastic constants of vdW heterostructures are larger than those of the corresponding monolayers, which means that the 2D monolayers will strain more than the vdW heterostructures under the same applied force. Herein, under strain or deformation, the vdW heterostructures with greater elastic constant or elastic modulus can act as the scaffold material or substrate material, where the interlayer vdW interaction can be employed to introduce a driving force to deform the 2D monolayers with smaller elastic stiffness values.

In order to get a further understanding of the mechanical properties of the BP/MS3 vdW heterostructures, we calculated Young’s modulus E(θ) and Poisson’s ν(θ) ratio along the arbitrary in-plane direction θ (θ identifies the angle relative to the armchair direction) from the elastic constants according to the following equations:where Δ = C11C22 – C122, c = cos θ, and s = sin θ. The corresponding polar diagrams are presented in Figure . For the BP monolayer, Young’s modulus and Poisson’s ratio along the y direction are about 3–4 times larger than those along the x direction. It is notable that the small negative Poisson’s ratio around 45° for the BP monolayer well represents the previous theoretical[57] and experimental results.[58] For the MS3 monolayers, Young’s modulus along the y direction is about 2 times larger than those along the x direction. In addition, Poisson’s ratio around 45° is about 3–4 times larger than those along the axial directions. Anyway, all the monolayers show strong anisotropic mechanical properties according to the plots of Young’s moduli and Poisson’s ratios. Interestingly, although the plots of BP and MS3 monolayers show very different patterns, Young’s moduli and Poisson’s ratio plots of the BP/MS3 vdW heterostructures follow the feature of MS3 monolayers very well. It is because that the rigidity of the MS3 monolayers is stronger than that of the BP monolayer. Anyway, compared with Young’s moduli of typical 2D flexible materials, such as graphene (342.2 N/m)[59,60] and BN (275.8 N/m),[60,61] the BP/MS3 vdW heterostructures show anisotropic flexibility in different directions.

Figure 2

Polar diagrams of (a) Young’s modulus E(θ) and (b) Poisson’s ratio ν(θ) for the BP and MS3 monolayers and the BP/MS3 heterostructures, where the angle θ identifies the extension direction with respect to the armchair direction.

For the better understanding of the electronic structures of the BP/MS3 vdW heterostructures, we started from the initial screening of the band structure of the monolayers. The calculated band structure of BP and MS3 monolayers using the HSE06 hybrid functional based on the optB86-vdW wave functions are illustrated in Figure S1. The results indicate that the BP and TiS3 monolayers are direct gap semiconductors, where both conduction band minimum (CBM) and valance band maximum (VBM) locate at the Γ point of the first Brillouin zone, whereas the HfS3 monolayer is an indirect gap semiconductor, where CBM locates at the Γ point and VBM locates between the Γ and K high-symmetry points. As is concluded in Table , the calculated band gaps using HSE06 hybrid functional calculations for BP, TiS3, and HfS3 monolayers are 1.41, 1.19, and 2.10 eV, respectively. All the electronic structure results agree well with previously published works.[26,27,62,63] It is interesting to note that both the BP/MS3 vdW heterostructures display direct band gap nature according to their HSE06 band structure plots in Figure . Both VBM and CBM of the two vdW heterostructures are located at the Γ point of the first Brillouin zone. The HSE06-predicted band gap values for BP–TiS3 and BP–HfS3 vdW heterostructures are 0.89 and 1.00 eV, respectively, which are also listed in Table . Further investigations on the band decomposition confirmed that the VBM of both the heterostructures is contributed by P atoms and CBM of BP–TiS3(BP–HfS3) vdW heterostructure is contributed by Ti(Hf) atoms. To provide vividly pictures for the better understanding of the role of the constituent layers in the vdW heterostructures, we have further illustrated the DFT wave functions for VBM and CBM in the real space in Figure as well. As can be seen, the VBM of both heterostructures is occupied by the P 3p electrons, whereas the CBM of the BP–TiS3 vdW heterostructure is originated from the Ti 3d electrons, and the CBM of the BP–HfS3 vdW heterostructure is originated from the Hf 5d electrons. Therefore, it can be concluded that the BP/MS3 vdW heterostructures are typical type-II vdW heterostructures where the CBM and VBM are localized in different monolayers of the heterostructures. It is worth noting that type-II heterostructures are highly desirable optical materials, where the photogenerated electrons and holes can be spatially separated from each other to different layers.[37] Such a phenomenon may efficiently reduce the electron–hole recombination probability and facilitate the efficiency for light detection and harvesting.[64]

Table 3

HSE06 Band Gap Eg (eV), Visible Light Absorption Spectrum Anisotropic Factor A, and TDHF Sunlight Energy Conversion Efficiency P (%) for the BP and MS3 monolayers and the BP/MS3 Heterostructures

system	Eg	A	Px	Py
BP	1.41	5.05	0.09	0.02
TiS3	1.19	0.37	0.07	0.21
HfS3	2.10	0.21	0.03	0.09
BP–TiS3	0.89	1.15	0.32	0.47
BP–HfS3	1.00	1.57	0.19	0.14

Figure 3

Band structure characterization plots and projected DFT wave functions of VBM and CBM for (a) BP–TiS3 heterostructure and (b) BP–HfS3 heterostructure. Herein, the size of the red, blue, yellow, and black circles illustrates the projected weight of electrons from Ti, Hf, S, and P atoms, respectively. The cyan and the violet isosurfaces present the positive and negative wave functions, respectively.

To understand how the excitation state and quantum confinement work on the optical properties, the optical responses were explored based on the real and imaginary parts of the time-dependent Hartree–Fock (TDHF) dielectric functions in Figure S2. TDHF simulations have been proved to provide a more accurate optical response with energy-distinguished peaks.[65] On the basis of the dielectric functions in Figure S2a–c, the optical absorption coefficients of the BP, TiS3, and HfS3 monolayers are presented in Figure a–c, respectively. It is obvious that all the monolayers show very strong optical anisotropy features. For the BP monolayer, the optical absorption along the xx direction is much stronger than the yy direction in the infrared and visible light range. In the ultraviolet range, the yy direction optical absorption peaks are higher than the xx direction. However, TiS3 and HfS3 monolayers show very opposite tendencies: the optical absorption along the xx direction is much weaker than the yy direction in the infrared and visible light range. In the ultraviolet range, the optical absorption properties are more or less isotropic. To achieve a quantitative understanding of the optical anisotropy features of the monolayers, we have integrated the optical absorption coefficients in the visible light range of 400–760 nm. The visible light absorption anisotropy factor A can be defined as A = S/S, where S and S are the area of the optical absorption coefficients in the visible light range along the xx and yy direction, respectively. For a 2D material, an A value of 1 denotes the optical isotropy, and any value that deviates from 1 represents the optical anisotropy. The calculated visible light absorption anisotropy factor A of the monolayers is also listed in Table . We observed that the BP, TiS3, and HfS3 monolayers possess strong optical anisotropy with A = 5.05, 0.37, and 0.21, respectively. We have further introduced the methods by Shi and Kioupakis[66] to obtain the sunlight energy conversion efficiency P to lowest energy excitons along different directions.[67] As the calculated results summarized in Table , the sunlight energy conversion efficiencies P of BP, TiS3, and HfS3 monolayers along the xx direction are 0.09, 0.07, and 0.03%, respectively, which are 0.02, 0.21, and 0.09% along the yy direction, respectively. The different energy conversion efficiencies along diffractions indicate strong optical anisotropy of the monolayers as well.

Figure 4

Absorption coefficient α of (a) BP, (b) TiS3, and (c) HfS3 monolayers; (d) BP–TiS3 heterostructure; and (e) BP–HfS3 heterostructure obtained using TDHF@HSE06 calculations.

We have further illustrated the optical absorption coefficients of the BP/MS3 vdW heterostructures in Figure d,e according to the TDHF dielectric functions in Figure S2d,e. It is obvious that the optical absorption of the vdW heterostructures is not simply added by that of the monolayers. Interestingly, the formation of the vdW heterostructures leads to a spectral blue shift along the xx direction and a spectral red shift along the yy direction. As a result, the optical absorption curves are homogenized in different directions. Especially in the visible light range, the difference of the optical absorption of the BP/MS3 vdW heterostructures along the xx direction and yy direction is much less than that of the monolayers. For quantitative understanding of the optical anisotropy features of the heterostructures, we have shown the visible light absorption anisotropy factor A and the sunlight energy conversion efficiency P in Table as well. We have got the visible light absorption anisotropy factor A values of 1.15 and 1.57 for the BP–TiS3 and BP–HfS3 vdW heterostructures, respectively, which show very high isotropic features. The sunlight energy conversion efficiencies P of the BP–TiS3 and BP–HfS3 vdW heterostructures along the xx direction are 0.32 and 0.47%, respectively, which are 0.19 and 0.14% along the yy direction, respectively. We found that the formation of the vdW heterostructures not only leads to optical isotropy but also enhances the solar light energy conversion efficiencies.

Conclusions

In summary, based on DFT calculations, we have systematically studied the mechanical properties, electronic structures, and optical properties of BP, TiS3, and HfS3 monolayers together with the corresponding heterostructures. The most energetic favorable stacking configurations of the BP–TiS3 and BP–HfS3 heterostructures have been determined by global total energy minimum searching. We have found that the BP–TiS3 and BP–HfS3 heterostructures belong to typical vdW heterostructures but show very different cohesive energy mapping features and stable stacking configurations. The elastic constants and mechanical property analyses indicate that the BP/MS3 vdW heterostructures show anisotropic flexibility in different directions. However, further optical properties guarantee their high isotropic visible light absorption features. Our findings will provide valuable insights into the exploration of these remarkable elastic anisotropic and optic isotropic 2D materials and vdW heterostructures and shed light on their applications in flexible optical and photoelectrical devices.

Computational Details

Our DFT calculations of the heterostructures were performed using the Vienna Ab initio Simulation Package (VASP)[68] in conjunction with the projector-augmented wave[69] generalized gradient approximations[70] of Perdew–Burke–Ernzerhof[71] pseudopotentials. The optB88-vdW functional[72] was used for all calculations to enhance the description of the vdW interactions.[73] The valence electron configurations for, P, S, Ti, and Hf were 3s23p3, 3s23p4, 3d34s1, and 5d26s2, respectively. We introduced at least 20 Å thick vacuum in the z-axis direction in order to prevent the interaction between periodic images.[51] The automatically generated k-point set of 14 × 20 × 1 with the Γ symmetry was used for self-consistent calculations. The relaxation convergence for ions and electrons were 1 × 10–5 and 1 × 10–6 eV, respectively, which were achieved with the cutoff energy of 600 eV. The Heyd–Scuseria–Ernzerhof (HSE06) hybrid functional[74] was applied to better describe the band gaps and dielectric functions. To obtain accurate optical dielectric functions comparable with experimental results,[37,75] we employed the TDHF calculations[76] based on HSE06 wave functions by including the excitonic effects. The crystal structures and isosurfaces were visualized using the VESTA package.[77]