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Mechanical and electronic properties of monolayer and bilayer phosphorene under uniaxial and isotropic strains
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Nanotechnology Nanotechnology 25 (2014) 455703 (9pp)
doi:10.1088/0957-4484/25/45/455703
Mechanical and electronic properties of monolayer and bilayer phosphorene under uniaxial and isotropic strains Ting Hu, Yang Han and Jinming Dong Group of Computational Condensed Matter Physics, National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, People’s Republic of China E-mail: [email protected] Received 29 April 2014, revised 5 August 2014 Accepted for publication 14 August 2014 Published 21 October 2014 Abstract
The mechanical and electronic properties of both the monolayer and bilayer phosphorenes under either isotropic or uniaxial strain have been systematically investigated using first-principles calculations. It is interesting to find that: 1) Under a large enough isotropic tensile strain, the monolayer phosphorene would lose its pucker structure and transform into a flat hexagonal plane, while two inner sublayers of the bilayer phosphorene could be bonded due to its interlayer distance contraction. 2) Under the uniaxial tensile strain along a zigzag direction, the pucker distance of each layer in the bilayer phosphorene can exhibit a specific negative Poisson’s ratio. 3) The electronic properties of both the monolayer and bilayer phosphorenes are sensitive to the magnitude and direction of the applied strains. Their band gaps decrease more rapidly under isotropic compressive strain than under uniaxial strain. Also, their direct-indirect band gap transitions happen at the larger isotropic tensile strains compared with that under uniaxial strain. 4) Under the isotropic compressive strain, the bilayer phosphorene exhibits a transition from a direct-gap semiconductor to a metal. In contrast, the monolayer phosphorene initially has the direct-indirect transition and then transitions to a metal. However, under isotropic tensile strain, both the bilayer and monolayer phosphorene show the direct-indirect transition and, finally, the transition to a metal. Our numerical results may open new potential applications of phosphorene in nanoelectronics and nanomechanical devices by external isotropic strain or uniaxial strain along different directions. Keywords: phosphorene, electronic properties, strain (Some figures may appear in colour only in the online journal) 1. Introduction
stacked together by van der Waals interactions, and the 2D phosphorene is another 2D stable elemental material that can be mechanically exfoliated [8] in addition to the graphene. Unlike the zero-gap semimetal graphene, phosphorene exhibits a direct band gap that can be modified from 1.51 eV of a monolayer to 0.59 eV of a five-layer [11]. Moreover, it has been reported that phosphorene has a high carrier mobility up to 1000 cm2 V−1 · s [7] and an appreciably high on/off ratio of up to 104 [8], making this material of great interest for future nanoelectronic applications. Furthermore, the phosphorene has a characteristic puckered structure, leading to a substantial anisotropy of its mechanical behavior, electric
Two-dimensional (2D) nanosheets with a hexagonal structure, such as graphene [1, 2], hexagonal boron-nitride (h-BN) [3, 4] and the transition metal dichalcogenides (TMD) [5, 6], have been intensively investigated in recent years due to their distinctive physical properties, which are caused by their low dimensionality and quantum confinement effect. Most recently, another 2D material, few-layer black phosphorus (phosphorene), was successfully fabricated and immediately attracted considerable attention [7–10]. Black phosphorus is a 3D layered material in which individual atomic layers are 0957-4484/14/455703+09$33.00
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[17, 18] in which the projected augmented wave method [19] and the Perdew–Burke–Ernzerhof (PBE) [20] exchange-correlation functional are employed. The 3 s and 3 p orbitals of the phosphorus atom are treated as valence ones. The layered structures are placed in the xy plane, and a large vacuum region is added in the z direction; therefore, the closest distance between the two adjacent phosphorenes is 15 Å. The geometric structures of the phosphorene are optimized by the conjugated-gradient minimization scheme. A plane-wave cutoff of 350 eV was used in our calculations, and the energies were converged to 10−5 eV atom−1. Both the atomic positions and the lattice constant along the ribbon axis were fully relaxed until the maximum residual forces on the atoms were less than 0.001 eV Å−1. The 1D Brillouin integration was sampled with a 30 × 30 × 1 Monkhorst–Pack grid. For the bilayer phosphorene, we incorporated the van der Waals interactions between its two layers by adding a semiempirical dispersion potential to the conventional Kohn–Sham DFT energy through a pair-wise force field, following Grimme’s DFT-D2 method [21].
conductance and optical responses [8, 12–14], which distinguishes this material from many other isotropic 2D crystals. On the other hand, the ability to externally and effectively control the electronic properties of a material is highly desirable. Strain engineering is known to be an effective method to tune the electronic and magnetic properties of a system. Recent work has reported that monolayer phosphorene demonstrates superior mechanical flexibility and can hold up to 30% [15] critical strain. By applying an appropriate uniaxial or biaxial strain, the anisotropy of the electron effective mass and the corresponding mobility direction can be rotated 90 degrees [14]. The first-principles calculation has predicted that the band gap of phosphorene experiences a direct-indirect-direct transition when an axial strain is applied [16]. A negative Poisson’s ratio in the out-of-plane direction under uniaxial strain is also predicted by first-principles calculations [13]. However, the phosphorene’s response to an isotropic (biaxial) strain has not yet been studied, which is expected to exhibit a different effect from that of the uniaxial strain listed above. In this paper, we have systematically studied the strainmodulated mechanical and electronic properties of both the monolayer and bilayer phosphorenes under either isotropic or uniaxial strain using first-principles calculations. It is interesting to find that the mechanical behavior of phosphorene is closely related to the layer number under isotropic strain. The electronic properties of monolayer and bilayer phosphorenes are sensitive to the magnitude and direction of applied strains. The band gap decreases more rapidly under isotropic compressive strain than under uniaxial strain. However the directindirect band gap transitions of both the monolayer and bilayer phosphorenes happen at a larger isotropic tensile strain compared with that under uniaxial strain. The remainder of this paper is organized as follows: In section 2, the geometrical structure and computational details are described. In section 3, the main numerical results and some discussions are given. Finally, in section 4, a conclusion is presented.
3. Results and discussions 3.1. Mechanical properties under isotropic strain
We have firstly studied the geometric structures of the monolayer and bilayer phosphorenes without an applied strain by first-principles calculations. The optimized geometric structure of the monolayer phosphorene with four P atoms per unit cell is shown in figure 1(a), which shows a puckered honeycomb-like lattice structure. Thus, we can define the phosphorene plane as the one lying at the pucker’s middle, as indicated by the blue dashed line in figure 1(a). The relaxed lattice constants of the monolayer phosphorene are a1 = 4.619 Å and a2 = 3.298 Å, which is in good agreement with other theoretical calculations [16]. There are two inequivalent P-P bonds in the relaxed structure, which are l1 = 2.221 Å and l2 = 2.259 Å, respectively. The pucker distance of the monolayer phosphorene is d = 2.104 Å. The geometric structure of the bilayer phosphorene is also optimized, as shown in figure 1(b) in which the lattice parameters are a1 = 4.503 Å and a2 = 3.313 Å, while the interlayer distance D between the two phosphorene planes is 5.235 Å. Then, an in-plane isotropic strain is applied on the monolayer phosphorene, while its pucker distance variation due to the Poisson’s ratio effect is automatically taken into account in the calculations. The stress-strain relation is presented in figure 2(a) from which one can determine that under compressive strain, its stress-strain curve loses its linear character beyond the strain value of about −13%; this indicates that the applied compressive strain beyond −13% is in the plastic deformation range, while for the tensile strain, the stress-strain relationship becomes a parabolic curve. One notices that when the isotropic compressive strain is applied, a wrinkle perpendicular to the monolayer plane emerges to relieve the in-plane stress, as shown in figure 2(b). The pucker distance d varies linearly with the isotropic strain
2. Model and method Black phosphorus is structurally similarly to layered graphite in which individual atomic layers are stacked together by van der Waals interactions. However, unlike the graphene’s flat structure, the single-layer black phosphorus (phosphorene) has a characteristic puckered honeycomb lattice, with each phosphorus atom covalently bonded with three adjacent atoms. Adjacent layers are stacked in an AB stacking order. The optimized geometric structures of the monolayer and bilayer phosphorenes are shown in figures 1(a) and (b); the first Brillouin zone is schematically illustrated in figure 1(c). The x and y axes in the phosphorene are taken to be along the armchair and zigzag directions of the phosphorene, respectively. The geometric and electronic structures are calculated using the density functional theory in the generalized gradient approximation (GGA), implemented by the VASP code 2
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Figure 1. Optimized geometrical structure of: (a) monolayer phosphorene and (b) bilayer phosphorene, viewed from both the in-plane y axis
(up) and out-of-plane z axis (down). The second layer of bilayer phosphorene in (b) is colored blue. The a1 × a2 unit cell is outlined with black dashed lines, while the phosphorene plane is represented by blue dashed lines. (c) The first Brillouin zone for both the monolayer and bilayer phosphorene is schematically illustrated.
in the range of −10% to 23%, as shown in figure 2(e) in which dε the corresponding Poisson’s ratio ν = − dε z of about 0.57 is
with inter-distance L will bond together. In this case, the corresponding geometric structure (side view along its y direction) is presented in figure 2(d).
xy
obtained where εx is the variation of pucker distance d, and εxy is the variation of the in-plane lattice parameter. A strongly nonlinear behavior of the pucker distance is observed under large tensile strain (>23%), showing a sharp reduction of pucker distance d; finally, when the tensile strain increases to 29%, the monolayer phosphorene eventually evolves into a flat hexagonal structure (d = 0) with its lattice constants of l1 = 2.261 Å and l2 = 2.214 Å, as indicated in figures 2(c) and (e). However, the situation is quite different for the bilayer phosphorene for which the pucker distance d of each layer varies between 2.303 Å and 1.775 Å, while interlayer distance D experiences a sharp reduction when the tensile strain is larger than 12%, as indicated in figure 2(f). The D reduction is mainly caused by the shrinkage of distance L between the bottom P atom of the 1st phosphorene layer and the upper P atom of the 2ndsecond one, as indicated in figure 1(b). It is interesting to find that even when neglecting the van der Waals interaction between the two monolayers, the L reduction can still be observed under the same tensile strain, indicating that the overlap of electronic wave functions between the two monolayers plays the more dominant role in the sharp reduction of interlayer distance D. By further increasing the tensile strain up to 22%, those two P atoms
3.2. Mechanical properties under uniaxial strain
We have also studied the uniaxial strain effects on the geometric structures of both the monolayer and bilayer phosphorenes, which are applied along either the x (armchair) or y (zigzag) direction. One notices that the monolayer phosphorene would never evolve into a flat plane structure under uniaxial tensile strain along either the armchair or zigzag direction even at a very high tensile strain of up to 35%, which is much different from the situation under isotropic tensile strain. For the bilayer phosphorene under the uniaxial strain, the dε in-plane Poisson’s ratios ν = − dtransverse are calculated, and the εaxial obtained results are shown in figure 3(a) in which the data are fitted to a function of y = −0.193 x + 0.004 x2 under uniaxial strain εx, and y = −0.798 x − 0.013 x2 under uniaxial strain εy. Here, 0.193 and 0.798 can be regarded as the elastic Poisson’s ratio under εx and εy, respectively, which are close to the values of about 0.2 and 0.7 in the monolayer phosphorene [16]. There are two types of out-of-plane Poisson’s ratios in the bilayer phosphorene: One is for the variation of pucker dε distance d in its each sublayer, defined as νd = − dε d , and the axial
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Figure 2. (a) The stress-strain relation of a monolayer phosphorene under isotropic strain. (b) The side view of the wrinkle in monolayer
phosphorene under compressive strain. (c) The top view of the monolayer phosphorene in its flat structure under 29% tensile strain. (d) The side view of the bilayer phosphorene under 22% tensile strain in which the inner two sublayers are bonded together. (e) The variation of pucker distance d of the monolayer phosphorene with isotropic strain. (f) The interlayer distance D and distance L, defined in figure 1(b), and the pucker distance d of each monolayer in the bilayer phosphorene versus the external strain.
Figure 3. (a) The transverse strain response in the y(x) direction when the uniaxial strain along the x(y) direction is applied on the bilayer
phosphorene. (b) The variations of the pucker distance d of each monolayer and the total thickness d + D in the bilayer phosphorene (both are counted in their z-direction strains) versus the external strain.
other one is for the variation of its total thickness D + d in the dε z-direction, defined as νd + D = − dεd + D . The intrinsic negative axial Poisson’s ratio in the out-of-plane direction can still be observed in each sublayer of the bilayer phosphorene in the
strain range from −5% to 15% under uniaxial strain εy, as shown in figure 3(b). However, the variation of total thickness D + d under both εx and εy exhibits a positive Poisson’s ratio. In addition, the two inner sublayers would not bond together 4
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Figure 4. Band structures of the monolayer phosphorene under different isotropic strains: (a) −7%, (b) −3%, (c) 0%, (d) 4%, (e) 12%, (f) 14% and (g) 22%. The negative values indicate the compressive strain. The variations of the B, C and D states with strain are marked by the dashed black, green and blue lines, respectively. (h) The band gap of the bilayer phosphorene varies with the external isotropic strain.
It can be found from figure 4 that the isotropic tensile strain can raise the pristine CBM position at Γ, labeled as state B, while the energies of the next two conduction bands above state B at Γ, labeled as states C and D, respectively, would decrease with the increasing strain. The variations of all of the B, C and D states with the applied isotropic strains are shown in figure 4(a) by the dashed black, green and blue lines, respectively. It is clearly seen from the dashed green line in figure 4(a) that the energy of state C will be lower than that of state B and will become a new CBM at a strain of 4%. Also, as shown by the dashed blue line in figure 4(a), further increasing the strain to ε = 17% will decrease the energy of state D until it is lower than that of state C. At the same time, the dispersion relation of the highest valence band between Γ–Y will gradually deform with an increase of the tensile strain, making the pristine VBM point at Γ move to another energy point between Γ–Y at ε = 14%, labeled as state E. The combined movements of these states in both the conduction and valance bands will induce a transition from a direct to an indirect gap at ε = 14% and finally a SM transition at a tensile strain of about 22%.
under a uniaxial tensile strain up to 30%, which is quite different than the situation under isotropic tensile strain. 3.3. Variation of electronic properties with isotropic strain
Our DFT calculations found that the monolayer phosphorene without the applied strain is a semiconductor with a direct band gap of about 0.91 eV at the Γ point, which is consistent with the previous studies [8, 12, 16]. In addition, the bilayer phosphorene is also a semiconductor with a direct band gap of about 0.41 eV at the Γ point. Then, an isotropic strain is applied on the monolayer phosphorene to modulate its electronic properties. It is interesting to find that the isotropic strain could tune its direct band gap into an indirect one and even induce a semiconductor-metal (S-M) transition. The detailed analysis of the band structure variation under different isotropic strains are presented in figure 4, in which only the band structures along two particular k directions, Γ to X (0.5, 0, 0) and Γ to Y (0, 0.5, 0), are given because of the remarkable strain effects on the band structure along these two directions. 5
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On the other hand, when isotropic compressive strain is applied, the CBM always locates at state B at the Γ point, which moves down close to the Fermi level with an increase of compressive strain, while the VBM experiences a transition from state A to state E equal to the transition under tensile strain. Thus, the direct band gap can also change into an indirect one under isotropic compressive strain at the much smaller value of about ε = −3%. A further increase of the compressive strain to ε = −7% will turn the semiconductor into a metal. The energy gap variation of monolayer phosphorene with isotropic strain is shown in figure 4(h). One can determine that the energy gap increased with the increasing tensile strain from the original measurement of 0.91 eV to 1.193 eV at ε = 4%, at which point the CBM was replaced by state C. Increasing the tensile strain further leads to a band gap reduction until the gap closure, which occurs at a tensile strain of 22%. However, under isotropic compressive strain, the band gap decreases linearly with the strain, falling sharply from 0.91 eV to 0 eV in the compressive strain range from 0% to −7%. To understand the band gap engineering via isotropic strain, let us study the bonding statuses of the related conduction band states B, C and D because their energy shift with the strain is closely related to their orbital statuses, e.g. bonding, anti-bonding and non-bonding [16], which then causes the corresponding band gap variation with the strain. For example, the external tensile strain would increase the atomic distance between the two bonding atoms, causing the bonding energy to increase but the anti-bonding energy to decrease while keeping the non-bonding energy unchanged. The variations of the B-, C- and D-state energies with the external strain are depicted in figure 5(a), while the corresponding partial charge densities of these states are shown in figures 5(b)–(d), respectively. Their wavefunction characteristics are also presented in figure 5. It can be seen from figure 5(a) that the energy of state B increases linearly with the strain, while the energies of state C and D display a nearly linear reduction with the strain. State B is mainly contributed by the coupling between the pz and px orbitals, which exhibit a bonding nature along the y direction while non-bonding in the x direction, as clearly seen from figure 5(b). Both states C and D have similar partial charge densities, which are mainly dominated by the px orbitals of the P atoms and show an anti-bonding character along the x direction and non-bonding in the y direction, as clearly seen from figures 5(c) and (d), respectively. Thus, when the atom distance increases under the tensile strain, the energy of state B is expected to increase, while the energy of states C and D decrease with the increasing strain. In the same way, the moving tendency of these states should be reversed under the compressive strain. The isotropic strain effect on the electronic structures of the bilayer phosphorene is also studied, and the obtained results under different strains are presented in figure 6. It can clearly be seen from figure 6 that the band gap transition from a direct to an indirect one, as well as the S-M transition, also exist, but the situation is different from that in monolayer
phosphorene. As in the case of monolayer phosphorene, the tensile strain can raise the energy of the CBM state B′ at Γ, while it can lower the energy of the third conduction band state C′. The variations of states B′ and C′ with applied isotropic strains are marked by the dashed black and green lines, respectively, in figure 6. However, as clearly seen from the dashed black line in figure 6, another state, F’, that lies between Γ–Y on the next lowest conduction band moves down faster than state C’, causing it to become a new CBM at only ε = 1%; this induces the direct-to-indirect band gap transition at that point. With a further increase of the tensile strain to 4%, the energy of state C' becomes lower than that of state F′, making the CBM return to the Γ point, which turns the indirect band gap back into the direct band gap. Interestingly, by increasing the tensile strain further, the energy of the conduction band at a point between Γ-X, labeled state G′, decreases rapidly, becoming lower than that of state C′ at ε = 8%. The variation of state G’ with the strain is also given in figure 6(a) by the dashed blue line. Meanwhile, the energy of state E′ lying on the highest valance band at a point between Γ–Y increases gradually with the tensile strain, becoming higher at ε = 8% than that of the pristine VBM at Γ, labeled as state A′, which leads to a transition from the direct band gap into an indirect one. Increasing the tensile strain further will cause the band gap closure, giving rise to a metallic state at ε = 14%. On the other hand, when the bilayer phosphorene is subjected to compressive strain, both the CBM and VBM at the Γ point will move close to the Fermi level, thereby reducing its band gap quickly. Finally, an S-M transition occurs when the compressive strain increases to ε = −3%. The band gap variation of the bilayer phosphorene with the isotropic strain is given in figure 6(i), which shows a similar variation tendency to that of monolayer phosphorene, but it also shows some differences with that. For example, the S-M transition of bilayer phosphorene happens at the smaller critical strains under both tensile and compressive strains, which is found to be about ε = 14% (−3%) under the tensile (compressive) strain. The band gap of the bilayer phosphorene reaches its maximum value of 0.667 eV at 4% tensile strain, at which point state C′ replaces state B′ to become the CBM. 3.4. Variation of electronic properties with uniaxial strain
We have also investigated the effect of uniaxial strain on the electronic properties of both the monolayer and bilayer phosphorenes. The calculated band gap variations with the uniaxial strains εx and εy for the monolayer and bilayer phosphorene are shown, respectively, in figures 7(a) and (b). It can be found from figure 7(a) that the band gap variation of the monolayer phosphorene in the uniaxial strain range from −12% to 12% is in good agreement with the previous theoretical work [16]. Any further increase of the compressive strain will induce the semiconductor-metal transition at εx = −14% and εy = −13%, respectively. On the other hand, the band gap of the monolayer phosphorene does not go to zero even when the uniaxial tensile strain εx 6
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Figure 5. (a) The energies of the B, C and D states vary with the isotropic strain. The corresponding partial charge densities of (b) state B, (c)
state C and (d) state D are also depicted. The schematics of the wavefunction character of states C and D are given on the right side of their charge densities.
increases up to εx = 30%. While under the tensile strain along the y direction, the band gap increases linearly from εy = 15% to 25% and then reduces to zero at εy = 28%. As for the bilayer phosphorene, the tensile strain along the x direction can initially increase the band gap from 0.413 eV to its maximum value of 0.667 eV at εx = 6% and then reduce the band gap to zero at εx = 25%. The compressive strain along the x direction gives rise to a linear reduction of the band gap of the bilayer phosphorene to zero at εx = −7%, which is in contrast to that of the monolayer one at a much lower value of −14%. The bilayer phosphorene shows direct semiconducting character under the uniaxial strain εx in the range of −6% to 2% and then turns into an indirect one when the strain is larger than 2% until the S-M transition happens at 25% strain. When the uniaxial strain is taken along the y direction, the band gap of the bilayer phosphorene increases to 0.496 eV as the uniaxial tensile strain increases up to εy = 4%, at which point the band gap changes from direct to indirect. Any further increase of the tensile strain will quickly reduce the indirect band gap to zero at εy = 12%. However, the band gap will open up again at the larger tensile strain of 16%; finally, the system again transforms into a metal when the tensile strain εy increases to 19%. On the other hand, when the uniaxial compressive strain along the y direction is applied, the band gap experiences a reduction until the S-M transition occurs at εy = −12%,
accompanied by a direct-indirect band gap transition at εy = −4%.
4. Conclusions We have systematically investigated the strain-modulated mechanical and electronic properties of both the monolayer and bilayer phosphorenes under either isotropic or uniaxial strain using first-principles calculations. We found that: (1) A sufficiently large isotropic tensile strain on the monolayer phosphorene would make its pucker structure transform into a flat hexagonal plane structure at ε = 29%, while that on bilayer phosphorene leads to the contraction of its interlayer distance, causing its two inner sublayers to be bonded at ε = 22%. (2) The negative out-of-plane Poisson’s ratio can also be obtained in each sublayer of bilayer phosphorene under uniaxial εy, but the variation of its total thickness D + d exhibits a positive Poisson’s ratio under both εx and εy. (3) Under isotropic compressive strain, the band gaps of monolayer and bilayer phosphorene decrease more rapidly than under the uniaxial one. Also, the bilayer phosphorene exhibits a transition from a direct gap semiconductor to a metal at −3%, which is a very small percentage. This is in contrast to the initial directindirect transition and then to the metal transition for the monolayer phosphorene at the larger −7%. (4) Under isotropic tensile strain, the direct-indirect band gap transitions of both the mono- and bilayer phosphorenes happen at the larger 7
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Figure 6. Band structures of the bilayer phosphorene under different strains: (a) −8%, (b) −3%, (c) 0%, (d) 2%, (e) 5%, (f) 9%, (g) 12% and (h) 14%. The variations of states B′, C′ and G′ with the strain are marked out by the dashed black, green and blue lines, respectively. (i) The band gap of the bilayer phosphorene varies with the external isotropic strain.
Figure 7. The band gap of the (a) monolayer phosphorene and (b) bilayer phosphorene varies with the external uniaxial strain applied along
either the x or y direction.
strain values of 14% and 8%, respectively, compared with those under a uniaxial one, which are 7% (8%) under εx (εy) for the monolayer and 2% (4% ) under εx (εy) for the bilayer. (5) However, under the isotropic tensile strain, the band gaps of both increase firstly and then decrease, showing a transition from a direct gap to an indirect one and finally to a metal. Our
numerical results demonstrate a possible way to effectively tune the mechanic and electronic properties of both the monoand bilayer phosphorenes by an external isotropic strain or a uniaxial one along different directions, which may open new potential applications of phosphorene in nanoelectronics and nanomechanical devices. 8
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Acknowledgments
[8] Liu H, Neal A T, Zhu Z, Tomanek D and Ye P D 2014 Phosphorene: a new 2D material with high carrier mobility ACS Nano 8 4033–41 [9] Xia F, Wang H and Jia Y 2014 Rediscovering black phosphorus: a unique anisotropic 2D material for optoelectronics and electronics Nat. Commun. 5 4458 [10] Reich E S 2014 Nature 506 19 [11] Qiao J, Kong X, Hu Z-X, Yang F and Ji W 2014 Few-layer black phosphorus: emerging direct band gap semiconductor with high carrier mobility Nat. Commun. 5 4475 [12] Tran V, Soklaski R, Liang Y and Yang L 2014 Phys. Rev. B 89 235319 [13] Jiang J-W and Park H S 2014 Negative poisson’s ratio in single-layer black phosphorus Nat. Commun. 5 4727 [14] Fei R and Yang L 2014 Strain-engineering anisotropic electrical conductance of phosphorene Nano Lett. 14 2884 [15] Wei Q and Peng X 2014 Superior mechanical flexibility of phosphorene and few-layer black phosphorus Appl. Phys. Lett. 104 251915 [16] Peng X, Copple A and Wei Q 2014 Strain engineered directindirect band gap transition and its mechanism in 2D phosphorene Phys. Rev. B 90 085402 [17] Kresse G and Hafner J 1993 Phys. Rev. B 48 13115 [18] Kresse G and Furthmüller J 1996 Comput. Mater. Sci. 6 15 [19] Blöchl P E 1994 Phys. Rev. B 50 17953 [20] Perdew J P, Burke K and Ernzerhof M 1996 Phys. Rev. Lett. 77 3865 [21] Grimme S 2006 J. Comp. Chem. 27 1787
This work is supported by the State Key Program for the Basic Research of China through Grant Nos. 2010CB630704 and 2011CB922100Q. Our numerical calculations were performed in the High Performance Computing Center of Nanjing University.
References [1] Geim A K and Novoselov K S 2007 Nat. Mater. 6 183–91 [2] Neto A H C, Guinea F, Peres N M R, Novoselov K S and Geim A K 2009 Rev. Mod. Phys. 81 109 [3] Jin C, Lin F, Suenaga K and Iijima S 2009 Phys. Rev. Lett. 102 195505 [4] Topsakal M, Akturk E and Ciraci S 2009 Phys. Rev. B 79 115442 [5] Mak K F, Lee C, Hone J, Shan J and Heinz T F 2010 Phys. Rev. Lett. 105 136805 [6] Splendiani A, Sun L, Zhang Y, Li T, Kim J, Chim C, Galli G and Wang F 2010 Nano Lett. 10 1271–5 [7] Li L, Yu Y, Ye G-J, Ge Q, Ou X, Wu H, Feng D, Chen X-H and Zhang Y 2014 Black phosphorus field-effect transistors Nat. Nanotechnol. 9 372–7
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Mechanical and electronic properties of monolayer and bilayer phosphorene under uniaxial and isotropic strains
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 Nanotechnology 25 455703 (http://iopscience.iop.org/0957-4484/25/45/455703) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 137.207.120.173 This content was downloaded on 18/11/2014 at 18:03
Please note that terms and conditions apply.
Nanotechnology Nanotechnology 25 (2014) 455703 (9pp)
doi:10.1088/0957-4484/25/45/455703
Mechanical and electronic properties of monolayer and bilayer phosphorene under uniaxial and isotropic strains Ting Hu, Yang Han and Jinming Dong Group of Computational Condensed Matter Physics, National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, People’s Republic of China E-mail: [email protected] Received 29 April 2014, revised 5 August 2014 Accepted for publication 14 August 2014 Published 21 October 2014 Abstract
The mechanical and electronic properties of both the monolayer and bilayer phosphorenes under either isotropic or uniaxial strain have been systematically investigated using first-principles calculations. It is interesting to find that: 1) Under a large enough isotropic tensile strain, the monolayer phosphorene would lose its pucker structure and transform into a flat hexagonal plane, while two inner sublayers of the bilayer phosphorene could be bonded due to its interlayer distance contraction. 2) Under the uniaxial tensile strain along a zigzag direction, the pucker distance of each layer in the bilayer phosphorene can exhibit a specific negative Poisson’s ratio. 3) The electronic properties of both the monolayer and bilayer phosphorenes are sensitive to the magnitude and direction of the applied strains. Their band gaps decrease more rapidly under isotropic compressive strain than under uniaxial strain. Also, their direct-indirect band gap transitions happen at the larger isotropic tensile strains compared with that under uniaxial strain. 4) Under the isotropic compressive strain, the bilayer phosphorene exhibits a transition from a direct-gap semiconductor to a metal. In contrast, the monolayer phosphorene initially has the direct-indirect transition and then transitions to a metal. However, under isotropic tensile strain, both the bilayer and monolayer phosphorene show the direct-indirect transition and, finally, the transition to a metal. Our numerical results may open new potential applications of phosphorene in nanoelectronics and nanomechanical devices by external isotropic strain or uniaxial strain along different directions. Keywords: phosphorene, electronic properties, strain (Some figures may appear in colour only in the online journal) 1. Introduction
stacked together by van der Waals interactions, and the 2D phosphorene is another 2D stable elemental material that can be mechanically exfoliated [8] in addition to the graphene. Unlike the zero-gap semimetal graphene, phosphorene exhibits a direct band gap that can be modified from 1.51 eV of a monolayer to 0.59 eV of a five-layer [11]. Moreover, it has been reported that phosphorene has a high carrier mobility up to 1000 cm2 V−1 · s [7] and an appreciably high on/off ratio of up to 104 [8], making this material of great interest for future nanoelectronic applications. Furthermore, the phosphorene has a characteristic puckered structure, leading to a substantial anisotropy of its mechanical behavior, electric
Two-dimensional (2D) nanosheets with a hexagonal structure, such as graphene [1, 2], hexagonal boron-nitride (h-BN) [3, 4] and the transition metal dichalcogenides (TMD) [5, 6], have been intensively investigated in recent years due to their distinctive physical properties, which are caused by their low dimensionality and quantum confinement effect. Most recently, another 2D material, few-layer black phosphorus (phosphorene), was successfully fabricated and immediately attracted considerable attention [7–10]. Black phosphorus is a 3D layered material in which individual atomic layers are 0957-4484/14/455703+09$33.00
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[17, 18] in which the projected augmented wave method [19] and the Perdew–Burke–Ernzerhof (PBE) [20] exchange-correlation functional are employed. The 3 s and 3 p orbitals of the phosphorus atom are treated as valence ones. The layered structures are placed in the xy plane, and a large vacuum region is added in the z direction; therefore, the closest distance between the two adjacent phosphorenes is 15 Å. The geometric structures of the phosphorene are optimized by the conjugated-gradient minimization scheme. A plane-wave cutoff of 350 eV was used in our calculations, and the energies were converged to 10−5 eV atom−1. Both the atomic positions and the lattice constant along the ribbon axis were fully relaxed until the maximum residual forces on the atoms were less than 0.001 eV Å−1. The 1D Brillouin integration was sampled with a 30 × 30 × 1 Monkhorst–Pack grid. For the bilayer phosphorene, we incorporated the van der Waals interactions between its two layers by adding a semiempirical dispersion potential to the conventional Kohn–Sham DFT energy through a pair-wise force field, following Grimme’s DFT-D2 method [21].
conductance and optical responses [8, 12–14], which distinguishes this material from many other isotropic 2D crystals. On the other hand, the ability to externally and effectively control the electronic properties of a material is highly desirable. Strain engineering is known to be an effective method to tune the electronic and magnetic properties of a system. Recent work has reported that monolayer phosphorene demonstrates superior mechanical flexibility and can hold up to 30% [15] critical strain. By applying an appropriate uniaxial or biaxial strain, the anisotropy of the electron effective mass and the corresponding mobility direction can be rotated 90 degrees [14]. The first-principles calculation has predicted that the band gap of phosphorene experiences a direct-indirect-direct transition when an axial strain is applied [16]. A negative Poisson’s ratio in the out-of-plane direction under uniaxial strain is also predicted by first-principles calculations [13]. However, the phosphorene’s response to an isotropic (biaxial) strain has not yet been studied, which is expected to exhibit a different effect from that of the uniaxial strain listed above. In this paper, we have systematically studied the strainmodulated mechanical and electronic properties of both the monolayer and bilayer phosphorenes under either isotropic or uniaxial strain using first-principles calculations. It is interesting to find that the mechanical behavior of phosphorene is closely related to the layer number under isotropic strain. The electronic properties of monolayer and bilayer phosphorenes are sensitive to the magnitude and direction of applied strains. The band gap decreases more rapidly under isotropic compressive strain than under uniaxial strain. However the directindirect band gap transitions of both the monolayer and bilayer phosphorenes happen at a larger isotropic tensile strain compared with that under uniaxial strain. The remainder of this paper is organized as follows: In section 2, the geometrical structure and computational details are described. In section 3, the main numerical results and some discussions are given. Finally, in section 4, a conclusion is presented.
3. Results and discussions 3.1. Mechanical properties under isotropic strain
We have firstly studied the geometric structures of the monolayer and bilayer phosphorenes without an applied strain by first-principles calculations. The optimized geometric structure of the monolayer phosphorene with four P atoms per unit cell is shown in figure 1(a), which shows a puckered honeycomb-like lattice structure. Thus, we can define the phosphorene plane as the one lying at the pucker’s middle, as indicated by the blue dashed line in figure 1(a). The relaxed lattice constants of the monolayer phosphorene are a1 = 4.619 Å and a2 = 3.298 Å, which is in good agreement with other theoretical calculations [16]. There are two inequivalent P-P bonds in the relaxed structure, which are l1 = 2.221 Å and l2 = 2.259 Å, respectively. The pucker distance of the monolayer phosphorene is d = 2.104 Å. The geometric structure of the bilayer phosphorene is also optimized, as shown in figure 1(b) in which the lattice parameters are a1 = 4.503 Å and a2 = 3.313 Å, while the interlayer distance D between the two phosphorene planes is 5.235 Å. Then, an in-plane isotropic strain is applied on the monolayer phosphorene, while its pucker distance variation due to the Poisson’s ratio effect is automatically taken into account in the calculations. The stress-strain relation is presented in figure 2(a) from which one can determine that under compressive strain, its stress-strain curve loses its linear character beyond the strain value of about −13%; this indicates that the applied compressive strain beyond −13% is in the plastic deformation range, while for the tensile strain, the stress-strain relationship becomes a parabolic curve. One notices that when the isotropic compressive strain is applied, a wrinkle perpendicular to the monolayer plane emerges to relieve the in-plane stress, as shown in figure 2(b). The pucker distance d varies linearly with the isotropic strain
2. Model and method Black phosphorus is structurally similarly to layered graphite in which individual atomic layers are stacked together by van der Waals interactions. However, unlike the graphene’s flat structure, the single-layer black phosphorus (phosphorene) has a characteristic puckered honeycomb lattice, with each phosphorus atom covalently bonded with three adjacent atoms. Adjacent layers are stacked in an AB stacking order. The optimized geometric structures of the monolayer and bilayer phosphorenes are shown in figures 1(a) and (b); the first Brillouin zone is schematically illustrated in figure 1(c). The x and y axes in the phosphorene are taken to be along the armchair and zigzag directions of the phosphorene, respectively. The geometric and electronic structures are calculated using the density functional theory in the generalized gradient approximation (GGA), implemented by the VASP code 2
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Figure 1. Optimized geometrical structure of: (a) monolayer phosphorene and (b) bilayer phosphorene, viewed from both the in-plane y axis
(up) and out-of-plane z axis (down). The second layer of bilayer phosphorene in (b) is colored blue. The a1 × a2 unit cell is outlined with black dashed lines, while the phosphorene plane is represented by blue dashed lines. (c) The first Brillouin zone for both the monolayer and bilayer phosphorene is schematically illustrated.
in the range of −10% to 23%, as shown in figure 2(e) in which dε the corresponding Poisson’s ratio ν = − dε z of about 0.57 is
with inter-distance L will bond together. In this case, the corresponding geometric structure (side view along its y direction) is presented in figure 2(d).
xy
obtained where εx is the variation of pucker distance d, and εxy is the variation of the in-plane lattice parameter. A strongly nonlinear behavior of the pucker distance is observed under large tensile strain (>23%), showing a sharp reduction of pucker distance d; finally, when the tensile strain increases to 29%, the monolayer phosphorene eventually evolves into a flat hexagonal structure (d = 0) with its lattice constants of l1 = 2.261 Å and l2 = 2.214 Å, as indicated in figures 2(c) and (e). However, the situation is quite different for the bilayer phosphorene for which the pucker distance d of each layer varies between 2.303 Å and 1.775 Å, while interlayer distance D experiences a sharp reduction when the tensile strain is larger than 12%, as indicated in figure 2(f). The D reduction is mainly caused by the shrinkage of distance L between the bottom P atom of the 1st phosphorene layer and the upper P atom of the 2ndsecond one, as indicated in figure 1(b). It is interesting to find that even when neglecting the van der Waals interaction between the two monolayers, the L reduction can still be observed under the same tensile strain, indicating that the overlap of electronic wave functions between the two monolayers plays the more dominant role in the sharp reduction of interlayer distance D. By further increasing the tensile strain up to 22%, those two P atoms
3.2. Mechanical properties under uniaxial strain
We have also studied the uniaxial strain effects on the geometric structures of both the monolayer and bilayer phosphorenes, which are applied along either the x (armchair) or y (zigzag) direction. One notices that the monolayer phosphorene would never evolve into a flat plane structure under uniaxial tensile strain along either the armchair or zigzag direction even at a very high tensile strain of up to 35%, which is much different from the situation under isotropic tensile strain. For the bilayer phosphorene under the uniaxial strain, the dε in-plane Poisson’s ratios ν = − dtransverse are calculated, and the εaxial obtained results are shown in figure 3(a) in which the data are fitted to a function of y = −0.193 x + 0.004 x2 under uniaxial strain εx, and y = −0.798 x − 0.013 x2 under uniaxial strain εy. Here, 0.193 and 0.798 can be regarded as the elastic Poisson’s ratio under εx and εy, respectively, which are close to the values of about 0.2 and 0.7 in the monolayer phosphorene [16]. There are two types of out-of-plane Poisson’s ratios in the bilayer phosphorene: One is for the variation of pucker dε distance d in its each sublayer, defined as νd = − dε d , and the axial
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Figure 2. (a) The stress-strain relation of a monolayer phosphorene under isotropic strain. (b) The side view of the wrinkle in monolayer
phosphorene under compressive strain. (c) The top view of the monolayer phosphorene in its flat structure under 29% tensile strain. (d) The side view of the bilayer phosphorene under 22% tensile strain in which the inner two sublayers are bonded together. (e) The variation of pucker distance d of the monolayer phosphorene with isotropic strain. (f) The interlayer distance D and distance L, defined in figure 1(b), and the pucker distance d of each monolayer in the bilayer phosphorene versus the external strain.
Figure 3. (a) The transverse strain response in the y(x) direction when the uniaxial strain along the x(y) direction is applied on the bilayer
phosphorene. (b) The variations of the pucker distance d of each monolayer and the total thickness d + D in the bilayer phosphorene (both are counted in their z-direction strains) versus the external strain.
other one is for the variation of its total thickness D + d in the dε z-direction, defined as νd + D = − dεd + D . The intrinsic negative axial Poisson’s ratio in the out-of-plane direction can still be observed in each sublayer of the bilayer phosphorene in the
strain range from −5% to 15% under uniaxial strain εy, as shown in figure 3(b). However, the variation of total thickness D + d under both εx and εy exhibits a positive Poisson’s ratio. In addition, the two inner sublayers would not bond together 4
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Figure 4. Band structures of the monolayer phosphorene under different isotropic strains: (a) −7%, (b) −3%, (c) 0%, (d) 4%, (e) 12%, (f) 14% and (g) 22%. The negative values indicate the compressive strain. The variations of the B, C and D states with strain are marked by the dashed black, green and blue lines, respectively. (h) The band gap of the bilayer phosphorene varies with the external isotropic strain.
It can be found from figure 4 that the isotropic tensile strain can raise the pristine CBM position at Γ, labeled as state B, while the energies of the next two conduction bands above state B at Γ, labeled as states C and D, respectively, would decrease with the increasing strain. The variations of all of the B, C and D states with the applied isotropic strains are shown in figure 4(a) by the dashed black, green and blue lines, respectively. It is clearly seen from the dashed green line in figure 4(a) that the energy of state C will be lower than that of state B and will become a new CBM at a strain of 4%. Also, as shown by the dashed blue line in figure 4(a), further increasing the strain to ε = 17% will decrease the energy of state D until it is lower than that of state C. At the same time, the dispersion relation of the highest valence band between Γ–Y will gradually deform with an increase of the tensile strain, making the pristine VBM point at Γ move to another energy point between Γ–Y at ε = 14%, labeled as state E. The combined movements of these states in both the conduction and valance bands will induce a transition from a direct to an indirect gap at ε = 14% and finally a SM transition at a tensile strain of about 22%.
under a uniaxial tensile strain up to 30%, which is quite different than the situation under isotropic tensile strain. 3.3. Variation of electronic properties with isotropic strain
Our DFT calculations found that the monolayer phosphorene without the applied strain is a semiconductor with a direct band gap of about 0.91 eV at the Γ point, which is consistent with the previous studies [8, 12, 16]. In addition, the bilayer phosphorene is also a semiconductor with a direct band gap of about 0.41 eV at the Γ point. Then, an isotropic strain is applied on the monolayer phosphorene to modulate its electronic properties. It is interesting to find that the isotropic strain could tune its direct band gap into an indirect one and even induce a semiconductor-metal (S-M) transition. The detailed analysis of the band structure variation under different isotropic strains are presented in figure 4, in which only the band structures along two particular k directions, Γ to X (0.5, 0, 0) and Γ to Y (0, 0.5, 0), are given because of the remarkable strain effects on the band structure along these two directions. 5
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On the other hand, when isotropic compressive strain is applied, the CBM always locates at state B at the Γ point, which moves down close to the Fermi level with an increase of compressive strain, while the VBM experiences a transition from state A to state E equal to the transition under tensile strain. Thus, the direct band gap can also change into an indirect one under isotropic compressive strain at the much smaller value of about ε = −3%. A further increase of the compressive strain to ε = −7% will turn the semiconductor into a metal. The energy gap variation of monolayer phosphorene with isotropic strain is shown in figure 4(h). One can determine that the energy gap increased with the increasing tensile strain from the original measurement of 0.91 eV to 1.193 eV at ε = 4%, at which point the CBM was replaced by state C. Increasing the tensile strain further leads to a band gap reduction until the gap closure, which occurs at a tensile strain of 22%. However, under isotropic compressive strain, the band gap decreases linearly with the strain, falling sharply from 0.91 eV to 0 eV in the compressive strain range from 0% to −7%. To understand the band gap engineering via isotropic strain, let us study the bonding statuses of the related conduction band states B, C and D because their energy shift with the strain is closely related to their orbital statuses, e.g. bonding, anti-bonding and non-bonding [16], which then causes the corresponding band gap variation with the strain. For example, the external tensile strain would increase the atomic distance between the two bonding atoms, causing the bonding energy to increase but the anti-bonding energy to decrease while keeping the non-bonding energy unchanged. The variations of the B-, C- and D-state energies with the external strain are depicted in figure 5(a), while the corresponding partial charge densities of these states are shown in figures 5(b)–(d), respectively. Their wavefunction characteristics are also presented in figure 5. It can be seen from figure 5(a) that the energy of state B increases linearly with the strain, while the energies of state C and D display a nearly linear reduction with the strain. State B is mainly contributed by the coupling between the pz and px orbitals, which exhibit a bonding nature along the y direction while non-bonding in the x direction, as clearly seen from figure 5(b). Both states C and D have similar partial charge densities, which are mainly dominated by the px orbitals of the P atoms and show an anti-bonding character along the x direction and non-bonding in the y direction, as clearly seen from figures 5(c) and (d), respectively. Thus, when the atom distance increases under the tensile strain, the energy of state B is expected to increase, while the energy of states C and D decrease with the increasing strain. In the same way, the moving tendency of these states should be reversed under the compressive strain. The isotropic strain effect on the electronic structures of the bilayer phosphorene is also studied, and the obtained results under different strains are presented in figure 6. It can clearly be seen from figure 6 that the band gap transition from a direct to an indirect one, as well as the S-M transition, also exist, but the situation is different from that in monolayer
phosphorene. As in the case of monolayer phosphorene, the tensile strain can raise the energy of the CBM state B′ at Γ, while it can lower the energy of the third conduction band state C′. The variations of states B′ and C′ with applied isotropic strains are marked by the dashed black and green lines, respectively, in figure 6. However, as clearly seen from the dashed black line in figure 6, another state, F’, that lies between Γ–Y on the next lowest conduction band moves down faster than state C’, causing it to become a new CBM at only ε = 1%; this induces the direct-to-indirect band gap transition at that point. With a further increase of the tensile strain to 4%, the energy of state C' becomes lower than that of state F′, making the CBM return to the Γ point, which turns the indirect band gap back into the direct band gap. Interestingly, by increasing the tensile strain further, the energy of the conduction band at a point between Γ-X, labeled state G′, decreases rapidly, becoming lower than that of state C′ at ε = 8%. The variation of state G’ with the strain is also given in figure 6(a) by the dashed blue line. Meanwhile, the energy of state E′ lying on the highest valance band at a point between Γ–Y increases gradually with the tensile strain, becoming higher at ε = 8% than that of the pristine VBM at Γ, labeled as state A′, which leads to a transition from the direct band gap into an indirect one. Increasing the tensile strain further will cause the band gap closure, giving rise to a metallic state at ε = 14%. On the other hand, when the bilayer phosphorene is subjected to compressive strain, both the CBM and VBM at the Γ point will move close to the Fermi level, thereby reducing its band gap quickly. Finally, an S-M transition occurs when the compressive strain increases to ε = −3%. The band gap variation of the bilayer phosphorene with the isotropic strain is given in figure 6(i), which shows a similar variation tendency to that of monolayer phosphorene, but it also shows some differences with that. For example, the S-M transition of bilayer phosphorene happens at the smaller critical strains under both tensile and compressive strains, which is found to be about ε = 14% (−3%) under the tensile (compressive) strain. The band gap of the bilayer phosphorene reaches its maximum value of 0.667 eV at 4% tensile strain, at which point state C′ replaces state B′ to become the CBM. 3.4. Variation of electronic properties with uniaxial strain
We have also investigated the effect of uniaxial strain on the electronic properties of both the monolayer and bilayer phosphorenes. The calculated band gap variations with the uniaxial strains εx and εy for the monolayer and bilayer phosphorene are shown, respectively, in figures 7(a) and (b). It can be found from figure 7(a) that the band gap variation of the monolayer phosphorene in the uniaxial strain range from −12% to 12% is in good agreement with the previous theoretical work [16]. Any further increase of the compressive strain will induce the semiconductor-metal transition at εx = −14% and εy = −13%, respectively. On the other hand, the band gap of the monolayer phosphorene does not go to zero even when the uniaxial tensile strain εx 6
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Figure 5. (a) The energies of the B, C and D states vary with the isotropic strain. The corresponding partial charge densities of (b) state B, (c)
state C and (d) state D are also depicted. The schematics of the wavefunction character of states C and D are given on the right side of their charge densities.
increases up to εx = 30%. While under the tensile strain along the y direction, the band gap increases linearly from εy = 15% to 25% and then reduces to zero at εy = 28%. As for the bilayer phosphorene, the tensile strain along the x direction can initially increase the band gap from 0.413 eV to its maximum value of 0.667 eV at εx = 6% and then reduce the band gap to zero at εx = 25%. The compressive strain along the x direction gives rise to a linear reduction of the band gap of the bilayer phosphorene to zero at εx = −7%, which is in contrast to that of the monolayer one at a much lower value of −14%. The bilayer phosphorene shows direct semiconducting character under the uniaxial strain εx in the range of −6% to 2% and then turns into an indirect one when the strain is larger than 2% until the S-M transition happens at 25% strain. When the uniaxial strain is taken along the y direction, the band gap of the bilayer phosphorene increases to 0.496 eV as the uniaxial tensile strain increases up to εy = 4%, at which point the band gap changes from direct to indirect. Any further increase of the tensile strain will quickly reduce the indirect band gap to zero at εy = 12%. However, the band gap will open up again at the larger tensile strain of 16%; finally, the system again transforms into a metal when the tensile strain εy increases to 19%. On the other hand, when the uniaxial compressive strain along the y direction is applied, the band gap experiences a reduction until the S-M transition occurs at εy = −12%,
accompanied by a direct-indirect band gap transition at εy = −4%.
4. Conclusions We have systematically investigated the strain-modulated mechanical and electronic properties of both the monolayer and bilayer phosphorenes under either isotropic or uniaxial strain using first-principles calculations. We found that: (1) A sufficiently large isotropic tensile strain on the monolayer phosphorene would make its pucker structure transform into a flat hexagonal plane structure at ε = 29%, while that on bilayer phosphorene leads to the contraction of its interlayer distance, causing its two inner sublayers to be bonded at ε = 22%. (2) The negative out-of-plane Poisson’s ratio can also be obtained in each sublayer of bilayer phosphorene under uniaxial εy, but the variation of its total thickness D + d exhibits a positive Poisson’s ratio under both εx and εy. (3) Under isotropic compressive strain, the band gaps of monolayer and bilayer phosphorene decrease more rapidly than under the uniaxial one. Also, the bilayer phosphorene exhibits a transition from a direct gap semiconductor to a metal at −3%, which is a very small percentage. This is in contrast to the initial directindirect transition and then to the metal transition for the monolayer phosphorene at the larger −7%. (4) Under isotropic tensile strain, the direct-indirect band gap transitions of both the mono- and bilayer phosphorenes happen at the larger 7
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Figure 6. Band structures of the bilayer phosphorene under different strains: (a) −8%, (b) −3%, (c) 0%, (d) 2%, (e) 5%, (f) 9%, (g) 12% and (h) 14%. The variations of states B′, C′ and G′ with the strain are marked out by the dashed black, green and blue lines, respectively. (i) The band gap of the bilayer phosphorene varies with the external isotropic strain.
Figure 7. The band gap of the (a) monolayer phosphorene and (b) bilayer phosphorene varies with the external uniaxial strain applied along
either the x or y direction.
strain values of 14% and 8%, respectively, compared with those under a uniaxial one, which are 7% (8%) under εx (εy) for the monolayer and 2% (4% ) under εx (εy) for the bilayer. (5) However, under the isotropic tensile strain, the band gaps of both increase firstly and then decrease, showing a transition from a direct gap to an indirect one and finally to a metal. Our
numerical results demonstrate a possible way to effectively tune the mechanic and electronic properties of both the monoand bilayer phosphorenes by an external isotropic strain or a uniaxial one along different directions, which may open new potential applications of phosphorene in nanoelectronics and nanomechanical devices. 8
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Acknowledgments
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This work is supported by the State Key Program for the Basic Research of China through Grant Nos. 2010CB630704 and 2011CB922100Q. Our numerical calculations were performed in the High Performance Computing Center of Nanjing University.
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