Size and strain effects on mechanical and electronic properties of green phosphorene nanoribbons

Size and strain effects on mechanical and electronic properties of green phosphorene nanoribbons Evan Garrison,1 Candace K. Chan,2 and Xihong Peng1,3,a 1Department of Physics, Arizona State University, Tempe, Arizona 85287, USA 2Materials Sciences and Engineering, School for Engineering of Matter, Transport and Energy, Arizona State University, Tempe, Arizona 85287, USA 3College of Integrative Sciences and Arts, Arizona State University, Mesa, Arizona 85212, USA


I. INTRODUCTION
Graphene, 1,2 a two-dimensional (2D) layered crystal of carbon atoms, has initiated tremendous interest in searching for similar materials made of different elements.Graphene exhibits high carrier mobility for great potential in electronic applications, but its gapless band structure has limited such applications. 3][10] While black phosphorene was found to be the most stable form of 2D phosphorus, 11 and another isomer blue phosphorene has been realized by molecular beam epitaxy. 12Yet another newer isomer, green phosphorene, has been recently theoretically predicted, and the band gap, in the 0.7-2.4eV range, has been shown to be tunable by changing the number of layers. 13Monolayer 2D phosphorene has high in-plane anisotropy for optical absorption and transport properties, and investigations on the effects of temperature and substrate on which the phosphorene is synthesized have been conducted. 13n addition to these methods of tuning the properties of phosphorene, recent computational work focusing on the effects of strain on the electronic band structure of green phosphorene has been reported 14 and the mechanism of the resulting direct-indirect band gap transition was explored. 15ther studies have shown that the electronic properties of one dimensional (1D) phosphorene nanoribbons obtained from the 2D sheets could be further tuned by varying the width of the ribbons and the species used to passivate the edge's dangling bonds, 16,17 making these ribbons even richer in potential for possible applications.When applying strain to tune its properties, it is important to know how the structure holds up with strain.9][20] Black phosphorene can withstand up to 30% tensile strain because of its puckered surface, 15 and green phosphorene can sustain an impressive 35%. 14sing first principles density-functional theory (DFT) calculations, this paper focuses on the effects of ribbon width and mechanical strain (both tensile and compressive) on the mechanical and electronic properties of armchair green phosphorene nanoribbons (AGPNR) and zigzag green phosphorene nanoribbons (ZGPNR) with ribbon width up to 57 Å.

II. METHODOLOGY
First principles DFT 21 calculations with the Perdew-Burke-Ernzerhof (PBE) exchangecorrelation functional 22 and projector-augmented wave (PAW) potentials 23,24 was implemented for all calculations using the Vienna Ab-initio Simulation Package (VASP). 25,26The kinetic energy cutoff for the plane wave expansion was set at 500 eV, and the convergence criteria for the structural optimization were set to 10 -5 eV and 10 -4 eV for the electronic and ionic iterations, respectively.A snapshot illustrating the puckering of the 2D green phosphorene sheet is shown in Fig. 1(a).the ribbon.Tensile (positive) and compressive (negative) strains were then applied to the ribbon by rescaling the relaxed lattice constant to study strain effects.It is noted that there exists a significant asymmetry in strain induced mechanical instability of 2D structures such as graphene, 27,28 where compression usually inevitably cause buckling and ripple formation of graphene to relieve the compressive strain energy.Therefore, the threshold compression (i.e. the largest compression the system can take while still maintains a flat 2D monolayer) is much smaller than the threshold value of tensile strain.However, this situation was not found in the green phosphorene under the compression range studied in this work because the green phosphorene structure itself is already puckered and has the freedom to minimally adjust its pucker configuration to relieve the strain energy.

III. RESULTS AND DISCUSSION
Four different widths (7 Å, 16 Å, 26 Å, and 33 Å) of AGPNRs and five widths (10 Å, 14 Å, 24 Å, 33 Å, 57 Å) of ZGPNRs were studied.The size effect on the structural, mechanical and electronic properties are presented in Fig. 2. Fig. 2(a) shows the relaxed lattice constant along the 1D axial direction of the ribbon for the AGPNRs and ZGPNRs with various widths.The lattice constant for the AGPNRs decreases with increasing widths, approaching to its value of 14.295 Å for the 2D sheet, whereas the lattice constant for the ZGPNRs remains nearly unchanged with increasing width.
We also calculated the Young's modulus for all of the studied ribbons.The Young's modulus was calculated according to the formula Y = σ , where σ is the lengthwise stress in GPa, and = a−a 0 a 0 is the fractional lengthwise strain, with a and a 0 being the magnitude of the lattice constant for the strained and relaxed ribbons, respectively.Note that in a 1D or 2D system, the stress calculated from DFT must be modified to avoid the force being averaged over the entire simulation cell including the vacuum space.For the 2D green phosphorene sheet, we rescaled the DFT reported stress by Z/d 0 to obtain the equivalent stress, where Z is the cell length in the z direction and d 0 is the effective thickness of the monolayer (i.e.5.71 Å, the interlayer spacing of bulk green phosphorus).For the 1D green phosphorene ribbon, we rescaled the DFT reported stress by ZL/(d 0 L 0 ), where Z and d 0 are same as mentioned above, L 0 is the width of the ribbon and L is the cell size in the ribbon width direction.After eliminating the vacuum effect, our calculated Young's modulus for AGPNRs and  ZGPNRs were in the range of 10-35 GPa and 160-170 GPa, respectively, which is fairly close to the corresponding Young's moduli of 2D black phosphorene, i.e., 44 GPa for the armchair direction and 166 GPa for the zigzag direction. 18Young's moduli for the 2D green phosphorene sheet along the armchair and zigzag directions were 31.27GPa and 122.59 GPa, respectively, as found in previous work. 14Young's modulus values for all studied widths of AGPNRs and ZGPNRs are shown in Fig. 2(b).It demonstrates that the Young's modulus of AGPNRs increases as the width increases to approach the value of the 2D green phosphorene, whereas the Young's modulus for ZGPNRs stays approximately the same with increasing width.
The electronic band structure was calculated for each ribbon at its relaxed lattice constant and it was found that all ribbons studied in this work are semiconductors with a finite band gap.The band gap E G , which is defined as the energy difference between the conduction band minimum (CBM) and the valence band maximum (VBM), was calculated and is shown in Fig. 2(c).The band gap increases when the ribbon width decreases due to the effect of quantum confinement.Figure 3 shows the band structures of the relaxed AGPNRs and ZGPNRs at multiple widths.For the 2D green phosphorene sheet, DFT predicts a direct band gap of 1.21 eV and hybrid functional HSE06 predicts a 1.96 eV band gap located at Γ. 14 For the 1D nanoribbons, the DFT predicted band gap stays indirect for all widths of AGPNRs, whereas the band gap for ZGPNRs starts as indirect for the 10 Å wide ribbon but then changes to direct at 14 Å and stays that way with wider ribbons.This indirect-direct band gap transition via changes in size might be due to the combined effects of quantum confinement and the different effective masses between two competing states which define the band gap. 29,30For both AGPNRs and ZGPNRs, the band gap is seen to get smaller with increasing width as a result of the VBM becoming higher in energy, and the CBM becoming lower in energy.
In addition to size, the effects of strain on the electronic band structure were also explored.Fig. 4 presents the band gap as a function of strain for all studied ribbon widths and orientations.The general trend for all AGPNRs and ZGPNRs is that the band gap decreases in magnitude with both tensile and compressive strain.Any slight variations from this general trend can be explained by the details of the band structure.As an example, Fig. 5 shows the band structures of the AGPNR with width of 16 Å and that of the ZGPNR with width of 14 Å at different values of strain.For the AGPNR, the band gap is indirect at 0% strain, and experiences an indirect-to-direct transition with both expansive and compressive strains.For the ZGPNR, the band gap is direct at Γ for the relaxed ribbon.It stays direct but becomes narrower with tensile strain, while the band gap transitions to indirect with compression.With significant large degrees of the compression, DFT predicted the band gap to close up.
The direct-indirect band gap transition can be understood by performing a detailed examination of the spd orbital projection and bonding/antibonding characteristic of the CBM and VBM, combined with the conclusion obtained from Heitler-London's exchange energy model, 15 where a bonding state has a positive linear dependence on strain, while an antibonding state a negative linear dependence on strain. 15In Fig. 5(d), the indirect band gap of the relaxed AGPNR is determined by the VBM at Γ (denoted as state F) and the CBM along the direction Γ to X (indicated as state J).The conduction band at Γ (noted as state G) has higher energy than that of J for the relaxed ribbon.Under tensile strain, the energy of state G becomes lower than that of J and the VBM remains at Γ, resulting in a direct band gap.The larger downward energy shift of state G compared to that of state J is due to the antibonding character of state G (dominated by 36% s and 34% p y orbitals).Under compression, two conduction bands at Γ (i.e.states G and H) experience an energy flip over; state H has a lower energy which thus defines the CBM.VBM is still at Γ giving a direct band gap.Energy of State H (dominated by 52% px orbital) decreases rapidly with compression due to its bonding nature.
On the other hand, the ZGPNR shows a direct band gap for the relaxed ribbon in Fig. 5 C could close up the band gap of the ribbon.This is a result of significant structural deformation.The angle α defined in Figure 1(b) is 94.4 o for the relaxed ribbon; however, α becomes 79.1 o when a -15.5% strain is applied.The charge density contour plot of state D reveals that its p y orbitals align and connect to form a conducting channel.

IV. CONCLUSION
The size and strain effects on the properties of green phosphorene nanoribbons along the armchair and zigzag directions were investigated.It was found that the axial lattice constant of AGPNR was expanded compared to that of 2D green phosphorene and this expansion becomes more prominent with reduced ribbon widths.On the other hand, the lattice constant of ZGPNR was slightly smaller compared to that of 2D sheet.Young's moduli were in the range of 10-35 GPa for AGPNRs and 160-170 GPa range for ZGPNRs, respectively, which were on the same order of magnitude as the corresponding values for 2D green phosphorene.The increase of the band gap as the ribbon width decreased resulted from the quantum confinement effect.A similar trend was also found for the work function.The band gap is indirect for all widths of AGPNR.A direct-to-indirect band gap transition occurs when the width of ZGPNR gets smaller.It was also found that either an indirect-to-direct or a direct-to-indirect band gap transition can be triggered by applying mechanical tensile and compressive strain to either AGPNRs or ZGPNRs, respectively.Under large degrees of compression, the band gap of the ZGPNR could close up to form a conducting channel through the p y orbital of the phosphorus atoms.
FIG. 1. Snapshots of green phosphorene 2D sheet and nanoribbons.(a) Side view showing puckered structure of 2D green phosphorene.(b) Top view of 2D green phosphorene.The dashed diamond shape and rectangle indicate the unit and conventional cells, respectively.The Brillouin zone corresponding to the conventional cell is on right.Arrows indicate lattice vectors.(c) An example of an AGPNR.The blue atoms on the edges are H atoms passivating the dangling bonds on the edges and the unit cell is indicated by the rectangle.(d) An example of H-passivated ZGPNR.

FIG. 2 .
FIG. 2. Size effects on (a) the lattice constant in the direction along the length of the ribbon, (b) Youngs modulus, (c) the band gap, and (d) the work function.

Fig. 2 (
d) shows the work function W e = E vac −E F , i.e., the vacuum energy minus the Fermi energy, was in the range of 5-5.7 eV for the AGPNRs and ZGPNRs.The calculated work function for the 2D sheet was 5.25 eV.

FIG. 4 .
FIG. 4. Band gap of (a) AGPNRs and (b) ZGPNRs as a function of strain.The gap generally decreases with increased strain.
FIG. 5. Band structures of 16 Å wide AGPNR (top) and 14 wide Å ZGPNR (bottom) at various strains.The competing states for band edges are colored.Energies are referenced to vacuum.For the AGPNR, the band gap transitions from indirect to direct with stretching or compression.For the ZGPNR, a direct-to-indirect transition occurs with stretching or compression.