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Semiconductor to Metal Transition, Dynamical Stability and Superconductivity of Strained Phosphorene

Very recently, field-effect transistors based on few-layer phosphorene crystals with thickness down to a few nanometres have been successfully fabricated, triggering interest in this new functional two-dimensional material. In this work, we apply the first-principles calculations to studying the evolution of electronic structures and lattice dynamics with vertical strain for monolayer and bilayer phosphorenes. It is found that, by changing the thickness of phosphorene or the strain applied on it, its band gap width can be well tuned, and there will appear a transition from semiconductor to metal. In particular, the bilayer phosphorene may become a good BCS superconductor by adjusting the interlayer distance, in which the interlayer van der Waals coupling is favorable to the dynamical stability against strain.


I. INTRODUCTION
Black phosphorus (BP) is the most stable allotrope of the element phosphorus and has many interesting physical properties. BP can server as the electrode material for Lithium-Ion batteries. 1,2 Several structural phase transitions were found under pressure accompanied by semiconductor-semimetal-metal transition. 3,4 Furthermore, it was reported that the BP single crystal shows superconductivity with T C higher than 10K under high pressure. 5,6 The structure of BP consists of puckered double layers, which are held together by weak van der Waals (vdW) force. This peculiar layer structure permits to employ mechanical exfoliation to extract thin black phosphorus from a bulk crystal. Recently, the few-layer black phosphorus (phosphorene) had been successfully exfoliated and used them to create field-effect transistors, which exhibit large current on-off ratios and high mobilities. [7][8][9] The successful fabrication of this novel two-dimensional (2D) semiconducting material was soon paid amounts of attention and predicted to have a great potential for practical applications. 10,11 Bulk BP is a direct-gap semiconductor with a 0.33 eV band gap. 12 However, the band gap of phosphorene is predicted to be highly sensitive to its thickness and strain. Previous first-principles calculations show that the band gap ranges from 1.5 eV for a monolayer to 0.6 eV for a 5-layer. 13 A 3% in-plane strain can change phosphorene from a direct-gap to an indirect-gap semiconductor, 8 while a vertical compression can induce the semiconductor to metal transition. 14 Strain is an effective method to tailor electronic properties for 2D semiconductors. For monolayer phosphorene (MLP) and bilayer phosphorene (BLP), whether there are different strain effects on electronic properties is one of the motivation of this study.
Furthermore, due to the interlayer coupling bonded by weak van der Waals force, it is also interesting to see how bilayer phosphorene responses to the change of interlayer distance.
While there have been substantial works on the electronic properties of phosphorene, investigations on the vibrational properties of phosphorene are scarce. The Raman spectra was measured for the BP flake with thicknesses ranging from 1.6 nm to 9 nm. 15 The phonon dispersion of MLP was calculated by first-principles calculations. 10,16 The lattice vibrational modes of MLP with in-plane strain were also reported by Fei et al. through first-principles simulations. 16 However, phosphorene has an anisotropic structure, whether it has different dynamical behavior for phosphorene with out-of-plane strain? This problem is our concern.
We will pay particular attention to the dynamical stability of strained phosphorene in order to better use them for practical application.
In this work, geometric structure, electronic structure and lattice dynamics of MLP and BLP under vertical strain are studied and compared by first-principles calculations. The calculated results show that the evolution behavior of band structure for MLP and BLP under vertical strain is different. The dynamical stable range of the strained BLP is wider than that of the strained MLP. Furthermore, we also find that BLP may become a good BCS superconductor by adjusting the interlayer distance.  chains along x and y, respectively. Each phosphorus atom is covalently bonded to three neighbors within the puckered layer. Our optimized bond lengthes, bond angles (as labeled in Fig. 1) and lattice constants for BP are listed in Table 1. For comparison, the results without vdW correction for bulk BP are also listed in Table 1. It can be clearly seen that the inclusion of vdW interaction changes a and b parameters slightly, but reduces the c parameter substantially (about 0.6Å). The overall best agreement with experimental 19 and previous computational results 20 is achieved with inclusion of vdW correction.
For the free MLP and BLP (see Table 2), lattice constant b and bond lengthes are very close to those in the bulk BP, while a and bond angles θ 2 are larger than those for the bulk BP. This fact shows that the increase of lattice constants a is not due to the change of bond lengthes, but is due to the flattening of puckered layer. Next we concentrate mainly on the strained phosphorene. The vertical strain is modeled by constraint z for the outmost layer atoms, the unit cell and other atomic positions are then relaxed. The strain is defined where h and h 0 are the thickness of the strained and the free phosphorene, respectively. The positive (negative) of σ corresponds to the tensile (compressive) strain.
The obtained geometric parameters for MLP and BLP under vertical strain are listed in Table 2. For the BLP, when tensile strain is applied, the distance between puckered layers increases, while the covalent bonds within the puckered layer change slightly. And we also find that the geometric parameters of BLP are tend to those in the MLP when tensile strain is increased to 40%. This result is reasonable, since the coupling between puckered layers may be ignored at so large interlayer distance (about 6Å). With increase of compressive strain applied, the change of geometric parameters of BLP is qualitatively same as in the case of MLP but with smaller magnitude. Unlike to that for the MLP, the lattice constant a of the BLP always increases whatever compressive or tensile strain is applied along z direction. This peculiar property is same to that observed in the bulk BP. 21 This result implies that BLP has the same unusual mechanical response as bulk BP. For the BLP, when tensile strain is applied, the band gap E g keeps direct and increases until tends to the value in the free MLP. The corresponding band structure (Fig. 2i) is also similar to that of the free MLP (see Fig. 2c). Under small compressive strain (1%), the band gap soon turns into indirect and decreases with increasing the compressive strain. It turns into zero when the compressive strain reaches about 8% (Fig. 2g), and finally converts into metal (Fig. 2f).
The evolution of the band gap E g with the strain for the MLP and BLP is summarized in Fig. 2j. The change of band gap is very extensive. So we can find that the electronic structure of thin BP is very sensitive to its thickness and the strain applied. This properties help to tailor materials in electronics. In order to better understand this novel 2D materials and use them for potential applications, next we turn to discuss the dynamical stability of the strained phosphorene.

C. Lattice vibration under vertical strain
The symmetry of phosphorene is described by the D 2h point group. For the MLP, the primitive cell contains four atoms, leading to 12 vibrational modes, i.e. nine optical and three acoustic phonons branches. The Γ point modes can be decomposed as and Among optical modes, the A u mode is silent, B  Table 3. Our calculated frequencies are slightly smaller than those in Ref. [16]. For the free BLP, the frequencies of corresponding Raman active modes are also listed in Table 3. We find that the frequency shift between MLP and BLP is small except for A 1 g and B 2 2g modes, which vibrate mainly along the z direction. The frequencies of this two modes in BLP is smaller that those in MLP probably due to the attractive vdW interaction existed in BLP.
The phonon dispersion of free MLP calculated by the density functional perturbation theory is presented in Fig. 3 (a), which compares very well with previous calculations. 10,16 Near the Γ point, two in-plane acoustic modes exhibit linear dispersions, while the offplane acoustic mode exhibits a parabolic dispersion due to the 2D character of phosphorene.
The vibrations of the bond-breathing modes occupy the high-frequency region, while the vibrations of layer-breathing modes and the acoustic modes occupy the low-frequency region. Fig. 3 (b) shows the calculated phonon dispersion of free BLP which has 24 phonon branches.
The optical branches have small splitting and are nearly double degenerate due to the weak interlayer interaction. Near the Γ point, the splitting has an appreciable magnitude. Three branches from the opposite vibrations of two puckered layers (i.e. layer-breathing modes) deviate from three acoustic branches. From Fig. 3a and 3b, we can see that there is no imaginary frequency in the full phonon spectra, indicating the dynamical stability for both free-standing MLP and BLP. This conclusion is in good agreement with the experiment.
Raman spectroscopy and transmission electron microscopy measurements show that the exfoliated flakes of BP are stable even in free-standing form. 15 When vertical strain is increased to a certain value, we find phosphorene becomes unstable. For MLP, there appears imaginary frequency in the phonon dispersions when tensile strain reaches 15% and compressive strain reaches 4%. The appearance of imaginary frequency of phonon modes indicates the dynamical instability under such strain. The dynamical stable range of MLP for strain σ is about (−3%, 12%). Within this range, the frequencies of Raman active modes under several strains are also listed in Table 3. The response of each Raman mode to strain is not the same. Four modes ( B 1g , A 1 g , B 1 2g , B 2 2g ) exhibit monotonic change: their frequencies are red shifted under tensile strain and blue shifted under compressive strain. Among them, A 1 g , B 2 2g modes which are vibrated mainly along the z direction have prominent shifts. The other two Raman modes exhibit nonmonotonic change. B 3g mode exhibits an abnormal red shift under small compressive strain. This is possibly due to the fact that the flattening of the puckered layer under compressive strain is in favor of the opposite vibrations of two layers along the zigzag chain. A 2 g mode exhibits a nonmonotonic shift under tensile strain. Its frequency decreases at small tensile strain and then increases with increasing tensile strain. This abnormal behavior can be understood from the following: A 2 g mode corresponds to the vibration of d 1 bond-breathing mainly along x direction. The lattice constant a is contracted when vertical tensile strain is applied. When the effect of this contraction of a dominates over the effect of increase of thickness of phosphorene, it will enhance the interatomic interactions, resulting in the increase of frequency of A 2 g mode. For BLP, we find its dynamical stable range for strain σ is wider than that of MLP. When compressive strain reaches 8%, imaginary frequency appears in the phonon dispersion along Γ-X direction. Within the stable dynamical range for strain σ, the frequencies of Raman active modes are also listed in Table 3. Under compressive strain, B 3g mode exhibits an abnormal red shift as in the case of MLP, while the frequencies of other Raman modes are blue shifted as expected as usual. BLP is still stable when tensile strain reaches 40%.
As discussed above, large tensile strain only results in large interlayer distance with other geometric parameters in close to those in the MLP. Then BLP may be regarded as two nearly independent MLPs, which are stable in free-standing form as discussed above. Under the wide range of tensile strain, the changes of frequencies of four modes ( B 3g , B 1g , A 2 g , B 1 2g ) are all very small. This shows that the vertical tensile strain has little effect on the inplane vibrational modes. With the increase of tensile strain, A 1 g , B 2 2g modes vibrated mainly along the z direction are red shifted at first as expected, but they are then blue shifted.
This abnormal blue shift may be due to the decrease of attractive vdW interaction at large interlayer distance.

D. Superconductivity by adjusting the interlayer distance
The above phonon calculations show that BLP is dynamical unstable when it achieves the transition from semiconductor to metal. Here we propose that stable metal phase or even a BCS superconductor can be achieved by adjusting the interlayer distance of BLP. We fix the interlayer distance at different given values, the unit cell and other atomic positions are then relaxed. Since the results with increasing the interlayer distance are very similar to those in the tensile strained BLP, next we only discuss the opposite case.
When the interlayer distance 0.6d 0 < d < d 0 (d 0 is the interlayer distance of free BLP), our calculations show that the lattice constant a increases slightly, d 2 and θ 2 show only a weak variation, while the lattice constant b, d 1 and θ 1 hardly change. This means that the puckered character of BLP remains and changes little at this range of interlayer distance.
When d < 0.6d 0 , the structure of BLP starts to change substantially. The calculations of band structure show that the band gap E g goes to zero when d = 0.85d 0 . Keeping on decreasing the interlayer distance will result in the transition from semiconductor to metal.
The calculations of phonon show that BLP is dynamical unstable when d is 0.65d 0 or less.
So the range of d for the stable metal phase of BLP is about 0.65d 0 < d < 0.85d 0 .
The Eliashberg spectral function α 2 F (ω) depends directly on the EP matrix element, g qν k+qj kj , which can be determined self-consistently using linear response theory. By plotting α 2 F (ω) we can estimate the relative strength of the EP coupling. The phonon density of states F (ω) and a 2 F (ω) are plotted in Fig. 4 for three different interlayer distances, which are within the range of stable metal phase of BLP. When d = 0.8d 0 , d = 0.75d 0 and d = 0.7d 0 , F (ω) is very similar, while a 2 F (ω) is different. With smaller interlayer distance, the Eliashberg spectral function a 2 F (ω) has a significantly enhanced peak at the low-frequency side. The characteristic phonon modes which dominate the EP coupling are just from the opposite vibrations of two puckered double layers. So we speculate that the interlayer vdW attractive interaction may play an important role in enhancing the EP coupling of BLP with smaller interlayer distance.
The superconducting temperature T c can be estimated from the Allen-Dynes modified McMillan equation 22 : where ω ln is the logarithmically averaged frequency, µ * is the Coulomb repulsion parameter. The calculated electronic densities of states at Fermi level N (E F ), EP coupling constant λ, ω ln and the estimated T C when taking µ * = 0.1 are summarized in Table 4. With the decrease of interlayer distance, BLP apparently becomes more metallic and results in the increase of N (E F ). Most remarkably, we see λ increases dramatically. The increase of both N (E F ) and λ is favorable to enhance T C . When d = 0.7d 0 , λ reaches as high as 1.45 and the estimated T C is about 10K. The superconductivity of BP was reported early under high pressure. 5,6 Our results suggest that BLP may become a good BCS superconductor by adjusting the interlayer distance.

IV. SUMMARY
In conclusion, MLP and BLP under vertical strain are studied by density functional and density-functional perturbation theory. The results show that the electronic structure of thin BP is very sensitive to its thickness and the strain applied. The change is extensive.
It can increase direct band gap, decrease indirect band gap or realize the transition from semiconductor to metal, which help to tailor materials in a variety of settings, from infrared optoelectronics to high-mobility quantum transport. The shift of zone-center Raman active modes for strained phosphorenes is analyzed and explained combining the relaxation of geometric structure. Our study clearly show that both MLP and BLP in free-standing form are dynamical stable. However, material becomes dynamical unstable with large vertical strain.
We find that the dynamical stable range for strain σ in BLP is wider than that in MLP due to add the interlayer weak vdW coupling, this additional freedom. This information is essential for future device fabrication and potential applications of phosphorene. Furthermore we also find that BLP may become a good BCS superconductor by adjusting the interlayer distance.      for BLP at three different interlayer distances.