Modulation of the Electronic Properties of Ultrathin Black Phosphorus

Article

The Modulation of the Electronic Properties of Ultrathin Black Phosphorus by Strain and Electrical Field Yan Li, Shengxue Yang, and Jingbo Li J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/jp506881v • Publication Date (Web): 24 Sep 2014 Downloaded from http://pubs.acs.org on September 30, 2014

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The Journal of Physical Chemistry

The Modulation of the Electronic Properties of Ultrathin Black Phosphorus by Strain and Electrical Field Yan Li, Shengxue Yang, and Jingbo Li∗ State Key Laboratory of Superlattice and Microstructures, Institute of Semiconductors, Chinese Academy of Sciences, Beijing 100083, China. E-mail: [email protected]

Abstract The structural and electronic properties of the bulk and ultrathin black phosphorus and the effects of in-plane strain and out-of-plane electrical field on the electronic structure of phosphorene are investigated using first-principles methods. The computed results show that the bulk and few-layer black phosphorus from monolayer to six-layer demonstrate inherent direct bandgap features ranging from 0.5 to 1.6 eV. Interestingly, the band structures of the bulk and few-layer black phosphorus from X point via A point to Y point present degenerate distribution, which shows totally different partial charge dispersions. Moreover, strong anisotropy in regard of carrier effective mass has been observed along different directions. The response of phosphorene to in-plane strain is diverse. The bandgap monotonically decreases with increasing compressive strain, and semiconductor-to-metal transition occurs for phosphorene when the biaxial compressive reaches -9 %. Tensile strain firstly enlarges the gap until the strain reaches around 4%, after which the bandgap exhibits descending relationship with tensile strain. The bandgaps of the pristine and deformed phosphorene can also be continuously ∗ To

whom correspondence should be addressed

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modulated by the electrical field and finally close up at about 15 V/nm. Besides, the electron and hole effective mass along different directions exhibit different responses to the combined impact of strain and electrical field. Keywords: Phospherene First-principles methods Anisotropy Carrier effective mass Semiconductor-to-metal transition Band structure

Introduction The discovery of graphene 1 has intrigued strong interest in two-dimensional (2D) materials, such as graphyne, 2–5 silicene, 6,7 boron nitride, 8,9 ultrathin transition metal dichalcogenides (TMDs), 10–16 and few-layer III-VI semiconductors. 17,18 The honeycomb network and sp2 hybridized orbitals bring graphene many unique electronic and mechanical properties, like the massless Dirac fermion, half-integer quantum Hall effect, 19,20 high mobility of up to 106 cm2 V−1 s−1 , 21,22 the laurel of the strongest material ever tested, 23 etc.. All these appealling features suggest that graphene has great potential for ultrahigh-speed electronics. However, the absence of intrinsic bandgap poses graphene a major obstacle for the adaptability in field-effect devices. The emergence of ultrathin TMDs makes compensation for the shortage of graphene, as many of TMD monolayers have nature direct bandgap. 12 Among the various TMDs, MoS2 has been widely investigated for device applications. Recently, the successful fabrication of sigle-layer MoS2 -based field-effect transistor (FET) has been reported with a high ON/OFF ratios of 108 and carrier mobility of 200 cm2 V−1 s−1 , 24 which is orders of magnitude lower than that of graphene. GaS and GaSe are two stable III-VI wide bandgap semiconductors, and their ultrathin layer transistors have been fabricated with moderate ON/OFF ratios and low mobility. 17 Therefore, unremitting endeavor has been still carried on to research for layered semiconductors with reasonable direct bandgap and high 2 ACS Paragon Plus Environment

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carrier mobility. Very recently, the monolayer of black phosphorus, named as phosphorene, has been mechanically exfoliated from its bulk materials. 25–27 More exciting is the experimental and theoretical discoveries that phosphorene has inherent direct bandgap with high hole mobility of up to 103 ∼ 104 cm2 V−1 s−1 and exhibits high sensitivity to external perturbations, 28–37 which bring new hope for both the fundamental investigation and industrial application. In this paper, we report on the structural and electronic properties of the bulk and few-layer black phosphorus using first-principles calculations based on the density functional theory (DFT). The bulk and few-layer black phosphorus from single-layer to six-layer are found to be direct bandgap semiconductors, which indicates the potential application in the optoelectronic devices. The electronic properties of phosphorene are very sensitive to external perturbations, including inplane strain and out-of-plane electrical field. On the one hand, direct-to-indirect bandgap transition has been observed when the in-plane strain reaches different critical values for different forms of strain. On the other hand, the bandgap values of the pristine and deformed phosphorene can be continuously modulated by vertical electrical field. In addition, strong anisotropic feature has been found for both electron and hole effective mass along different directions, and the carrier effective mass exhibits different responses to the intercoupling of external perturbations.

Computational method For our research, we performed first-principles calculations using the Vienna ab initio simulation package (VASP), 38 which is based on the DFT in a plane-wave basis set with the projectoraugmented wave (PAW) method. 39 The generalized gradient approximation of Perdew-BurkeErnzerhof (GGA-PBE) 40 and the Heyd-Scuseria-Ernzerhof (HSE06) hybrid functional 41 both including van der Waals corrections proposed by Grimme 42 was chosen for the exchange-correlation functional. Energy cutoff for plane-wave expansion was set to 500 eV. A vacuum space of 11 Å was added to all 2D nanosheets to prevent the interaction between adjacent images. Brillouin zone sampling was performed with Monkhorst Pack (MP) special k points meshes. 43 K-points grids of



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12×8×4 and 12×8×1 were chosen for the calculations of the bulk and few-layer black phosphorus systems, respectively. All the structures were fully relaxed using the conjugated gradient method until the Hellmann-Feynman force on each atom was less than 1 meV/Å.

Results and discussion Black phosphorus is the most stable allotrope of phosphorus, 44 which packs into an orthorhombic lattice with a puckered honeycomb structure resulting from the sp3 hybridization, as illustrated in Fig.1. The intralayer phosphorus atoms are coupled by strong covalent bond, while the individual layers are held together by weak van der Waals forces. The two adjacent phosphorene layers slide relative to each other by 1/2 a along the x direction, forming the AB stacking. Thus, an accurate description of the dominated dispersive forces (vdW) of the interlayer interaction is important. Here, based on the dispersion-corrected PBE functional, the theoretical equilibrium crystal structures of the bulk and few-layer black phosphorus have been obtained by full structural optimization. The calculated basic physical parameters of the bulk and few-layer black phosphorus are listed in Tab.1. Taking the experimental results 45,46 as reference, we find that our vdW-corrected lattice parameters a = 3.32 Å, b = 4.42 Å and c = 10.48 Å are only 0.3%, 1.1% and 0.1% larger than the experimental values in order, as well as in good agreement with other theoretical works. 25,28,47–49 The difference of constant a between the bulk and few-layer black phosphorus is small, while the variation of constant b from the bulk to few-layer black phosphorus is significant, as much as 4.76% between the bulk and phosphorene. Moreover, the P-P distance shows only subtle changes for both in-plane bond length din and out-of-plane bond length dout from three dimensional materials to its 2D counterparts, while the bond angle changes a lot, especially for the out-of-plane angles θout . Such changes suggest that the van der Waals forces have much more influence on the bond angle than on the bond length, in other words, the weak forces are only able to change the bond angles but not strong enough to alter the distances between phosphorus atoms. The in-plane bond angles

θin increase from monolayer to six-layer black phosphorus, while the out-of-plane bond angles


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Figure 1: (a) Top and side view of the crystal structure of the bulk black phosphorus. The cyan and yellow balls denote A and B layers, respectively.(b) and (c) Band structure of the bulk and monolayer black phosphorus, respectively. The black and red lines present PBE and HSE results, respectively. The inset in (b) denotes the first Brillouin zone of black phosphorus.

θout exhibit the opposite change tendency, from which it can be inferred that the absence of van der Waals forces gives rise to the the contraction of the plane along the x direction and extension of the plane along the y direction, which are in-line with the changes of constants a and b, respectively. Table 1: The calculated basic properties of the bulk and few-layer black phosphorus: number of layer (NL), lattice constants a (Å), b (Å), in-plane bond length din (Å), out-of-plane bond length dout (Å), in-plane bond angle θin , out-of-plane bond angle θout , bandgap calculated by PBE GP (eV) and by HSE GH (eV), electron effective mass at the Γ point along the Γ–X direction m∗ex (m0 ) and along the Γ–Y direction m∗ey (m0 ), hole effective mass along the Γ–X direction m∗hx (m0 ) and along the Γ–Y direction m∗hy (m0 ) calculated by HSE. NL 1 2 3 4 5 6 bulk

a 3.30 3.31 3.32 3.32 3.32 3.32 3.32

b 4.63 4.50 4.48 4.47 4.46 4.45 4.42

din 2.22 2.22 2.22 2.22 2.22 2.22 2.23

dout 2.26 2.26 2.26 2.26 2.26 2.26 2.26

θin 95.94 96.34 96.38 96.41 96.43 96.43 96.53

θout 104.22 103.14 102.91 102.80 102.73 102.69 102.41

GP 0.92 0.43 0.19 0.07 0.04 0.05 0.04

GH 1.60 1.04 0.78 0.63 0.55 0.50 0.27

m∗ex 1.16 1.20 1.17 1.17 1.17 1.17 1.14

m∗ey 0.22 0.24 0.25 0.26 0.26 0.27 0.39

m∗hx 3.24 1.33 0.97 0.82 0.77 0.73 0.62

m∗hy 0.19 0.20 0.20 0.21 0.21 0.21 0.29

Band structure is the fundamental information of electronic properties to describe the relationship between energy and electron wave vectors, which has promoted the development of semiconducting science and made contributions to the micro-electronic industry. In the present work, we have calculated the electronic properties for the bulk and few-layer black phosphorus within



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vdW-corrected PBE and HSE approximations. The band structure of the bulk black phosphorus is shown in Fig.1. The general features of the band structures calculated by PBE and HSE are similar with minor exceptions. The PBE result shows that the bulk black phosphorus is nearly direct bandgap semiconductor with conduction band minimum (CBM) and valence band maximun (VBM) located near the Γ point, while the HSE result gives the direct bandgap of 0.27 eV at the Γ point, which is better than the value of PBE (0.038 eV) in reference to the experimental value about 0.3 eV. 50,51 In addition, both PBE and HSE results reveal the thickness-dependent relationship between the bandgap values and the number of phosphorene layers. More specifically, the bandgap decreases with the increasing number of layers. All the few-layer black phosphorus considered in this work are direct bandgap semiconductors with CBM and VBM both located at the Γ point. Beyond that, it is interesting to find that, from the lowest valence band to the highest conduction band, every two adjacent energy bands exhibit degenerate distribution along the high symmetry direction from X point via A point to Y point, as illustrated in Fig.1 (b) and (c). Since every unit cell of phosphorene contains four phosphorus atoms, and every atom possesses five valence electrons, there are total twenty valence electrons in the unit cell and the highest occupied band is the tenth one. As displayed in Fig.1 (c), the tenth band superposes with the ninth from X point via A point to Y point. Fig.2 (a) and (b) illustrate the partial charge distribution of the ninth and the tenth bands of phosphorene at A point, respectively. It is noted that for the ninth band, the charges mainly distribute between P1 and P3, P2 and P4, there is no charge distribution between P1 and P4, P2 and P3. In the case of the tenth band, however, the charge distribution is just opposite to that of the ninth band, the charges are mainly located between P1 and P4, P2 and P3. This situation holds true for other degenerate band alignments for the bulk and few-layer black phosphorus along the high symmetry direction from X point via A point to Y point. Then, we calculated the carrier effective mass from the band structure by the formula (m∗i j )−1 =

1 ∂ 2E , h¯ 2 ∂ ki ∂ k j

where h¯ is the reduced

Planck constant, ki (k j ) is the wave vector along i (j) direction and E is the energy eigenvalue. The detailed calculation process is supplied in the Supporting Information. As expected, the bulk and few-layer black phosphorus exhibit strong anisotropic feature in respect of the carrier effective


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mass. As illustrated in Tab.1, both the electron and hole effective mass along the Γ–X direction are substantially heavier than that of along the Γ–Y direction, respectively, which can be inferred from the flat band alignment from the Γ to X and the dispersive band alignment from the Γ to Y point, as shown in Fig.1 (b) and (c). The anisotropic character of phosphorene becomes stronger than the bulk, especially for the hole effective mass. More specifically, the hole effective mass along the Γ–X direction and along the Γ–Y direction are 3.24 m0 and 0.19 m0 (m0 denotes the electron rest mass) respectively, while those of the bulk are 0.62 m0 and 0.29 m0 , respectively. The VBM and CBM of black phosphorus are mainly composed by P 3pz orbitals, which are more localized than 3px and 3py orbitals and vertical to the puckered plane . For the highest occupied band and the lowest unoccupied band along the Γ–X direction, the component of 3pz orbitals changes almost nothing, at the same time, the 3pz orbitals keeps the dominant component among other orbitals components. Therefore, the states along the Γ–X direction are greatly localized and the band dispersions are much flat, resulting in the much larger carrier effective mass; In the case of the two bands along the Γ–Y direction, the component of 3pz orbitals shows appreciable variations, thus the states along the Γ–Y direction are much more nonlocalized and the corresponding band alignments are quite dispersive, leading to much smaller carrier effective mass.

Figure 2: (a) and (b) The partial charge distribution of the ninth and tenth band of phosphorene at A point. Strain engineering has been proved to be an efficient way to tune the electronic, optical and 7 ACS Paragon Plus Environment


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(a)

HSE

PBE

(b)

(c) σ∗ π∗

π σ

Figure 3: (a) Variations of the bandgap and strain energy ∆E of phosphorene as functions of the biaxial strain. The shadowed area denotes the strain range of direct bandgap region. (b) Variations of energy eigenvalues of the CBM and VBM as functions of the biaxial strain. (c) Variations of energy eigenvalue of σ , σ ∗ , π , π ∗ states as functions of the biaxial strain magnetic properties of graphene and other 2D materials. 52–55 In order to have an insight into the response of phosphorene to external strain, uniaxial and biaxial strains ranging from -10% to 16% in steps of 2% were applied to the plane of phosphorene. The electronic structure of phosphorene was computed for each deformed configuration by means of PBE and HSE06. The strain energy is defined as ∆E = Ed - E p , where Ed and E p denote the total energy of the deformed and pristine phosphorene, respectively. Fig.3 (a) demonstrates the strain energy ∆E as a function of the biaxial strain. It is shown that ∆E increases monotonously with the increasing strain, suggesting that the range of in-plane strain (-10% to 16%) is in the range of elastic deformation, among which the deformed structure can return to its pristine geometry after the strain is removed. By checking the electronic structures of the deformed monolayer systems, we note that the compressive strain firstly turns phosphorene into indirect bandgap semiconductors with the VBM slightly shifting along the 8 ACS Paragon Plus Environment

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Γ–X direction, and the maximum difference value between the energy eigenvalues at VBM and at the Γ point is less than 0.05 eV, therefore the band structure under compressive strain can be viewed as nearly direct bandgap. The bandgap presents nearly linearly decreasing relationship with the increasing compressive strain, and finally closes up at -9%, bringing about the semiconductorto-metal transition. For the biaxial tensile strain, the bandgap firstly increases until the tensile strain reaches 4%, after which the bandgap monotonously decreases. At the same time, the tensile strain keeps the direct bandgap feature of phosphorene with the CBM and VBM both located at the Γ point until the strain reaches 14%. The variations of the energy eigenvalues of CBM and VBM as functions of the biaxial strain are displayed in Fig.3 (b). It is observed that VBMs of the deformed systems calculated by both PBE and HSE shift down with the increasing tensile strain or decreasing compressive strain, while the CBMs move up with decreasing compressive strain and increasing tensile strain until it reaches 4% and then shift down with larger tensile strain, and the shifting-down rate of CBM is much quicker than that of VBM, resulting the decrease of the bandgap. When stretched by the in-plane strain, the in-plane bond length becomes longer, while the out-of-plane bond length becomes shorter, therefore the superposition of px or py orbitals of P1 and P3, P2 and P4 has been reduced, in contrast, that of pz orbitals of P1 and P4, P2 and P3 has been increased. As a result, the energy of the σ state formed by the interaction between pz orbitals has been lowered, while that of corresponding σ ∗ state has been enhanced, analogously, the energy of the π state formed by the coupling between px or py orbitals have been enhanced, while that of corresponding π ∗ state has been lowered. It can be deduced that the case for the compressive strain is exactly the opposite. For the pristine phosphorene, the VBM and CBM, located at the Γ point, are mainly composed by pz orbitals and a small part of s and py orbitals, therefore the VBM exhibits the feature of σ state, and the CBM possesses the character of σ ∗ state. The state of ninth band at the Γ point is mainly composed by py orbitals as well as a fraction of pz and s orbitals, and the eleventh band at the Γ point possesses the same orbitals component with the ninth band, therefore, the states of the ninth band and the eleventh band at the Γ point form the π state and π ∗ state, respectively. Fig.3 (c) illustrates the variations of energy eigenvalues of above



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four states as functions of the biaxial strain, which all verify the above analysis. The σ state is pushed down (up) by increasing tensile (compressive) strain, while the σ ∗ varies oppositely. The

π state is moved up (down) by increasing tensile (compressive) strain, and the π ∗ state changes just reversely. When the tensile strain further increases, the repulsion between the σ state and σ ∗ state becomes increasingly strong, the σ ∗ state further shifts up, consequently, the energy of the

σ ∗ state exceeds that of the π ∗ state, and the orbital states of CBM changes from main pz orbital feature to main py orbital feature. Therefore the greater tensile strain makes the new CBM move down, which accounts for the decreasing tendency of bandgap when the tensile strain becomes larger than 4%.

HSE

PBE

HSE

PBE

Figure 4: Variations of the bandgap and strain energy ∆E of phosphorene as functions of the uniaxial strain along (a)the x direction and (b) the y direction. The shadowed area denotes the strain range of direct bandgap region. There are two types of in-plane uniaxial strain: one is along the x direction (εx ) and another one is along the y direction (εy ). The calculated strain energies under the two kinds of uniaxial strains obey monotonic ascending trend as the strain increases, suggesting that the deformed systems are in the range of elastic regime. While the strain energy along the x direction at each strain value is approximate four times as many as that of along the y direction, implying that phosphorene can be much easier compressed or stretched along the y direction than along the x direction. This is due 10 ACS Paragon Plus Environment

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to the different deformation response to the strains along different directions. More specifically, when the strain is applied along the x direction, only the in-plane bond angle θin and the in-plane bond length din are changed, while for the strain along the y direction, the in-plane and out-ofplane bond angles and bond lengths are all changed, therefore the deformation energy can be much more released. Fig.4 also shows the variation trend of the bandgap versus uniaxial strain. For the (a)

x

0

= -0.06

(b)

x

(e)

y

= 0.04

(c)

x

= 0.14

Energy (eV)

-2 -4 -6 -8 -10 0

(d)

y

= -0.1

= 0.02

(f)

= 0.14

y

-2 Energy (eV)

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-4 -6 -8 -10 X

A

Y

X

A

Y

X

A

Y

Figure 5: The band structures of phosphorene under uniaxial strain εx of (a) -0.06 (b) 0.04 and (c) 0.14 and εy of (d) -0.1 (e) 0.02 and (f) 0.14 calculated by HSE06. The vacuum level is aligned for all systems. The green arrows point from VBM to CBM. case of εx , the bandgap decreases with increasing compressive strain while increases with tensile strain before the strain reaches 4%, then the bandgap descends with increasing tensile strain. As illustrated in Fig.5, in the range of compressive strain, phosphorene becomes indirect bandgap material with the VBM shifting gradually away from the Γ point along the Γ–X direction while the CBM located at the Γ point. In the range of tensile strain, the direct bandgap feature is maintained until the strain reaches 10%, then the CBM moves away from the Γ point along the Γ–Y direction. For the case of εy , the turning point for the bandgap changing from ascending to descending occurs at 6%. The VBM and CBM keep unchanged at the Γ point in the range of -9 % to 7 %, and the VBM moves away from the Γ point along the Γ–Y direction at εy = -10 %, as shown in Fig.5 (d), 11 ACS Paragon Plus Environment


the CBM moves away from the Γ point to X point when εy is larger than 8 %, as presented in Fig.5 (f). 1.2

(a) PBE

(b) HSE

1.8

1.0 1.5

0.8

Bandgap (eV)

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1.2

0.6

0.9

0.4

0.6

1L

u 0.2

0.3

0.0

0.0

x y

0

3

6

9

12

15

0

3

6

9

12

15

E-field (V/nm)

Figure 6: The bandgap of the pristine and deformed phosphorene as functions of the electrical field calculated by (a) PBE and (b) HSE. Next, we turn to investigate the response of the pristine and deformed monolayer to external vertical electrical field. The deformed structures under biaxial strain εu = 0.02, uniaxial strain εx = 0.02 and εy = 0.02 are chosen as typical models to investigate how phosphorene responds to combined influence of strain and electrical field. Different from other 2D monolayer materials, such as graphene, MoS2 , phosphorene distinctly responds to the applied vertical electrical field, as illustrated in Fig.6. The bandgaps of the pristine and deformed monolayer exhibit monotonic decreasing relationship with increasing electrical field and close up at 13 V/nm (15 V/nm) calculated by PBE (HSE). Moreover, the direct bandgap structures of the three models are all kept regardless of the existence of the electric field. It can be deduced that other deformed phosphorene can also present similar changes under electrical field with these three cases. Fig.7 and Fig.8 illustrate the variations of the electron and hole effective mass of the pristine and deformed phosphorene under vertical electrical field, respectively. The electron effective mass m∗e along the Γ–X direction reduces slightly firstly with the increasing electrical field and then decreases substantially from 5 V/nm, as shown in Fig.7 (a) and (b), while m∗e along the Γ–Y direction exhibits the opposite change tendency, that is increasing slightly with the increasing vertical electrical field and then ascending 12 ACS Paragon Plus Environment

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significantly from 5 V/nm, as presented in Fig.7 (c) and (d). Both the uniaxial and biaxial strain have increased m∗e along the Γ–X direction, and εu has the greatest impact on m∗e , followed by εy , and the last is εx , as can be seen in Fig.7 (a) and (b). In addition, the difference in the influence of the three types of in-plane strains on m∗e has been gradually reduced by the incremental electrical field, quantitatively, under zero electrical field, ∆m∗e is equal to 0.1 m0 between the pristine one and the deformed one under εu , while in the case under 9 V/nm, ∆m∗e decreases to 0.03 m0 . εy has very small impact on m∗e along the Γ–Y direction, while εx has similar and appreciable impact on m∗e with εu , that is m∗e under εu or εx becomes heavier than the one without strain or under εy . In addition, the strong anisotropy between m∗e along the Γ–X direction and the Γ–Y direction has been weakened by the electrical field. In the case of m∗h , the incremental electrical field progressively reduces m∗h along both directions ( the Γ–X and Γ–Y), as shown in Fig.8. εy increases m∗h along the Γ–X direction, while εu and εx remarkably decrease it. For the case along the Γ–Y direction, all three types of in-plane strains increase m∗h , and εu increases the most, next is the εx and the third comes εy . The strong anisotropy in respect of m∗h along the Γ–X and the Γ–Y direction has been maintained under the applied electrical field. 2.6

(b) HSE

(a) PBE 2.3

2.5

mex

1L

2.4

2.2

u x

2.3

y

2.2

2.1

0.7

(c) PBE

0.7

0.6

mey

(d) HSE

0.8

0.6

0.5

0.5 0.4 0.4 0.3 0

2

4

6

8

10

0

2

4

6

8

10

E-field (V/nm)

Figure 7: The variations of electron effective mass of the pristine and deformed phosphorene as functions of the vertical electric field along (a) the Γ–X direction calculated by PBE, (b) the Γ– X direction calculated by HSE, (c) the Γ–Y direction calculated by PBE, (d) the Γ–Y direction calculated by HSE.



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5.5

6.5 (a) PBE 6.0

mhx

(b) HSE 5.0

5.5

1L

5.0

u

4.5

x

4.0

y

4.5

4.0

3.5

3.5

3.0

3.0 0.34

(c) PBE

0.32

mhy

(d) HSE

0.36

0.35

0.34 0.30

0.33 0.28 0.32

0.26

0.31

0

2

4

6

8

10

0

2

4

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10

E-Field (V/nm)

Figure 8: The variation of hole effective mass of the pristine and deformed phosphorene as functions of the vertical electric field along (a) the Γ–X direction calculated by PBE, (b) the Γ–X direction calculated by HSE, (c) the Γ–Y direction calculated by PBE, (d) the Γ–Y direction calculated by HSE.

Conclusion In summary, we have investigated the structural and electronic properties of the bulk and ultrathin black phosphorus from monolayer to six-layer and the influence of in-plane strain and out-of-plane electric field on the electronic properties of phosphorene using first-principles calculations. The lattice constant a increases slightly with the increasing number of phosphorene layers, while the constant b shrinks distinctly. The band alignments of few-layer black phosphorus are all direct and the bandgap shows thickness-dependent relationship with the number of phosphorene layers ranging from 0.5 to 1.6 eV calculated by HSE. Strong anisotropy along the Γ–X and Γ–Y direction has been observed in respect of carrier effective mass. Besides, it is found that the adjacent bands present degenerate distribution from X point via A point to Y point but with totally different partial charge distributions. Biaxial and uniaxial strains (εu , εx and εy ) in range of elastic deformation have been demonstrated to exert different influences on the electronic structures of phosphorene. The bandgap can be continuously modulated by the strains from zero to 1.97 eV, which is expected to be promising materials for solar cell and other optical electronic devices. Moreover, it is noted 14 ACS Paragon Plus Environment

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that phosphorene also responds to external vertical electrical field. In the range of 0 to 5 V/nm, the bandgap decreases slightly, while it descends significantly with further larger electrical field, and finally closes up at 13 V/nm (15 V/nm) calculated by PBE (HSE). In addition, the electron effective mass decreases with the increasing field along the Γ–X direction while it presents the opposite variation tendency in the case along the Γ–Y direction, therefore the strong anisotropy has been weakened by the vertical electrical field. The hole effective mass along both directions descends with the ascending field. The fact that the bulk and few-layer black phosphorus possess inherent direct bandgaps and strong anisotropic carrier effective masses, together with the ultrasensitivity to the in-plane strain and out-of-plane electrical field, offers a wide range of physical properties and great possibilities to the electronic science and technology.

Acknowledgement This work was supported by the National Natural Science Foundation of China under Grant No.91233120 and the National Basic Research Program of China.

Supporting Information Available This material is available free of charge via the Internet at http://pubs.acs.org/.

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Modulation of the Electronic Properties of Ultrathin Black Phosphorus

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