Article pubs.acs.org/JPCC
Strain Modulation of Electronic Properties of Monolayer Black Phosphorus Zhe Zhang, Yipeng Zhao, and Gang Ouyang* Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Synergetic Innovation Center for Quantum Effects and Applications (SICQEA), Hunan Normal University, Changsha 410081, China ABSTRACT: Recent advances in the fabrication of monolayer black phosphorus (MBP) call for a detailed understanding of the physics underlying the electronic structure and related modulation by the method of strain engineering. Here, we present an analytic study to explore the uniaxial strain effect of band structure in MBP based on the first-principles calculations and atomic-bond-relaxation correlation mechanism. It was found that the stress responses of MBP show evident anisotropy due to different edge type structures. The electronic band structure of MBP can be tuned by the applied strain. Moreover, we propose an analytic expression for the variation of the bandgap induced by the uniaxial strain from the perspective of atomistic origin, which suggests an effective bridge between the measurable quantities and the atomic bond identities of MBP. The underlying mechanism on the strain-dependent band offset can be attributed to the variation of crystal potential induced by the changes of bond length, strength, and angle, providing a better understanding of the modulation of electronic properties with strain engineering.
1. INTRODUCTION Black phosphorus (BP), a layered semiconductor material which is covalently bonded to nearest three atoms by means of sp3 hybridization, has been recognized as a new kind of materials with unique properties that is critical to the application of electronics and optoelectronics.1,2 Since the successful fabrication of few-layer phosphorene field-effect transistors (FET) in 2014,3 BP has attracted intense research interest owing to its fantastic electronic and optical properties such as narrow intrinsic gap and high carrier mobility.4−6 BP, the most stable allotrope of phosphorus, shows a distinctive thickness-dependent bandgap varying from 0.3 to 2.0 eV when the thickness thins from bulk to monolayer.4,7−13 Recently, layered BP nanosheets have been successfully synthesized through liquid-phase exfoliation14,15 and chemical vapor deposition,16 which are more preferred for spectroscopic and electrical investigations. In general, the modulation of physical properties in electronic materials can be approached by the method of “strain engineering”. So far, strain has been proved to be an effective method for tuning the electronic and optical properties of carbon nanotubes, 17 graphene, 18,19 MoS 2 , 20,21 and BP,13,22−31 etc. For example, Rudenko et al.27 provide a tightbinding (TB) model parametrization for BP with an arbitrary number of layers to explore the realistic problems related to the electronic properties of multilayer BP. Also, the MBP possesses a superior flexibility and can be sustained a tensile strain up to 30% in the zigzag direction and 27% in the armchair direction, allowing exceptional control of optical and electronic properties in practical strain engineering.32−35 In particular, the in-plane uniaxial strain along the armchair and zigzag directions have been used to modify the bandgap of MBP,30,36−39 while the relative efficacy of uniaxial and biaxial strains have been © 2017 American Chemical Society
comparatively studied for their effects on the electronic band structure.29,40−42 Physically, the electronic bandgap of semiconductor is a crucial quantity which determines its applications in electronic and optoelectronics. Strain engineering, which is used as an effective way, has been introduced to tune electronic properties of semiconductors in recent years. However, two questions have emerged: how does the electronic band structure change when the MBP is approached under strain? What is the theoretical relationship between the change of bond parameters and applied strain? Furthermore, the underlying mechanism on the strain-depenent electronic properties in MBP from the perspective of atomistic origin remains unclear. Accordingly, a systematic analysis and understanding for the strain effect on the band offset for a general strain type will be necessary based on manipulation of the electronic properties in MBP, which comprises the focus of the present work. Although it is now well built that the bandgap exhibits a forceful response to the strain in bulk BP,43 few dependable data and theories are available for the corresponding respond to in-plane strain in MBP system. Therefore, for these issues, in this contribution we explore the electronic and structural response of MBP to in-layer strain based on the first-principles calculations44 and atomic-bondrelaxation (ABR) correlation mechanism. Our calculations show that the Poisson’s ratio and strain energy exhibit an intense anisotropy with respect to the uniaxial strain direction. In the x direction (i.e., applied in the zigzag direction), it is interesting to note that there is a transformation from direct-toReceived: June 28, 2017 Revised: August 2, 2017 Published: August 3, 2017 19296
DOI: 10.1021/acs.jpcc.7b06342 J. Phys. Chem. C 2017, 121, 19296−19304
Article
The Journal of Physical Chemistry C
Figure 1. Lattice and electronic structures of black phosphorus (BP): (a) a unit cell of BP; (b) top and (c) side view of the atomic structure of the MBP; (d) geometry parameters of BP; (e) uniaxial strain applied along x and y directions, respectively; (f) Brillouin zone path of MBP primitive cell and its electronic band structure.
indirect transition whether under uniaxial compressive or tensile strain. The values of bandgap achieve the maximum at εx = 3% tensile strain. Under uniaxial strain in the y direction (i.e., applied in the armchair direction), there is only a transformation that from direct-to-indirect transition under compressive strain. Moreover, we put forward an analytical model to address the strain-dependent band offset of MBP in terms of functional dependence on the bonding identities based on the ABR approach.45−48 Also, the physical mechanism on the strain modulation of electronic properties in MBP is clarified in detail through first-principles calculations and ABR consideration. Therefore, our method provides the information on the measurable quantities and bond identities, which infers
that the developed method is helpful for strain design on tunable electronic properties of MBP-based devices.
2. METHODS 2.1. First-Principles Calculations. Our calculations of the MBP’s electronic structure are performed by DFT.44 implemented in the Virtual NanoLab-Atomistic ToolKit (VNL-ATK) package. The VNL-ATK package in conjunction with linear combination of atomic orbitals (LCAO) method within the generalized gradient approximations (GGA)49 of Perdew−Burke−Ernzerhof (PBE) exchange-correlation functional is used. The electron wave function is expanded using a double-ζ polarized (DZP) basis set. The kinetic energy cutoff 19297
DOI: 10.1021/acs.jpcc.7b06342 J. Phys. Chem. C 2017, 121, 19296−19304
Article
The Journal of Physical Chemistry C basis set was chosen to be 180 Ry. The Monkhorst−Pack κpoint grid of 14 × 10 × 1 is employed in the calculations. The unit cell with periodic boundary condition was applied to simulate a monolayer of MBP. The layered structures are placed in the x−y plane, and a sizable vacuum of 18 Å is used to avoid interaction between periodic images of slabs in the z direction. The geometric structures of MBP are optimized by the limited-memory Broyden−Fletcher−Goldfarb−Shanno (LBFGS) algorithm. All atoms are relaxed, and the final forced exerted on each atom is less than 0.01 eV/Å for each ionic step and maximum stress tolerance smaller than 0.001 eV/Å3. In the calculations of band structure, we collected 100 points along each high symmetry line in reciprocal space. 2.2. Theoretical Methodologies. The structure of BP is the most stable among its phosphorus allotrope at room temperature.50 The bulk BP is orthorhombic with space group Cmca and there are 8 P atoms in each conventional unit cell (see Figure 1a).51,52 BP is structurally similarly to layered graphite in which individual atomic layers are stacked together by van der Waals interactions. The MBP has a puckered honeycomb structure with each phosphorus atom covalently bonded with three adjacent atoms (see the top and side views of MBP shown in Figure 1, parts b and c). Figure 1d shows the definition of in-plane bond length (d1) and bond angle (α), cross-plane bond length (d2) and bond angle (β), the distance of the vertical direction (h) and horizontal projection distance
(a(1 + ε)/2)2 + (b(1 + ενyx)/2 − (l(1 + ενyx))2
d1′ =
d 2′ =
(l(1 + ενyx))2 + (c(1 + ενzx))2 cos α′ = 1 −
(a(1 + ε)/2)2 2d1′ 2
cos β′ = {d1′ 2 + d 2′ 2 − [(b(1 + ενyx)/2)2 + (a(1 + ε)/2)2 + (c(1 + ενzx))2 ]}/2d1′d 2′ x ‐direction (1)
and d1′ =
(a(1 + ενxy)/2)2 + (b(1 + ε)/2 − (l(1 + ε))2 d 2′ =
(l(1 + ενxy))2 + (c(1 + ενzy))2 cos α′ = 1 −
(a(1 + ε)/2)2 2d1′ 2
cos β′ = {d1′ 2 + d 2′ 2 − [(b(1 + ε)/2)2 + (a(1 + ενxy)/2)2 + (c(1 + ενzy))2 ]}/2d1′d 2′ y‐direction (2)
Naturally, the relaxation of bond length and bond angle under uniaxial strain would induce the variations of deformation energy and total energy. For the case of MBP, the Stillinger−Weber (SW) potential is proved to be an available and effective type of interatomic potentials.56−58 The SW potential comprises two terms: a two-body term depicts the bond stretching interaction, and a three-body term is the bond angle bending interaction. Thus, the total potential energy could be expressed as59
(l = d 2 2 − h2 ), respectively. We obtained the lattice parameters of bulk BP, a = 3.3136 Å, b = 4.3763 Å, and c = 10.478 Å, which is the same as the experimental values7,51−53 and theoretical works.22,30,37 The relaxed lattice constants for a MBP are a0 = 3.3158 Å and b0 = 4.3806 Å. The value of bandgap is 0.87 eV, which is calculated by first-principles without any strain approach (see Figure 1f). Clearly, our results are in good agreement with theoretical calculations10,33 and less than other experimental values.54,55 We apply strain by directly tuning the in-plane lattice constant (a and b) of 2D puckered honeycomb crystal structure. Two types of strains are considered in our work: uniaxial compression or expansion in the x direction (zigzag strain) and in the y direction (armchair strain), as shown in Figure 1e. In order to explore the intraplanar strain for MBP, we have applied up to −5% compression and 5% stretch strain along x axis (the zigzag direction) and y axis (the armchair direction). Note that the strain is defined as εx = (ax − a0)/a0 and εy = (by − b0)/b0, where ax (by) and a0 (b0) are the lattice constants along the x (y) direction for the strained and original relaxed structures, respectively. The negative value of strain refers to the compression, while positive corresponds to the expansion. With each axial strain applied, the compressing or stretch of MBP is realized by the optimized lattice constant a0 (b0) to ax (by), the lattice constant in the transverse direction is fully relaxed through the method of energy minimization to ensure the force in the perpendicular direction is a minimum. According to Figure 1, parts d and e, the strain-induced P−P bond lengths and bond angles can be deduced as
E=
∑ V2(i , j) + ∑ i
