Effect of Carbon Doping on Electromechanical Response of Boron Nitride Nanosheets

Nanotechnology

PAPER Effect of carbon doping on electromechanical response of boron nitride nanosheets

To cite this article: S I Kundalwal et al 2020 Nanotechnology 31 405710

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Nanotechnology 31 (2020) 405710 (17pp) https://doi.org/10.1088/1361-6528/ab9d43 Effect of carbon doping on electromechanical response of boron nitride nanosheets

S I Kundalwal, V K Choyal, Nitin Luhadiya and Vijay Choyal

Applied and Theoretical Mechanics (ATOM) Laboratory, Discipline of Mechanical Engineering, Indian Institute of Technology Indore, Simrol, Indore 453 552 India

E-mail: kundalwal@ iiti.ac.in

Received 6 March 2020, revised 7 June 2020 Accepted for publication 16 June 2020 Published 16 July 2020

Abstract The electromechanical response of hexagonal-boron nitride nanosheets (h-BNSs) was studied via molecular dynamics simulations (MDS) with a three-body Tersoff potential force field using a charge-dipole (C-D) potential model. Carbon (C)-doped h-BNSs with triangular, trapezoidal and circular pores were considered. The elastic and piezoelectric coefficients of h-BNSs under tension and shear loading conditions were determined. The induced polarization in h-BNSs was found to depend on the local arrangement of C atoms around B and N atoms, and the polarization increases if C atoms are surrounded by N atoms. This is attributed to the generation of higher dipole moments due to C–N bonds compared with C–B bonds. At ∼5.5% C-doping concentration, the axial piezoelectric coefficient of doped h-BNSs with triangular and trapezoidal pores increased by 18.5% and 3.5%, respectively, while it reduced by 22.5% in the case of circular pores compared to pristine h-BNS. The shear piezoelectric coefficient of C-doped h-BNSs with triangular and trapezoidal pores increased by 20.5% and 1%, respectively, while it reduced by 7% in case of circular pores. Young’s moduli of C-doped h-BNSs with triangular, trapezoidal and circular pores increased by 9%, 7.5% and 5.5%, respectively, due to the C–C bonds being stronger than all other bonds. The respective improvements in shear moduli are 8.5%, 5% and 5%. The elastic and piezoelectric properties of armchair h-BNSs were found to be higher than zigzag h-BNSs. The results also reveal that the piezoelectric coefficient increases as doping increases; it reaches its maximum value around − 0.41 C m 2 at 12.6% C-doping concentration and then starts decreasing. The present work shows that we can engineer the electromechanical response of h-BNSs via novel pathways such as different types and size of pores as well as C-doping concentration to suit a particular nanoelectromechanical systems (NEMS) application. Keywords: boron nitride nanosheet, carbon doping, piezoelectric properties, elastic properties, molecular dynamics simulations

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1. Introduction subject of intense interest over the past few years, as h- BNSs possess a highly stable structure, superior mechan- A two-dimensional (2D) hexagonal-boron nitride nanosheet ical strength and functionalization capabilities that assist in (h-BNS) is made of boron (B) and nitrogen (N) atoms, altern- engineering their properties for nanoelectromechanical sys- atively arranged in a honeycomb pattern like carbon atoms tems (NEMS) applications as per requirement [5]. A h-BNS in a graphene sheet [1, 2]. Multilayer h-BNSs were first possesses a large band-gap around ∼5 to 6 eV, making it an synthesized by Pacil et al [3], and single- and bi-layer h- insulator with excellent physical properties, high chemical and BNSs were synthesized by Han et al [4]. This has been a thermal stabilities, and strong resistance to oxidation at higher

◦ temperatures (up to > 900 C) [6]. Moreover, h-BNSs pos- polarization [20]. Such a coupling is not restricted by a solid sess a non-centrosymmetric structure and partly demonstrate nanomaterial possessing crystallographic symmetry, but can ionic characteristic of B–N bonds due to the electronegativ- exist in a variety of nanomaterials ranging from liquid crystals, ity differences of B and N atoms, demonstrating piezoelectri- solid dielectrics, semiconductors and living membranes [20]. city activity, which has attracted a lot of attention in the field Unlike graphene, a h-BNS comprises different atoms, which of actuators and sensors [7], composite materials [8], hydro- breaks the latter’s inversion symmetry and produces piezoelec- gen storage [9], gas sensors, optoelectronic, optical devices, tricity in it. Dumitirica et al [21] performed the first-principles transistors and biological probes [10]. Therefore, the study of calculations to determine the atomic dipole moments induced elastic and piezoelectric properties of h-BNSs and their nano- in a h-BNS using direct electronic charge density integration. structures has been carried out by numerous researchers. Mele and Karl [22] examined the electric polarization in BN- Researchers used different techniques to study the mech- based nanostructures using the Berry phase quantum method, anical behavior of h-BNSs. For instance, Peng et al [11] and found the existence of piezoelectricity in BN-based nan- developed a non-linear continuum model based on density omaterials. Sai and Mele [23] performed DFT calculations functional theory (DFT) to investigate the mechanical beha- and reported that the BN-based nanomaterials possess piezo- vior of a h-BNS. They reported that the h-BNS experiences electric properties. An analytical study by Dorth et al [24] a non-linear elastoplastic deformation up to ultimate failure also proved the existence of piezoelectricity in a monolayer point, followed by a strain-softening behavior. Mortazavi and h-BNS via the use of a quantum geometric phase approach. Remond [12] investigated the thermomechanical response of Zhang et al [25] studied the piezoelectricity in molybdenum a h-BNS using molecular dynamics simulations (MDS) and disulfide (MoS2) and h-BNS layers under the influence of an reported that its Young’s modulus depends on the chirality and electric field, and compared their piezo-potentials using MDS. the converse is true for thermal conductivity. Han et al [13] They reported that MDS provides reliable results compared to studied the temperature and strain rate effects on the mech- classical electromechanical models. anical properties of a h-BNS using MDS and found that high The h-BNSs are being synthesized using several techniques temperature softens its mechanical behavior. Qi-Lin et al [14] such as ultrasonication [8], electron beam irradiation, micro- studied the fracture behavior of a monolayer h-BNS contain- mechanical cleavage, wet-ball milling and chemical vapor ing various defects using MDS. They found that the fracture deposition (CVD) [10, 26, 27]. Due to the limitation of syn- strength and strain of the h-BNS are higher in its armchair dir- thesizing and purification processes of h-BNSs, topological ection than the zigzag direction. Thomas et al [15] performed defects and pores inherently exist in them, that is, different MDS to determine the mechanical properties of a h-BNS using shapes of pores such as circular, triangular etc, atom vacancies a Tersoff interatomic empirical potential. They reported that [26], Thrower-Stone-Wales [28] and substitutional impurities the elastic properties of the h-BNS are size-dependent and [29]. These defects/pores make the local changes in the atomic increase with its size. Abadi et al [16] studied the effect of polarization and chemical bond orders of h-BNSs that lead to a temperature and grain size on the mechanical properties of a change in their electromechanical response. Under the specific polycrystalline h-BNS using MDS. They reported that a cent- requirements, defects and impurities in h-BNS are being intro- ral crack reduces the tensile strength and failure strain of the duced intentionally via irradiations with energetic particles h-BNS, and such reduction in properties is independent of the and chemical processes to engineer its properties [26, 27]. For initial crack length. Eshkalak et al [17] studied the mechan- example, Jin et al [26] effectively synthesized a monolayer ical properties of a pristine and defected hybrid graphene/BN h-BNS using an energetic electron beam irradiation method sheet using MDS. They found that the presence of defects in and revealed the existence of B and N monoatomic and trian- the hybrid sheet reduces its mechanical properties. Hosseini gular pores in it. They also found that the B atom vacancies et al [18] performed DFT calculations to study the effect of are energetically more stable than N atom vacancies. Suen- functionalization on the electromechanical response of a h- aga et al [30] analyzed monolayer h-BNSs with and without BNS. They reported that the functionalization of hydroxyl in point defects using electron energy-loss spectroscopy. They the h-BNS leads to a reduction in its mechanical properties, revealed that monovacancy at the N site is more prominent for and the converse is true for electrical properties. Using MDS, electronic properties compared to B site vacancies. Park et al Yao et al [19] studied the mechanical properties of vertically- [27] used the electron beam irradiation method to control the stacked BN/BN and BN/graphene nanostructures by consider- shape, size, and stability of 2D nanomaterials such as h-BNS, ing different shapes of nanopores in them. They reported that graphene and MoS2. They obtained the desired shape and size the failure strength and strain of hybrid nanostructure increase of pores in h-BNS experimentally, as shown in figure 1. with the strain rate. They also revealed that the mechanical In the existing experimental studies, several dopants such properties of hybrid nanostructures with nanopores in the BN as oxygen [31], fluorine [32], carbon (C) [33–35] and chro- region are more sensitive to the temperature than nanopores in mium [36] were used to modify the properties/behavior of other regions. h-BNS. Several experimental and theoretical studies showed Piezoelectricity is an important property of non- that an elemental C atom can be doped easily into BN-based centrosymmetric crystals, which is the basis for actuat- nanomaterials over a wide range of compositions, which is ors, sensors and transducers in multifarious NEMS applic- the preferred route to modify their physicochemical and elec- ations. The piezoelectricity phenomenon is recognized as tronic as well as other properties. For instance, Terrones et al the electromechanical coupling between strain and electric [37] performed the experimental study on C-doped BN-based

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(a) (b)

Figure 1. Existence of pores in h-BNS with different pore geometries: (a) triangular pore and (b) hexagonal pore [27] (Reproduced with permission). nanotubes, nanosheets and nanofillers. They reported signi- emphasis was placed on the study of the effect of different pore ficant improvement in the structural and electronic proper- geometries and C-doping on the electromechanical response ties of BN nanostructures. Du et al [38] studied the structural of h-BNS. and electronic properties of C-doped hexagonal boron nitride nanoribbons (BNNRs) via first-principles calculations. They 2. Atomistic modeling of h-BNSs found that a single C-substitution at the B or N atom site in BNNRs induces spontaneous magnetization. Bhattacharya MDS is the most widely used modeling technique for simu- et al [39] determined the electronic properties of h-BNSs lation and characterization of nanomaterials. It allows a com- by functionalizing them with several functional groups like prehensive study of interatomic interactions between critical H, F, CN, OH, CH3 and NH2 using first-principles calcula- phases at the atomic scale using Newtonian mechanics with tions. Gao et al [40] performed DFT calculations to study the a time integration approach [43]. This provides an edge over adsorption and catalytic activation of C-doped h-BNSs. They the classical models and conventional experiments with cost- found that even small C-doping concentration can functional- efficient simulations. The fundamental advantage ofMDS ize the large surface area of the h-BNS, making it a promising over classical models is that it can be used to obtain various catalytic material for oxygen-activation and -reduction reac- properties and study behavior of nanostructures [43]. There- tions. Zhang et al [41] studied the thermal conductivity of 2D fore, MDS was performed in the present study to investigate nanomaterials using MDS and reported a thorough compar- the electromechanical response of pristine and C-doped arm- ison between the thermal conductivities of hexagonal boron- chair and zigzag h-BNS. All MDS were performed in large- carbon-nitride (h-BCN), h-BN and graphene sheets. Beniwal scale atomic/molecular massively parallel simulator (LAM- et al [35] synthesized graphene, h-BN and h-BCN sheets, and MPS) [44]. The schematic representations of armchair and zig- performed first-principles calculations to predict the direct zag edges of h-BNSs are shown in figure 2. The interatomic electronic band gap of nanosheets. Most recently, Thomas interactions among B and N atoms were calculated using and Lee [42] studied the monolayers of h-BCN, h-BN and the Tersoff potential force field [45] as it is used by sev- graphene using MDS, and reported their mechanical, elec- eral researchers to study the electromechanical response of trical, optical and thermal properties. h-BNSs [15, 46–49]. The piezoelectric and elastic properties The literature review clearly indicates that numerous stud- of h-BNSs under tension and shear loadings were determined ies have been performed to study the (i) mechanical proper- using MDS coupled with the C-D model and Tersoff poten- ties of defective and (ii) piezoelectric properties of pristine h- tial force field. This is was achieved by using both MATLAB BNSs. To the best of the current authors’ knowledge, no single and LAMMPS codes. We developed MATLAB code to solve study exists that reports the electromechanical response of C- the tensorial equations and then the obtained parameters were doped h-BNSs with different pore geometries which inher- called in the MDS using LAMMPS code. Systematic steps ently occur during their fabrication and processing. This has involved in such hybrid simulation methodology are shown in inspired us to conduct this study. This work aims to study figure 3. the electromechanical response of different size h-BNSs with First, the initial structures of the h-BNS were created. Then, different pore geometries such as triangular, trapezoidal and the simulations were performed to minimize the energy of BN circular pores, using MDS coupled with a charge-dipole (C- structures using the conjugate gradient minimization method D) potential model. The elastic and piezoelectric properties to obtain their optimized structures. The minimized structure of pristine and C-doped h-h-BNSs under tension and shear of the h-BNS was treated as optimized when the difference loadings were calculated. In the present study, both types of in the total potential energy (PE) of its structure between the − − armchair and zigzag h-BNSs were considered. A particular consequent steps was less than 1.0 × 10 10 Kcal mol 1 [50].

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the non-bonded electro-chemical (E-c) interactions of charges and dipoles. The non-bonded interactions include the charge- dipole, charge-charge and dipole-dipole interactions. The total energy (Etot) can be expressed as − Etot = ETersoff + EE c (1)

− − − Etot = ETersoff + Ec c + Ec d + Ed d + Eext (2)

The Tersoff PE ((ETersoff)) between two neighboring atoms i and j can be expressed in the following form [55]: ∑∑ 1 ETersoff = f (r )[f (r ) + b f (r )] (3) 2 c ij A ij ij R ij i j≠ 1

where fA (r) and fR (r) are the attractive and repulsive pair potentials, respectively. The cutoff function fR (r) was defined to limit the range of potential and thus, to save the computational time in the MDS. Figure 2. Monolayer h-BNS with armchair and zigzag edges. The electro-chemical PE (EE–c) with a given combination of charges (c) and dipoles (d) placed at the atomic positions The MDS were performed in the constant temperature and (r) can be written as: volume canonical (NVT) ensemble with a time step of 0.5 fs ∑N ∑N − 1 − for 30 ps to equilibrate the h-BNS structure [51]. The velocity EE c = c Tc cc 2 i ij j Verlet algorithm was used to integrate the equations of motion i=1 j=1 in all MDS. The simulations were performed with a periodic ∑N ∑N ∑N ∑N 1 − − boundary condition in all directions of the sheet and the simu- − d Td dd c Tc dd 2 i ij j i ij j lation box was kept large enough to avoid the interlayer inter- i=1 j=1 j=1 j=1 actions. After the energy minimization, tensile and shear load- ∑N ∑N ings were applied on the sheets. Schematics of these loading − + ci (χi + Vi) diE (4) conditions are shown in figure 4. In the case of tensile load- j=1 j=1 ing, the constant strain was applied at the right edge whereas the left edge was fixed in the x-direction of the nanosheet. In where the first three terms are the interatomic interaction ener- the case of shear loading, the bottom edge was fixed and the gies between the charge–charge, dipole-dipole and charge– constant shear strain was applied to the top edge of nanosheet dipole [56], respectively. The fourth term accounts for the in the xy-plane. During the loading, the constant initial strain energy required to bring the charge (ci) in the presence of was applied on the respective strained edges for a period of external potential (Vi). It also accounts for the interaction of 1 ps, followed by a relaxation time of 30 ps; this constitutes a charge with electron of atom at position ri using electron affin- single load step [52]. A series of load steps were applied on the ity (χi). The factor 1/2 prevents the double accounting of some h-BNS to achieve a strain value of 0.0275, which is within its of these interatomic interactions. The respective electron affin- elastic limit. During the loading, the progressing atomic con- ities of B, C and N were considered as 0.279 eV [57], 1.262 eV figuration and equivalent charges and dipole moments were [58] and −0.07 eV [58]. The last term accounts for the inter- stored throughout the simulations. actions of dipoles and external electric field (E). In the calculation of interaction energy, the atoms of system were con- − sidered as a point charge [59]. The term Tc c is for the Cou- 2.1. A charge-dipole (C-D) potential model lombic interactions between atomic charges ci and cj separated by a distance ri, j which is given as [56, 60]: Figure 4(a) shows the primitive lattice vectors a1 and a2 of an undeformed h-BNS. The unit cell of the h-BNS was defined by c−c 1 1 a rhombus, as shown in figure 4. In the unit cell, a single B–N Tij = (5) 4πε0 ri,j covalent bond was considered. All atoms in the h-BNS were assigned charge c and dipole moment d [53]. The total atomic where ε0 is the dielectric permittivity of vacuum and ri,j is the interaction energy of a system is the sum of short- and long- distance between atoms i and j. range interactions. Short-range interatomic interactions com- The two other tensors for charge–dipole and prise the PE due to the chemical bonds, which was calculated dipole-dipole are: using the Tersoff potential force field [45]. The parameters for ( )( ) − 1 r , the bonded interactions between B, C and N atoms were taken Tc d = i j (6) i,j πε 3 from [54]. The long-range interactions comprise the PE due to 4 0 ri,j

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Initial coordinates of h-BNS

Minimization of energy of h- BNS structure

Solve total energy Eq. (1) LAMMPS ( )

MATLAB/ Solve C-D matrix (Eq. 13) using saved coordinates Mathematical solver

*Data were recorded at each Calculate forces using obtained charges time step and h-BNS was and dipole moments further deformed using the complete loop, and then only the final properties were determined.

*Is the targeted Yes Calculate the change in strain achieved? polarization with strain

No Calculate piezoelectric Add forces induced due to charges and & elastic coefficients dipole moments

Deform the h-BNS structure LAMMPS

Relax the h-BNS structure

Save the updated coordinates

Figure 3. Flow diagram of simulation methodology. ( ) erf √rij ( ) c−c 1 2R (( ) ) Tij = (8) d−d 1 2 5 4πε0 rij T = 3r , ⊗ r , − r I /r (7) i,j 4πε i j i j i,j i,j 0 Thus, the self-interaction energy of atoms reduces to where I is the 3 × 3 identity matrix. ( )√ − In equation (5), Tc c term contains 1 which diverges when c−c 1 2 1 ij rij T = (9) → ii 4πε π R rij 0 with charge-charge interactions.( ) In order to overcome 0 this, r is normalized with erf √rij and then equation (4) can The( width)√ of Gaussian distribution is determined as ij 2R c−c 1 2 1 T = ; where R , is the width of Gaussian be re-written as: ii 4πε0 π R A i

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(a) (b)

Figure 4. Schematic representations of h-BNS under (a) tensile and (b) shear loadings.

distribution for atom index i with type A. RB,j is the Gaus- and dipoles were calculated using the charge and dipole data sian distributed charge width for atom type B with index j obtained during simulations. The velocity-Verlet algorithm [61]. The values of R for B, C and N were taken as 0.35 Å, was used in the MDS to integrate forces (equation (10)) and 0.686 Å and 0.76 Å, respectively [60]. The equations (6) update atomic positions after each load step. Then, using the and (7) for charge-dipole and dipole-dipole, respectively, also updated atomic configuration, charges and dipoles were calcu- → diverge when rij 0 and to avoid divergence, normalized rela- lated using equation (13). If the initial atomic position at time t tions were used from references [56]. Using the above defined is considered then the charge dipole moment data of time step interatomic potentials, the trajectory of each atom after each (t − ∆t) is used, where ∆t is the time interval between two load step was calculated using the following relation: successive load steps [60].

− − − Polarization (Pm) for each unit cell was calculated using ∂ETersoff ∂Ec c ∂Ed d ∂Ec d − − − − the dipole moment data di and it is equal to the sum of dipole mi¨ri = (10) ∂ri ∂ri ∂ri ∂ri moments inside the single unit cell divided by its total volume, defined as follows: where the forces due to the Tersoff potential were incorpor- ated in LAMMPS code and the forces due to the C-D inter- ( ) ∑n actions were added separately. The governing equations were 1 Pm = di (14) obtained to determine the charge and dipole moment of each Vm i=1 atom. The governing equations for atomic charge ci were obtained by solving equations (11) and (12), as follows: where n is the number of atoms present in the unit cell (m) N N ∑ ∑ and Vm is the volume of the unit cell, calculated as V = At, in c−c − c−d c−c − c−d Tij cj Tij dj + Tij ci Tij di + χi = 0 which A and t are the surface area of the unit cell and thickness j=1, i≠ j j=1, i≠ j of the h-BNS, respectively. The thickness of the h-BNS was (11) considered as 3.35 Å [62]. The axial( strain) (ε) of the h-BNS − was calculated using relation, ε = Lf Li , in which L and Li i ∑N ∑N Lf are the initial and final lengths of the h-BNS, respectively, in c−d + d−d + c−d − d−d = Tij cj Tij dj Tij ci Tij di 0 (12) the direction of applied tensile loading. The shear( strain) (γ) of j=1, i≠ j j=1, i≠ j γ = ∆x the h-BNS was calculated using relation, W , in which ∆ These equations were used to determine the unknown val- x is the displacement of the h-BNS in the direction of applied ues of charge and dipole moment by arranging them into shear loading and W is the width of the h-BNS normal to the matrix-vector form, as follows: applied load. [ ][ ] [ ] The total polarization of the h-BNS can be expressed as the − − Tc cTc d c −χ sum of polarization of each unit cell, as follows: − − = (13) Tc dTd d d 0 ∑M Above relation provides the charge and dipole moment for P = Pm (15) the current atomic configuration. The forces due to charges m=1

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Figure 5. Effect of strain rates on (a) stress-strain behaviour and (b) Young’s modulus of pristine h-BNS (A_2560) under tension loading.

Table 1. Comparison of elastic properties of pristine h-BNSs predicted by different computational models. E (TPa) G (TPa) υ Present Ref. Present Ref. Present Ref.

ab-initio 0.706–0.900 0.778–0.781 [63]- 0.299–0.360 0.323–0.325 [63] 0.187–0.217 0.211 [63] (Armchair) 0.827–0.957 [64]- (Armchair) - (Armchair) 0.217 [11] 0.500–0.610 0.247–0.286 0.162–0.175 0.213 [64] DFT calculation (Zigzag) 0.800–0.850 [65] (Zigzag) 0.311–0.315 [65] (Zigzag) 0.216 [65] MD finite ele- 0.982–1.113 [66] - - ment method

The polarizations in the x-direction (Px) and y-direction and Z_4000. First, the elastic properties of armchair and zig- (Py) are due to the stretching of the h-BNS and the shearing zag h-BNSs were determined followed by the calculation of of the h-BNS in xy-plane, respectively, and are given by [59]: induced polarization and piezoelectric coefficients. The stress-strain relations are functions of applied strain rate and a system gets enough time to relax for reaching the Px = exxxεxx (16) equilibrium state at the lower values of strain rate. Therefore, to obtain the reliable and accurate results, the use of lower values of strain rate is preferred. However, we varied the values of − strain rate as 0.0001, 0.0003, 0.0005, 0.0007 and 0.001 ps 1 Py = eyxyεxy (17) to choose the optimal value to study the electromechanical ε ∂εxx response of the h-BNS. The tensile test was performed on the where, xx and ∂x are the respective strain and strain gradient terms; exxx and eyxy are the piezoelectric coefficients in tension pristine (A_2560) h-BNS by varying the strain rates and the and shear, respectively. variation of stress with strain for sheets under tensile loading is shown in figure 5. Figure 5(b) shows the effect of strain rate 3. Results and discussions on the Young’s modulus of the h-BNS and it can be observed that the modulus increases with the increase in strain rate. A 3.1. Electromechanical properties of h-BNSs similar observation was found by Han et al [13] and Yao et al [19]. It can also be observed from figure 5(b) that the change MD coupled with C-D potential models were developed to in Young’s modulus drastically increases when the strain rate − study the electromechanical response of armchair and zig- is greater than 0.0005 ps 1 and this is attributed to the dis- zag h-BNSs under tensile and shear loading conditions. The sipation of strain energy of the h-BNS at higher strain rates. different size of armchair and zigzag h-BNSs were con- Therefore, it can be concluded that the use of the strain rate − sidered for studying the electromechanical response. The fol- value of 0.0005 ps 1 is the optimal choice to investigate the lowing representation for h-BNSs is used: armchair (A)/zigzag electromechanical response of the h-BNS, and such a value (Z)_number of atoms. A total of eight cases were considered: was also used in the existing work to study the mechanical A_640, A_1440, A_2560, A_4000, Z_640, Z_1440, Z_2560 behavior of the h-BNS [50]. Therefore, we used the strain rate

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Figure 6. The variation of PE of different h-BNSs under uniaxial loading.

Figure 7. The stress-strain curves for h-BNSs under (a) tension and (b) shear loadings.

− of 0.0005 ps 1 in subsequent simulations. To verify the valid- that the values of maximum stresses in the armchair and zig- ity of current MDS, the elastic properties of h-BNSs reported zag h-BNSs are 23 GPa and 9 GPa at a strain value of 0.0275, by other researchers using different techniques were compared respectively. In the case of shear loading, the values of max- with our results, as summarized in table 1. The comparison is imum shear stresses are 9 GPa and 6.5 GPa in the armchair found to be in good agreement. and zigzag h-BNSs, respectively, at strain value of 0.0275. The The variation of PE of h-BNS subjected to tension load- elastic properties of different h-BNSs were obtained from the ing is shown in figure 6. As expected, it can be observed from stress-strain curves and the same are shown in figure 8. It can figure 6 that the larger size of h-BNS shows higher PE than be observed from figure 8 that Young’s and shear moduli of smaller ones. This is attributed to the larger volume of the h- armchair h-BNSs are higher than zigzag h-BNSs. In order to BNS, which eventually stores more PE. It can also be observed understand the physics behind such differences, the deforma- that the armchair h-BNS shows higher PE compared to the zig- tion mechanics of h-BNSs under the tension and shear load- zag one at the same strain value, which is in good agreement ings are explained in figures 9 and 9, respectively. During the with the existing results [67]. tension loading, it can be observed from figure 9 that the four The stress-strain response of armchair and zigzag h-BNSs types of deformed B-N bonds exist in the sheet: (i) a2 and z1 are under tension and shear loadings is shown in figures 7(a)–(b), slightly inclined to the direction of loading, and (ii) bonds a1 respectively. Young’s (Exx) and shear (Gxy) moduli of h-BNSs are aligned with the loading direction while bonds z2 are nor- under tension and shear loading conditions, respectively, were mal to the direction of loading. Under the tension loading, the determined from the slope of stress-strain curves within the bonds a1 and a2 are under tension while bonds z1 and z2 are elastic limit of the h-BNS. From figure 7(a) we can observe under tension and compression, respectively. Therefore, the

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Figure 8. Elastic and piezoelectric properties of pristine h-BNSs.

Figure 9. Deformation mechanics of (a) armchair and (b) zigzag h-BNSs under tension loading.

bonds a1 and a2 act as the stress-bearing bonds which carries dipole moment generates [68]. This results in the permanent more axial load than the bond z1. However, it can be observed polarization in BN-based structures. During the deformation, from figures 9 (a)–(b) that all six B–N bonds (a1 and a2) in B-N bond lengths increase that lead to change in the electronic the case of armchair h-BNSs carry the tensile load while in polarization and consequently, the piezoelectricity generates. the case of zigzag h-BNS, only four B–N bonds (z1) carry the The resultant dipole moments, atomic polarization and piezo- tensile load and rest of the B–N bonds are under compression. electric coefficients were calculated using the procedure dis- Therefore, the stress-bearing bonds in armchair h-BNSs are cussed in section 2.1. The slope of the polarization-strain curve higher than the zigzag h-BNSs and thus, Young’s modulus of during the tension and shear loadings defines the piezoelectric the former is slightly higher. During the shear loading, it can coefficients (exxx and eyxy) and these are shown in figure 11. It be observed from figure 10 that the two B–N bonds are paral- can be observed from figure 11 that the induced polarization lel to the loading direction in case of the armchair h-BNS and in the armchair h-BNS is higher than the zigzag case. This is no B–N bond is parallel to the loading direction in the case of attributed to the generation of higher resultant dipole moments the zigzag h-BNS. By resolution of applied forces, it can be in the x-direction of the armchair h-BNS compared to the zig- found that the shear stress value in the direction of B–N bond zag sheet (see figure 12). In the case of tensile loading, the of the zigzag direction is less than the armchair direction due piezoelectric coefficients of armchair and zigzag h-BNSs are −2 −2 to the applied force FR. approximately 0.38 C m and 0.19 C m , respectively, for Then, the size-dependent piezoelectric properties of h- a sheet having a total of 2560 atoms; in the case of shear load- BNSs were studied. The induced polarization in h-BNS was ing, piezoelectric coefficients of the same are approximately − − calculated using equation (14) considering a single unit cell 0.27 C m 2 and 0.14 C m 2, respectively. Note that the piezo- and the results are plotted in figures 11(a)–(b) for tension and electric coefficients of the h-BNS are a function of its chir- shear loadings, respectively. An atom shares its valence elec- ality and our results are in line with the results reported by tron with each neighboring atom. These shared valence elec- other researchers, as summarized in table 1. The compared trons between B and N atoms form a strong sigma bond. The and validated results prove that the C-D potential model can be contribution of electrons is higher from B to N atoms due to used for investigating the electromechanical response h-BNS. the higher electronegativity of N atom and thus, an in-plane It can be observed from figure 11(b) that the enhancement

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(a) (b)

Figure 10. Deformation mechanics of (a) armchair and (b) zigzag h-BNSs under shear loading.

Figure 11. The polarization-strain curves of h-BNSs under (a) tension and (b) shear loading conditions. of piezoelectric coefficient in shear loading is less compared enhance the piezoelectric properties of planar nanomateri- to tensile loading. This is due to the fact that the polariza- als [59, 60, 68]; therefore, further investigations were car- tion depends on the change in bond deformations, which are ried out to study the effect of C-doping on the electromechan- lower in the case of application of the shearing load [59]. The ical response of h-BNS under tension and shear loadings. To work was further extended to study the effect of C-doping on enhance the piezoelectricity in h-BNSs, C atoms were doped the piezoelectric and elastic properties of h-BNSs considering in such a way that can make them strong non-centrosymmetric only armchair sheets having 2560 atoms, because they show materials. To study the effect of C-doping and different shapes higher piezoelectricity per unit cell. of pores, further simulations were performed for four different types of armchair sheets: pristine h-BNS, and C-doped h-BNSs with triangular, trapezoidal and circular pores; schem- 3.2. Effect of C-doping on electromechanical response of atics of these structures are shown in figure 13. Note that a h-BNSs constant C-doping concentration of 5.625% was considered. Recently, piezoelectric h-BNSs have attracted a lot of atten- The doping concentration is the ratio of number of C atoms tion in nanotechnology applications to develop their NEMS at to the total number of atoms in the sheet. A certain number the nanoscale level. Among all other properties, polarization of B and N atoms were removed to create different pores in of h-BNS was found to be sensitive to defects and chemical h-BNSs. We fixed the size of pore in the h-BNS insucha doping, owing to interference and quantum mechanics effects way that it filled with the same number of C atoms irrespect- [69]. The modification of geometry and chemical structure ive of the shape of the pore to study the effect of C-doping on of h-BNSs with doping allows researchers to engineer their the electromechanical response of the h-BNS. Therefore, the electronic properties. In fact, the breaking of symmetry can resulting C-doping concentration leads to the value of 5.625%.

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Figure 12. Dipolar moment induced mechanism in the armchair and zigzag h-BNSs (red arrow shows the direction and magnitude of dipole moment).

Figure 13. Schematics of (a) pristine h-BNS and different C-doping arrangements in h-BNSs with (b) triangular pore (case 1), (c) trapezoidal pore (case 1), (d) circular pore (case 2), (e) triangular pore (case 2) and (f) trapezoidal pore (case 2).

First, the effect of position of C-dopants on the piezoelec- B atoms than N atoms (case 1, figures 13(b)–(c)) and (ii) C tric properties of h-BNS was studied by keeping the constant atoms surrounded by more N atoms than B atoms (case 2, fig- doping concentration. Two types of C-dopant arrangements in ures 13(e)–(f)). Figure 14 shows the variation of polarization the h-BNS were considered: (i) C atoms surrounded by more with the strain for two cases under tension and shear loadings.

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Figure 14. Variation of polarization in h-BNSs with different C-doping positions.

Figure 15. Stress-strain curves for armchair h-BNS (A_2560) under (a) tension and (b) shear loading conditions.

Figure 16. The variation elastic moduli of armchair h-BNS with the number of atoms (N).

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Figure 17. The variation of polarization-strain for h-BNSs under (a) tension and (b) shear loading conditions.

Figure 18. The variation of piezoelectric coefficients with number of atoms (N).

It may be observed from figure 14 that case 2 provides more Young’s moduli of the h-BNS with triangular, trapezoidal and polarization than case 1. This is attributed to the generation of circular pores are found to be 9 %, 7.5% and 5.5%, respect- more dipole moments due to C–N bonds than C–B bonds as ively, and the respective improvement in shear moduli are the former shows higher differences in electronegativity that 8.5%, 5% and 5%. It can be observed that the elastic moduli leads to the higher polarization. It is important to note that of C-doped sheets are higher than the pristine one, and this is out-of-plane movement of the sheet due to C-dopants was not attributed to the C–C bond being stronger than all other bonds. considered because C, B and N atoms have the same planar sp2 It may also be observed that the doped h-BNS with triangular hybridization [68]. The work was further extended to study the pores shows higher Young’s modulus compared to all other effect of C-doping on the piezoelectric and elastic properties of cases. This is due to the existence of a higher number of C– armchair h-BNSs with different percentage and types of pores C bonds in the triangular case for constant C-doping concen- considering case 2 only. tration. The same is true in the case of the shear modulus of Figures 15(a)–(b) show the stress-strain curves of C-doped the h-BNS as shown in figures 16(b). As expected, the elastic A_2560 h-BNS with different pores under tensile and shear properties of the h-BNS increase as its size increases and then loadings, respectively. We can observe from these plots that they become stabilized at a particular total number of atoms, the C-dopants improve the elastic behavior of h-BNS slightly that is, 2560 atoms (80 Å × 80 Å). This is due to the fact that compared to pristine one. Figure 16 shows the variation of the effect of applied forces on the system was found to be neg- Young’s and shear moduli of the h-BNS with C-doping con- ligible at some finite distance from the sections where they centration of 5.625%. The improvement in the values of were applied according to Saint Venat’s principle.

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(a) (b) (c)

1.406% 3.164% 5.625% (d) (e) (f)

8.789% 12.656% 18.906%

Figure 19. The schematics of h-BNSs with different C-doping concentrations.

Figure 20. The variation of (a) stress-strain (b) polarization-strain curves for h-BNSs containing triangular pores with different C-doping concentrations.

The effect of C-doping on the piezoelectric properties of the canceled out by each other. This phenomenon is also observed h-BNS can be seen in figures 17(a)–(b) under tension and shear for a graphene sheet with circular pores [59]. A C-doped h- loadings, respectively. The C-doping results in the increase BNS with different shaped pores shows a significant improve- of values of exxx of h-BNSs with triangular and trapezoidal ment in its piezoelectric coefficient when the tensile loading pores by 18.5% and 3.5%, respectively, and the reduction is applied due to the non-centrosymmetric nature of pores. of value exxx of h-BNSs with circular pores by 22.5% com- As a matter of fact, the piezoelectricity phenomenon is size- pared to a pristine h-BNS. During the shear loading, the val- dependent; therefore, we increased the number of B and N ues of eyxy of the h-BNS with triangular and trapezoidal pores atoms keeping the constant C-doping concentration. The vari- increased by 20.5% and 1.075%, respectively, and the value ation of piezoelectric coefficients of h-BNSs under tension eyxy of the h-BNS with circular pore reduced by 7%. The and shear loading conditions are shown in figure 18. As the piezoelectric coefficient of the C-doped h-BNS with circu- number of B and N atoms increases, by keeping the con- lar pore decreases and this is attributed to the fact that the stant C-doping, the values of exxx and eyxy increase up to the circular shape pore does not help in breaking the symmetry total number of 2560 atoms and then stabilize as the effect of of h-BNS and therefore, the induced dipole moments across presence of C-dopants becomes negligible. The same is true the circular edge of pore are found to be symmetric and get in the case of values of eyxy of h-BNS under shear loading.

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may provide a new platform for designing and developing their next-generation NEMS.

4. Conclusions

In this study, the electromechanical response of C-doped h- BNSs containing different shapes of pores was investigated comprehensively using MDS coupled with the charge-dipole (C-D) potential model. The interatomic interactions among B and N atoms were modeled using the Tersoff potential force field. First, the elastic and piezoelectric coefficients of pristine h-BNSs were compared with the available results, and good agreement was found between both sets of results. The axial and shear elastic/piezoelectric coefficients of h-BNSs were Figure 21. Variation of Young’s modulus and piezoelectric determined using the tension and shear loading conditions, coefficient of h-BNSs containing triangular pores with different respectively. The elastic and piezoelectric coefficients of arm- C-doping concentrations. chair h-BNSs were found to be higher than zigzag h-BNSs irrespective of the shape of the pores and C-doping concentration. The axial and shear piezoelectric coefficients of C-doped h-BNSs with triangular and trapezoidal pores increased, while The obtained results are in good agreement with the exist- they reduced in the case of circular pores compared to pristine ing results obtained by using the different techniques [23]. h-BNS. A h-BNS with triangular pores showed higher polar- Further, we studied the effect of different C-doping concen- ization than h-BNSs containing other shapes of pores. The C- trations on the electromechanical response of h-BNSs with doping with N atoms results in the generation of higher dipole triangular pores. moments, because the C-N bond is more electronegative than In the previous sets of results, elastic and piezoelectric the C-B bond. The piezoelectric coefficients of doped h-BNS properties of different size h-BNSs were determined consid- increase with the C-doping concentration up to 12.6% and then ering the constant C-doping concentration of 5.6%. However, they decrease. The elastic properties of doped h-BNSs increase the variation in C-doping concentration may influence the with the C-doping concentration up to 12.6% and then they elastic and piezoelectric properties of h-BNSs and therefore, stabilize and do not change beyond the total number of 2600 the C-doping concentration was varied for further analysis of BN atoms. The present work offers a theoretical framework for h-BNSs with triangular pores with 2560 atoms, as it showed predicting the elastic and piezoelectric properties of pristine, higher a electromechanical response under tension loading. A × defective and doped BN-based nanomaterials under different size of 80 Å 80 Å h-BNS was modeled and the C-doping types of loading conditions. concentration was varied from 1.46% to 18.9%. The schematics of C-doped h-BNSs with triangular pore are demonstrated in figure 19. The doping was done by attaching N atoms with C atoms. The stress-strain and polarization-strain curves of C- Acknowledgments doped h-BNSs are shown in figure 20. The obtained values of Young’s modulus and piezoelectric coefficient are plotted The work was supported by the Science Engineering Research in figure 21. It can be observed from figure 21 that Young’s Board (SERB), Department of Science and Technology, modulus of h-BNSs increases with C-doping concentration Government of India. S.I.K. acknowledges the generous and stabilizes at an approximate value of 12.6%. It can also support of the SERB Early Career Research Award Grant be observed from figure 21 that the piezoelectric coefficient (ECR/2017/001863). increases as C-doping concentration increases; it reaches its − maximum value around 0.41 C m 2 at 12.6% C-doping concentration and then starts decreasing. This behavior can be attributed to the reduction in the contribution from polarized Conﬂict of interest B and N atoms and increment in the non-polarized C atoms The authors declare that there is no conﬂict of interest in the to the total polarization. In case of trapezoidal and circular present study. pores, the respective values of maximum piezoelectric coef- − − ficients are 0.111 Cm 2 and 0.105 C m 2 at a strain range of 0.025–0.0275 with C-doping concentration of 18.9%, respectively. These results are not shown here for the sake of brev- ORCID iDs ity. The present study reveals that the C-doped h-BNSs with triangular pores show higher elastic and piezoelectric proper- S I Kundalwal  https://orcid.org/0000-0001-5237-5996 ties, and such improved electromechanical response of sheets Vijay Choyal  https://orcid.org/0000-0002-8202-9127

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