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Flexoelectric effect in boron nitride–graphene heterostructures

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Abstract

Hexagonal boron nitride and graphene layers offer an attractive way to build 2D heterostructures as their lattices are well-matched as well as they are isostructural and isoelectronic. In this work, the flexoelectric coefficients of monolayer boron nitride-graphene heterostructures (BGHs) are determined using molecular dynamics simulations with a Tersoff potential force field. This is achieved by imposing the bending deformation to the pristine BN sheet (BNS) and BGHs. Three shapes of graphene domains are considered: triangular, trapezoidal and circular. Overall polarization of BGHs was enhanced when the graphene domain was surrounded by more N atoms than B atoms. This enhancement is attributed to higher dipole moments due to the C–N interface compared to the C–B interface. The flexoelectric response for BGHs with 5.6% of triangular and trapezoidal graphene domains was enhanced by 15.2% and 7.83%, respectively, and reduced by 25% for the circular graphene domain. We also studied the bending stiffness of pristine BNS and BGHs using the continuum-mechanics approach. Our results also reveal that the bending stiffness of BGHs increases compared to the pristine BNS. Moreover, the enhancement in the flexoelectric coefficient and bending stiffness was more significant when the graphene domain breaks the symmetry of BGHs. Our fundamental study highlights the possibility of using BGHs in nanoelectromechanical systems (NEMS) such as actuators, sensors and resonators.

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Abbreviations

E:

Total energy

\(E_{i}\) :

Site energy

\(V_{{ij}}\) :

Bond energy

\(r_{{ij}}\) :

Distance between atoms i and j

\(b_{{ij}}\) :

Bond angle

\(f_{R}\) :

Repulsive potential

\(f_{A}\) :

Attractive pair potentials

\(f_{C}\) :

Cut-off function

R and D:

Model specific parameters

R:

Radius of curvature

K:

Strain gradient/bending curvature

p and q:

Curve fitting parameters

L x :

Length along the direction of x

\(S_{z}\) :

Bending displacement

\(P_{x}\) and \(P_{z}\) :

Polarizations along the x- and z-direction

\({\text{q}}_{{\text{i}}}\) :

Ion charge of the ith atom

\(r_{i}\) :

x-coordinate of the ith atom

N :

Number of atoms

A :

Surface area

\(D\) :

Bending stiffness

ΔE:

Bending potential energy

\(\theta _{{ijk}}\) :

Angle between bonds

\(\theta\) :

Angle between two vectors

\(\varepsilon _{{xz}}\) :

Strain in the x-direction due to deformation in the z-direction

\(e_{{ijk}}\) :

Piezoelectric coefficients

\(\mu _{{ijkl}}\) :

Flexoelectric coefficients

BN:

Boron nitride

BNS:

Boron nitride nanosheet

BGH:

Boron nitride-graphene heterostructure

BCN:

Carbon-doped boron nitride nanosheet

MD:

Molecular dynamics

DFT:

Density functional theory

CVD:

Chemical vapor deposition

EBR:

Electron beam irradiation

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Acknowledgements

The work was fully supported by the Science Engineering Research Board (SERB), Department of Science and Technology, Government of India. S.I.K. acknowledges the support of the SERB Early Career Research Award Grant (ECR/2017/001863) awarded to him.

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

    S. I. Kundalwal, V. K. Choyal & Vijay Choyal

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Appendix A: MD simulations parameters

Appendix A: MD simulations parameters

The parameters n, \(\beta\), \(\lambda _{{ij}}^{I}\), B, \(\lambda _{{ij}}^{{II}}\), and A are used for two-body interactions. The parameters m, \(\gamma\), \(\lambda _{3}^{m}\), c, d, and \(\text{cos}\,\theta _{0}\) are used for three-body interactions. R and D are adjustable parameters which can be used for both two- and three-body interactions. The value of m = 3,\(~\beta\) = 0, and \(\gamma\) = 1 are taken as constant. The different parameters \(\lambda _{{ij}}^{I}\), \(\lambda _{{ij}}^{{II}}\), \(A_{{ij~}}\), \(B_{{ij~}}\), \(R_{{ij~}}\), and \(S_{{ij~}}\) for species i and j can be calculated using the following mixing rules:

$$ \lambda _{{ij}}^{I} = ~\frac{1}{2}\left( {\lambda _{i}^{I} + \lambda _{j}^{I} } \right), $$
(A1)
$$ \lambda _{{ij}}^{{II}} = ~\frac{1}{2}\left( {\lambda _{i}^{{II}} + \lambda _{j}^{{II}} } \right), $$
(A2)
$$ A_{{ij~}} = ~\left( {A_{i} \times A_{j} } \right)^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}}, $$
(A3)
$$ B_{{ij~}} = ~\left( {B_{i} \times B_{j} } \right)^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}}, $$
(A4)
$$ R_{{ij~}} = ~\left( {R_{i} \times R_{j} } \right)^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}}, $$
(A5)
$$ S_{{ij~}} = ~\left( {S_{i} \times S_{j} } \right)^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}}. $$
(A6)

Refer to Table 3.

Table 3 MD parameters of Tersoff potentials for modeling B-B, N–N, B-N, C–C, C–N and C–B interactions [3, 62, 63]
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Kundalwal, S.I., Choyal, V.K. & Choyal, V. Flexoelectric effect in boron nitride–graphene heterostructures. Acta Mech 232, 3781–3800 (2021). https://doi.org/10.1007/s00707-021-03022-4

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