Inducing a Finite In-Plane Piezoelectricity in Graphene with Low Concentration of Inversion Symmetry-Breaking Defects

Abstract

We show that a finite in-plane piezoelectricity can be induced in graphene by breaking its inversion center with any in-plane defect, in the limit of vanishing defect concentration. We first consider different patterns of BN-doped graphene sheets of D 3h symmetry, whose electronic and piezoelectric (dominated by the electronic rather than nuclear term) properties are characterized at the ab initio level of theory. We then consider other in-plane defects, such as holes of D 3h or C 2v point-symmetry, and confirm that a common limit value (for low defect concentration) of the piezoelectric response of graphene is obtained regardless of the particular chemical or physical nature of the defects (e 11 ≈ 4.5 × 10 −10 C/m and d 11 ≈ 1.5 pm/V for direct and * To whom correspondence should be addressed † Equipe de Chimie Physique, IPREM UMR5254, Université de Pau et des Pays de l'Adour, 64000 Pau, France ‡ Chemistry Department, Faculty of Science, Minia University, Minia 61519, Egypt ¶ Dipartimento di Chimica and Centre of Excellence NIS (Nanostructured Interfaces and Surfaces), Università di Torino, via Giuria 5, IT-10125 Torino (Italy) 1 converse piezoelectricity, respectively). This in-plane piezoelectric response of graphene is one-order of magnitude larger than the out-of-plane previously investigated one.

Piezoelectricity of Functionalized Graphene: A Quantum-Mechanical Rationalization

“…21 Before addressing the out-of-plane response of the present functionalized graphenes, let us briefly discuss their in-plane features and prove once more the general validity of the above-mentioned "universal" behavior. To do so, the piezoelectricity of Si-, Ge-and Sn-doped graphene has been investigated as a function of the substitutional fraction x.…”

“…41 Our previous investigation of in-plane piezoelectricity of graphene documented the little dependence of the computed response on the particular adopted functional. 21 In the linear regime, direct e and converse d piezoelectric tensors describe the polarization induced by strain and the strain induced by an external electric field, respectively; a simple connection exists between the two (e = d C or d = e S) via the elastic tensor C and its inverse S = C −1 .…”

“…Table 1 reports the computed values of e 11 (total and purely vibrational contributions) as a function of defect concentration x for the three different chemical functionalizations. By looking at the total values of the direct piezoelectric constant, it is seen that for small dopant concentrations the common asymptotic value of approximately 5×10 −10 C/m is reached for all systems, 21 Let us just briefly recall that this is due to the fact that the band gap of the system varies with defect concentration according to two distinct behaviors depending on whether the considered superlattice is or not a multiple of 3 of the primitive cell of pristine graphene, as predicted by the energy band-folding model. [52][53][54] As a further consideration, we might notice that the vibrational contribution is always smaller in absolute value than the electronic one, which is found to dominate the in-plane piezoelectric response of functionalized graphene.…”

“…However, its inversion center can be broken and piezoelectricity engineered by several means including adsorption, hole creation, application of biaxial strain, chemical doping, etc. [16][17][18][19][20][21] A large out-of-plane piezoelectric response has recently been measured for a graphene single layer as deposited on a SiO 2 substrate. 20 Among other strategies to induce a piezoelectric response in graphene, chemical doping seems the most promising as it already represents an effective experimental mean for tuning its structural and electronic properties.…”

“…31 In a recent study, we have systematically investigated the in-plane piezoelectric response of graphene as induced by several inversion symmetry-breaking defects and found a peculiar "universal" behavior: a common finite in-plane piezoelectric response (characterized by a direct piezoelectric coefficient e 11 of about 5×10 −10 C/m) in the limit of vanishing defect concentration, thus highlighting an intrinsic nature of the piezoelectric activity of graphene. 21 The present investigation aims at providing a complete quantum-mechanical rationalization of the overall (in-plane and out-of-plane) piezoelectric effect as induced in graphene by any inversion symmetry-breaking defect. While confirming the "universal" in-plane behavior, the atomistic mechanisms behind a possible giant out-of-plane piezoelectricity are here addressed and under-stood.…”