Abstract

To determine whether piezoelectric properties can be engineered selectively into graphene by doping one side of the two-dimensional sheet with compatible adatoms. Density Functional Theory (DFT) calculations compare piezoelectric stress (d31) and strain (e31) coefficients for several adatom combinations to other piezoelectric materials, indicating possible applications in the dynamic control of motion and deformation within nanoelectromechanical systems and structures.

Introduction

The transition from the use of microelectromechanical systems (MEMS) to that of nanoelectromechanical systems (NEMS) poses many problems, notably that many of the methods employed to engineer properties such as transduction and strain response into MEMS begin to fail when carried into the nanoscale.1,2 One of the greatest challenges faced with implementing NEMS in consumer technology is generating the ability to exert dynamic control over the motion and deformation of nano-structures. Piezoelectric materials are already featured prominently as strain sensors for vibration detectors, electromechanical transducers, and probes for atomic force microscopes to achieve this kind of control.1,3 However, no promising analogue among known materials exists to afford this at the nanoscale.

Piezoelectricity is a remarkable property of some materials in which mechanical deformation results in the production of an electric field.4 The reverse piezoelectric effect is also true, and exposure to an electric field will cause a change in the dimensions of the piezoelectric material. In the modeling of the piezoelectric behavior of a material, d31 and e31 are among several of the coefficients used. The d31 coefficient relate in-plane strain to the electric field and the electrical polarization perpendicular to the plane, while e31 describes the magnitude of the piezoelectric effect displayed in the material for a small deformation or applied electric field.5,6

The ability to produce piezoelectric nanomaterials efficient at creating desired functionalities in a nano-device is an attractive prospect. To this end, it is thought possible to make nano-scale materials piezoelectric, even if they do not intrinsically possess this quality, by introducing nanoscale inhomogeneities into the material. In such systems, provided that they are non-centrosymmetric (i.e. they lack inversion symmetry), even uniform stress would induce polarization of the material which would indicate piezoelectric behaviour.2

Theoretical demonstrations have calculated that by inducing strain in piezoelectric materials it is possible to bring about a reversal in the magnetic fields of other materials.7 This effect is studied in a new field known as straintronics. In applications such as computation and signal processing, ambient energy alone is enough to generate this level of strain, resulting in devices requiring ultralow energy inputs to function as well as a minimization of energy dissipation. This effect has been computationally displayed in lead-zirconate-titanate (PZT) nanomagnets shown to have extraordinarily low energy dependence.8 The novelty of nanoscale films with these properties would allow for versatile and energy efficient control of nanodevices. Specific control over the dimensionality of thin films allows for precise control of the magnitude of their resultant properties. This is well understood for many materials through the transition from their bulk to their nanoscale analogs.

A two-dimensional sheet would constitute the idealized thin film, but it was initially believed that two-dimensional materials would be thermodynamically unstable and could not exist. This was based primarily on the observation that the melting temperature of thin film materials decreases rapidly with decreased thickness.9 The surprising discovery of graphene provided a material stronger than any other, that was still lightweight and flexible.10 Graphene exists as a monolayer of sp2 hybridized carbon atoms one atomic layer thick, making it a two-dimensional material.9 Graphene is traditionally isolated from graphite by a mechanical exfoliation process, but has also been formed using chemical vapor deposition and arc discharge techniques, which allow for the production of graphene with high electrical conductivities.11,12,13

Because graphene is well within the nanoregime, it shows much promise in aiding the transition from MEMS to NEMS and there has been a concerted effort to uncover the extent of its potential in technology. Chemical doping is one way to tailor graphene’s properties to suit various needs within device applications. Studies demonstrate that doping graphene with potassium changes its molecular symmetry and modifies its electronic properties.14 Understanding how dopants affect graphene could provide piezoelectric nanofilms for NEMS device control. Pertinent to this is the distribution density of adatoms, and the sites they reside in. The specific binding site on the graphene sheet may affect graphene’s symmetry and subsequently its properties, as this is seen in other molecules.15 This may be above a carbon atom (top site), in the center of the hexagon (hollow site) or over a bond joining carbon atoms (bond site) (see Fig. 4A).

Computer modeling techniques such as Density Functional Theory (DFT) are useful in this regard because they enable the use of theory to predict the behavior of matter under different conditions.16 The results of these calculations can then inform of the optimal synthesis routes to pursue in generating the desired reaction or products. DFT allows us to do this through the observation that a given wave function contains much more information about a system than is necessary to describe its behavior. Ground state molecular energy, the wave function, and all other molecular electronic properties are uniquely determined by the electron probability density ρ(x,y,z), which as a three-variable function is less computationally intensive. In principle, given the ground state electron density, the Hohenberg-Kohn theorem propounds that all ground state molecular properties can be calculated from ρ.17

Methods

Density functional theory implemented into the Quantum-ESPRESSO ab initio software package was used to carry out the calculations in this study. Ion cores were treated using Vanderbilt pseudopotentials in all cases except that of potassium, in which a norm-conserving pseudopotential treatment was utilized. A nonlinear core correction was included for potassium and fluorine. Electron exchange and correlation effects were described using the spin-polarized generalized-gradient corrected Perdew-Burke-Ernzerhof (PBE) approximation. All calculations were done using periodic boundary conditions and a primitive cell with one atom for every two carbon atoms except when indicated otherwise (Fig. 1C). The electronic wave function is expanded in a plane wave basis set with an energy cut-off of 60 Ry.

Single atom adsorption on one side gives rise to an asymmetric surface with a net electric dipole moment. A dipole correction was used to cancel out the artificial electrical field that arises from this dipole moment. Focus was placed upon calculating the d31 and e31 piezoelectric coefficients, where d31 is the transverse piezoelectric coefficient that describes the deflection normal to the direction of polarization. The e31 term signifies the intrinsic piezoelectric coefficient, where a large e31 value corresponds to a large electric charge induced at a small cost of mechanical strain, or conversely a large mechanical force generated in the presence of a small electric field.

Cases examined graphene doped with uniform coverages of lithium, potassium, hydrogen, and fluorine atoms. Consideration was given to situations involving two different atom dopants on opposite sides of the graphene sheet, such as fluorine and hydrogen, or fluorine and lithium. Several atom coverage densities were modeled using lithium by placing a single lithium atom in 1 X 1, 2 X 2, 3 X 3, and 4 X 4 graphene periodic supercells. A Löwdin analysis was used to calculate the partial charges on the lithium atoms at each of concentrations. Furthermore, the effect of adatom position on piezoelectric response was examined, in addition to the effect of crystallographic patterning of adatoms on the graphene surface for a fixed concentration of C32Li2.

Results

DFT calculations showed that Li and K preferentially bound to the central hollow sites of the graphene sheets, resulting in hexagonal (6mm) point group symmetry (Fig. 1B). H and F were found however, to bind at the top site which resulted in trigonal (3m) symmetry (Fig. 1B). In the cases involving two atoms, at least one of these atoms must bind to the top site, producing trigonal (3m) symmetry (Fig. 1B).

The 6mm, 32, and 3m point groups all result from the destruction of inversion symmetry, hence these materials will be non-centrosymmetric and display piezoelectric behavior. Point group symmetry enables the determination of those materials having non-zero d31 and e31 coefficients, which are common to all configurations in Fig. 1B. Upon application of a sawtooth potential (ensuring forces acting on the system are asymmetric, allowing the system center to experience force) with a width of 10 Å, an electric field is applied to the material. A roughly linear relationship is found between the electric field and the strain induced in the material when the field amplitude lies between -0.5 and 0.5 V/Å for many graphene-adatom combinations examined.

These behaviours have been experimentally achieved in devices containing graphene.18 The d31 coefficient is equal in magnitude to the gradient of the trend-lines (Fig. 2A & 2B) and Table 1 shows the d31 coefficients extracted from these lines, which display variability within three orders of magnitude. It was found that the binding of F to the graphene sheets resulted in only minor changes to the piezoelectric coefficient. This occurs similarly when H and F bind in an alternating manner and transverse to one another. The alkali metals Li and K however, produced much larger effects on the d31 coefficient. The greatest effect is achieved when F is added to the three top sites, with Li residing in the hollow on the opposite side of the sheet, yielding a d31 value of 3 X 10-1 pm/V. This is comparable to the theoretical value for the 3D piezoelectric boron nitride (BN), which is 3.3 X 10-1 pm/V.

The e31 coefficients were obtained by calculating the change in polarization normal to the surface as a function of the equibiaxial strain in the plane (from Fig. 2B). This gives a linear relationship in all cases between low strain values of -1% to 1%. A consequence of employing the equibiaxial in-plain strain is that the gradient of the trend-line has a value of twice the e31 coefficient for each atom. It was found that both alkali metals Li and K possessed the largest values for e31, indicating that they will deform the most in the presence of a small electric field, and they will generate the greatest charge under a given strain. Unlike the results for the d31 coefficient, e31 does not undergo a significant change when F is placed in the top sites with Li in the opposite hollow site.

To compare values for 2D graphene to those of traditional and well understood 3D materials, the e31 and d31 coefficients were divided by the 3.35 Å interlayer spacing yielding an e31,3D of 0.17 C/m2 and a similar d31 value of 0.19 C/m2. When compared to the e11,3D value of 0.731 C/m2 for two dimensional BN, it is smaller by a factor of 4, but it must be noted that e11 coefficients are generally much larger than those of e31. When the e31,3D is calculated this difference becomes much smaller, specifically as 0.31 C/m2 for wurtzite BN and -0.55 C/m2 for gallium nitride. It is important to note that graphene has the potential to have much larger polarization magnitudes than either of these materials, since graphene can undergo much more elastic strain before plastically deforming. This demonstrates the possibility of engineering piezoelectric graphene comparable to known materials.

It was found that varying Li coverage (Fig. 2C) causes deviation in the relationship between the electric field and the induced strain (Fig. 3A). By plotting the d31 coefficient as a function of Li coverage density, a maximum value is obtained when n = 8, corresponding to the unit cell C8Li. When this density is decreased the d31 value shows a steep decline (Fig. 3A inset). In contrast, the static dipole moment of Li on graphene increases as the Li coverage density decreases, producing a stronger interaction with the electric field (Fig. 3B bottom inset).

However, at low coverage densities this interaction is diminished, and a maximum is found between coverage density and d¬31 as a result of their competitive effects (Fig. 3A inset). The e31 coefficient varies in a similar manner to that of d31, also showing a maximum value for the C8Li unit cell, with decreasing values as coverage density decreases. This shows that the magnitude of the piezoelectric response doped graphene will display can be varied as a function of the adatom coverage density.

Considering only the C32Li system, there is nonlinear piezoelectric behavior in d31 when the electric field is less than -0.1 V/Å. When the field strength becomes less than this, nonlinear behavior is sharply displayed, and the strain exhibits a rapid linear decrease, with a trend-line gradient of 0.19 pm/V. This is because at approximately -0.1 V/Å, Li and graphene begin to undergo a charge transfer process. It should be noted that this charge transfer process could be used to very efficiently power the opening and closing of gates in transistor technologies.

Again considering all tested values of n, for electric field strengths greater than -0.1 V/Å, Li maintains a constant charge, but when this number is lesser, the charge transfer process comes into effect and this value decreases. Furthermore, when varying the height of Li above the graphene sheet as a function of the electric field, there is a minimum value obtained for -0.1 V/Å for the C32Li system, while the other coverage densities still display linear behavior. When the electric field value is larger, varying the distance between Li and the graphene sheet produces no change in charge. When the value is smaller, charge transfer occurs. The implication is that the charge transfer from graphene to Li is theoretically responsible for the large piezoelectric response to fields lesser than -0.1 V/Å.

We also investigated the effects of adatom position on the piezoelectric response of the doped graphene. While K and Li appear to diffuse moderately across the graphene sheet, it is not expected that this will affect the materials piezoelectric behaviour. This is because the doped graphene will be non-centrosymmetric regardless of the adatom location. The d31 coefficient was calculated when Li was placed at the hollow, top, and bond sites of the unit cell and the strain varies upon subjection to an electric field normal to the surface of the adatom positions (Fig. 4A). Again, the gradient of the trend-line gives the value of the d31 coefficient, and these are all found to be within 5% of one another, indicating that the piezoelectric response is independent of the position for the adatoms.

Crystallographic patterning of adatoms on graphene sheets with an adatom coverage density of C32Li has shown that by varying the relative position of the Li atoms in the unit cell, a 20% change is produced in the d31 coefficient, showing closer resemblance to a coverage density of C8Li (Fig. 4B). Despite this, it is unlikely that the pattern of adatoms on the sheet has a significant impact on the piezoelectric response strength of the graphene-adatom system based on the calculations carried out by the authors.

Discussion

Possessing the ability to calculate ground state molecular properties allows for an accurate determination of the behavior of materials under different theoretical conditions and permits the dynamic alteration of variables affecting the system. Widespread difficulty has been associated with producing specific desired behaviors in devices at the nanoscale (such as challenges of self-assembly).19,20 As such, being able to rapidly vary parameters in a given system and observe how the theoretical model responds to those changes is a very powerful tool. Undoubtedly, modeling techniques will play a large role in the creation of next generation devices and materials, allowing us to push the limits of technology beyond the bulk, and to finally enter the realm of nanoscience with devices such as sensors/actuators and transistors, among others.21,22

The authors’ work has done much to further this goal. This study paints graphene in an entirely new light as a potential host to a myriad of applications in nanoscience and nanotechnology previously inaccessible. The novelty of demonstrating for the first time that a material which is not intrinsically piezoelectric can be made so through careful chemical processes broadens our understanding of how to tailor nanomaterials so we can make them work for us. Given this, there are many topics that can be discussed in hindsight of this theoretical study and which the authors may consider.

The ability of graphene to display piezoelectric behavior could be exploited extensively in the fledgling field of straintronics.7,8 In the previously discussed theoretical study, PZT nanomagnets were shown to be viable power sources for low-energy devices. One of the limitations of using this kind of material is its rigid nature and relatively large thickness (relative to graphene sheets) on the order of ~50 nm.8 By partially hydrogenating a graphene sheet, a compound known as graphone, a ferromagnetic and hence magnetostrictive material is obtained.23 A small number of these could be stacked with a layer of piezoelectric graphene surrounding, within, or in between these, resulting in a setup comparable to the PZT nanomagnet studied. An advantage of this type of material stems from the enormous tensile strength and high flexibility of graphene films.24,25,26

Additionally, the excellent ability of graphene to conduct thermal energy only increases its appeal for use in molecular electronics.27 Nanodevices are currently expensive to manufacture, and their oftentimes integrated nature and small sizes impossible to manually fix necessitate that if a defect occurs, the device as a whole would need to replaced. With increasing power densities as devices are scaled down, thermal effects on devices must be given increasing importance. The high thermal conductance displayed by graphene may play a key role in maintaining the functionality of nanodevices under thermal stress by facilitating heat flow away from the delicate electronics within.28 Thermal expansion necessarily resulting from such applications as thermal conductance produces strain in the graphene that could further assist in power generation in straintronics applications.

Graphene has also found uses in supercapacitor technology as electrodes. A graphene nanofiber composite with conducting polymer polyaniline has shown capacitance values as high as 480 F/g at a current density of 0.1 A/g. This nanocomposite also shows good cycling stability during the charge-discharge process.29 Traditional piezoelectric materials display problems such as brittleness (as with PZT), among others.30 Graphene based piezoelectric devices would not have issues of brittleness as a result of their high flexibility. Furthermore, piezoelectric graphene would allow for efficient energy harvesting for its use in supercapacitors. Such technologies discussed here allude to the integration of graphene as various components of the same devices based on its treatment and subsequent properties. This theme may represent a future where carbon based materials completely replace silicon in fields such as molecular electronics. Graphene has the potential to revolutionize the development of all nanodevices as a result of its ability to take on so many chemical and physical roles as a material because of its chemical versatility and its exceptional mechanical and electrical properties.

Despite all of this, graphene is not centrosymmetric and as such intrinsically is not a piezoelectric material. If the calculations in this work lead to synthetic realization of the properties modeled then it will require processing to make it such. Since BN and GaN, among others, are well established piezoelectric materials that are clearly capable of being produced as nanotubes or monolayer sheets, why should graphene be a more attractive alternative?31,32,33 This appeal is outlined by the authors and it is exactly because graphene is not intrinsically piezoelectric that it holds such potential. Since piezoelectricity arises from adatom doping, graphene could be selectively doped to produce areas or patterns that are piezoelectric, and ones that are not. Furthermore, graphene has a zero band gap, but this can easily be varied by adding dopants to the sheet. Choosing BN nanosheets necessitates that the device has a band gap of 4.60 eV, and for GaN sheets of 3.4 eV, without significant evidence for altering these values appreciably.31,33 Graphene therefore possesses much more versatility in its applications, since it can be altered in various manners to serve a multitude of different tasks in the same device.

In future studies towards this goal, it would be valuable to examine this response at higher strain. The authors model the piezoelectric effect between -1% and 1% strain, noting that a rough conversion indicates graphene may be comparable to BN and GaN. They also mention that it may be able to surpass these materials since it is capable of enduring higher strain before failing. It would be interesting to probe the extent to which graphene can be deformed as a piezoelectric material and how this affects the magnitude of the piezoelectric response.

However, the authors’ model may oversimplify the assumptions leading to the proposed unit cells. They model every carbon as associated with an adatom for top site binding, or every hollow site as associated with an adatom for hollow site binding. Hydrogen is one of the adatoms calculated where all of the top sites are occupied on the graphene sheet. Contrary to this, there have been experimental studies that have demonstrated that at high hydrogen concentrations there is a problem with the clustering of hydrogen adatoms rather than their maintained homogeneous dispersion (Fig. 5).34 It has also been demonstrated that the binding energy for fluorine with this pattern is extremely low compared to more dispersed patterning and that if this were to occur it is likely to be a highly unstable compound.35 While this may be the case for low dosages of adatoms, it may be necessary to synthesize such piezoelectric graphene to say with any great conviction how adatoms will behave at high concentrations on the sheets surface.

The authors’ work has opened up many avenues of pursuit in efforts to shift to carbon based electronics. Of particular note, in the event of its synthesis, are the applications for straintronic devices and similar molecular electronics. There are undoubtedly a host of other applications in which piezoelectric graphene could serve as an invaluable material. Research in graphene will almost certainly provide the materials necessary to transition from current technologies to smaller, more efficient, and less energy intensive ones. Graphene possesses an apparent wealth of properties through manipulation of its structure and interactions, its insurmountable thinness, and its extraordinary strength. These are just some of the properties of graphene known to us that suggest it will play a major role in the implementation of nanotechnologies as the way of the future.

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