Research Article, J Nanomater Mol Nanotechnol Vol: 3 Issue: 4
Field Emission Simulations of Carbon Nanotubes and Graphene with an Atomic Model
Abstract
Field Emission Simulations of Carbon Nanotubes and Graphene with an Atomic Model
Most models of field emission from carbon nanotubes (CNTs) and graphene sheets consider the geometry of the emitter tip as a smooth spherical surface. In these models, indicated with the terms simplified geometry (SG), the electronic distribution is not taken into account. However, the electronic distribution is concentrated around the atoms nuclei and their bonds forming hexagonal patterns that affect the field enhancement and largely determines the field emission. In this paper we evaluate the field emission in a SGmodel and compare it with the emission in a ballstick model that better represents the atomic structure at the surface of the emitter. We found that the emission current from open and capped carbon nanotubes and for graphene sheets is typically 10 times larger in the ballstick geometry than in the SG. Furthermore, different morphologies of the CNT highlight the sensitivity of the emission current.
Keywords: Field emission simulation; Graphene emission; Carbon nanotubes emission; Hemisphere post emission; Ballstick emission
Keywords 

Field emission simulation; Graphene emission; Carbon nanotubes emission; Hemisphere post emission; Ballstick emission  
Introduction 

Few years after their discovery carbon nanotubes (CNTs) [1] were recognized as good electron emitters [2] and triggered abundant research on the application of CNTs in field emission (FE) devices. Since then, the most popular geometrical representation of capped CNTs for evaluating their field emission has been a hemisphereon apost [311] as in Figure 1a. For open single wall CNTs a toroidal edge is generally used [12,13] as shown in Figure 1b, while graphene sheets are represented as a wall with a cylindrical top [14] as represented in Figure 1c. Slightly different geometries such as pill capsules [15], floating spheres [16] and ellipsoids caps [17] have also been used to represent the CNT surface. Although these simplified geometries (SG) can give good insights in the qualitative behavior of the emission, the accuracy of the current obtained was never tested in geometries that have wrinkled surfaces that come into sight at atomic scales, indicated in Figures 1d1f. Thus, it may be expected that the FE capability in the range of atomic dimensions is better represented by ballstick geometry (BS), because field enhancement is very sensitive to the surface morphology [18] also at molecular dimensions. Field Emission Microscopy images demonstrate that electron emission occurs preferentially at the atoms and their bonds [1922]. However, modeling field emission from CNTs using BS is computationally difficult both memory and time wise; therefore, SG are preferred in most FE studies due to practical reasons. Yet, we can compare the FE from BS and SG in a few particular cases. Our objective is to evaluate the difference between FE calculated with SGmodels and corresponding BSmodels.  
Figure 1: CNT and graphene structures comparing SG with BS geometries. (a), (b) and (c) are SG for a capped CNT, an open CNT and a standing graphene sheet respectively, (d), (e) and (f) are the corresponding BS representations of the same structures. Few nanometers below the tip the field emission is negligible, therefore the balls and sticks need not to be modeled all the way down to the base of the emitters.  
Simulation Procedure 

We simulate the emission current with Comsol Multiphysics^{®} version 4.3b based on finite elements methods. Figure 2 shows a typical simulation domain CAD (computer aided design), where the bottom boundary and the emitting structure are grounded (V=0). This boundary condition implies that the structures are perfect conductors. Hence, the conductive nature of the structure (metallic or semiconductor) is not accounted for. Nevertheless, our procedure normalizes the effect of the CNT resistance as we will exemplify in the section Limitations of the BS model. The top boundary has a constant surface charge density σ =ε_{0}E_{appl}, where E_{appl} is a uniform vertically aligned electric field applied in the domain and ε_{0} is the dielectric permittivity of vacuum. The side boundaries are symmetry boundaries, which impose the electric field to be parallel to the boundaries (E.n=0). Symmetry boundaries work as mirrors, so the solution in the domain is equivalent to an infinite array. The simulator computes the field distribution in the domain.  
Figure 2: Typical simulation domain in a diode configuration: the side boundaries are symmetry planes that simulate an array of emitters; the bottom and the emitting structure are grounded; the top boundary has a surface charge density that generates a specified macroscopic electric field inside.  
The local current density is described by the FowlerNordheim equation [12,23]:  
(1)  
where A=1.54×10^{−6}A eV m^{2} V^{2}, B=6.83×10^{9} V m^{1} eV^{3/2}, φ is the work function (in eV) and E is the calculated local electric field (in V/m) at the surface of the emitter. The simulator integrates equation (1) over the emitter surface giving the total current. In all simulations we set φ =5eV. Let us first consider the validity of this assumption.  
The work function is a macroscopic parameter that works adequately in the SG model of CNTs. The uniformity of φ is justified because the electrons are delocalized in the πelectron molecular orbital covering a large part of the CNT. The Catoms have a sp^{2} hybridization, leading to an extended πmolecular orbital. In the atomic BS model, one may consider two types of BS Catoms 1) that are included in a regular Chexagon with sp^{2}hybridization and 2) that are in an incomplete hexagon and may be considered as dangling Catoms. The first type atoms are included in the πelectron system of the CNT or graphene sheet and the assumption of a uniform work function seems obvious. When the BS Catoms are not included in a regular hexagon as shown in Figures 1e and 1f, they may be bonded to hydrogen or oxygen and it is likely that the local work function in that case is larger than 5eV, taken here as default value for the πelectron system of CNTs and graphene sheets. The work function has been extensively studied theoretical and experimentally. Reported values are between 2 and 8eV, although most authors find values around 5eV. Su et al. made a review and a theoretical analysis using density functional calculations and determined φ of SWCNTs for many chiralities between (3,1) to (50,50) [24]. They found mostly 4.5eV, with some results reaching 6eV in small radii CNTs. In this work, the precise φ has minor importance, because the current ratio between the SG and the BS models is insensitive to φ, despite the emission current being sensitive to φ. In the section, Limitations of the BS model, we shall give a numerical example.  
In the BS geometry the radius and thickness of the emitters are constrained by the actual sizes of the carbon atoms, bond lengths and chirality. So, to compare the BS with the corresponding SG, both representations are built with same radius and thickness. The height is fixed at 5nm in all simulations. The radius of the carbon atom is 0.07nm and its sp^{2} bond length is 0.142nm.The bonds are important to be considered because they screen the atoms electrostatically hampering the emission as will be shown. The atoms coordinates and the CNTs diameters are obtained from the software Nanotube Modeler^{®} [25]. E_{appl} varies among the simulations in such a way to keep the total current I_{SG} constant at 10.00±0.01nA in the SG, whereas the current I_{BS} in the BS geometry, using the same E_{appl}, is the desired result.  
To obtain the current equation (1) must be integrated over the surface of the emitter. However, the surface of a molecule is not well defined. What is usually represented as the molecular boundary is the isodensity surface with the highest probability to find the outer electrons. We are aware that the electronic distribution extends farther from these surfaces and even depends on the electric field [26,27]. The full electronic distribution can be calculated using quantum mechanics approach [2331], but the complexity of evaluating FE with such a sophisticated model is impractical and is exactly what the SG tries to circumvent.  
Results 

FE from open CNTs  
Table 1 shows the analysis for open CNTs with chiralities: (6,6), (7,5), (8,4) and (10,1). These CNTs have almost the same diameter (~8.2 nm) as indicated in the third line, thus, any variation in the emission is due to their local atomic morphology. The pictures in the first line show the different morphologies and the field strength distribution on the tip of the CNTs, where the reddish regions^{1} indicate high field strength. Other chiralities, including the zigzag, were excluded in this analysis because their diameter differs too much from the average of 8.2 nm. The chiralities chosen highlight the effect of the atomic morphology, since radius, aspect ratio and spacing are the same in all four CNTs. Note that few nanometers below the tip the field strength is negligible, so the atoms no longer need to be represented and the rest of the CNT can be reduced to a cylinder. The corresponding SG geometry consists of a hollow tube with a toroidal edge, as in Figure 1b. By comparison, the toroidal edge geometry yields the same current 10nA in all cases, since the toroidal geometry cannot take into account the ridged shape in the edge that varies with the chirality. On the other hand, the BS model shows that the different tip morphologies cause the emission to vary considerably (~60 to ~110nA) and being roughly 8 times larger in average (5th line). The highest current is obtained for the SWNT with chirality (8,4), which has four isolated protruding C atoms at the top. The SWNT with chirality (10,1) has three protruding C atoms, but since these are neighbors, they shield each other better. The dangling bonds in the case of the armchair SWNT (6,6) obviously do not contribute particularly to the field emission current enhancement. We did not simulate the effect of a (10,0) zigzag CNT, because its radius is too small (0.783 nm) for a fair comparison.  
Table 1: Emission current from open cap emitters: Open CNTs with almost same diameter have different emitting current depending on the chirality. The first line is a snapshot of open CNTs with different chiralities presenting different tip morphologies; the red regions indicate high electric field strength. The fourth line is the macroscopic field necessary to generate 10nA in the corresponding simplified geometry with a toroidal edge. The last line is the emitting current simulated in these BS models.  
It is tempting to compare the results presented in table 1 with the conductive nature of the SWNTs. According to Saito et al. SWNTs with chirality (6,6) and (10,1) have metallic conductivity, whereas (7,5) and (8,4) are semiconducting [32]. However, in our simulations the CNTs are perfect conductors and the current variations are solely due to geometrical characteristics. The field enhancement and screening effect are responsible for the chirality emission dependency. In section 6 we shall discuss the conductivity effect.  
FE from standing graphene  
We simulated an infinitely long standing sheet with a chirality range varying from (8,0) to (8,8). Figure 3 illustrates the field distribution in graphenes with chirality (8,1), (8,3) and (8,5). Table 2 lists the current from all nine possible configurations. In this case we normalize the current of the SG to 1nA/nm. The results are similar to the open CNTs; the BS geometry yields a current that is, in average, 8 times larger than the corresponding SG shown in Figure 1c and the variation with chirality is roughly ± 25% of the average.  
Figure 3: Electric field distribution on the surface of the emitting structures for standing graphene sheets with different edge morphologies due to their chiralities.  
Table 2: Current as a function of the chirality: The simulation accounts for an array of infinitely long standing graphene sheets 50nm high separated by 50 nm.  
FE from flat graphene  
To evaluate the FE from the vicinity of the atoms in a capped CNT, we shall make use of the simulated emission from a flat graphene sheet. The emission current from a flat graphene sheet is considerably higher than that from a flat surface as shown in Table 3. In this table, the first line shows the top view of the unit cell adopted. As the side boundaries works as mirrors, the equivalent array replicates the flat graphene as shown in Figure 4. We vary the diameter of the bond between the carbon atoms, which results in less emission with increasing radius. This shows that the choice of unknown geometric parameters can make significant difference in the simulated current. For example, disregarding the bonds in the simulations of the open CNTs shown in Table 1, the emission current is higher, varying from 200nA to 300nA.  
Figure 4: Full graphene sheet generated by the symmetry boundaries of the triangular unit cell used to calculate the emission current.  
Table 3: Unit cells in a flat graphene: Simulation of the current with the bond radii.  
Although the detailed shape of the molecule is unknown, we can assume the diameter of the bonds to be close to the diameter of the atom as seen in the quantum mechanics models [2330]. Thus, we chose the radii of the bonds as 0.065nm in all simulations. In Table 3 the average emission from bonds between 0.06 and 0.07nm is 36.5nA, which is 2.2 times smaller than the emission of a sphere. In next section we will use this factor to estimate the emission in capped CNT where the simulator could not handle the bonds.  
FE from capped CNTs  
Table 4 shows results from capped CNTs. The cap is built from half C20, C60, C180 or C540 fullerenes respectively. In these simulations the cylinders that represent the bonds caused computational difficulties. To evaluate the effect of the bonds we just divided the results by the factor 2.2 obtained in the previous section.  
Table 4: Emission current from capped CNTs: The pictures in the first line show the half fullerene molecule used as a cap. The bonds are not represented in these cases, but results with bonds can be estimated as shown in the last line. The larger the CNT radius the more the emission departs from the hemisphereonapost as seen in the 5^{th} line.  
Note from the pictures in the first line that the surface of the cap looks more spherical as the radius increases, but IBS does not tend to the emission of a hemisphere (10nA). On the contrary, the more the radius increases the more the current departs from the hemisphereon apost. This counterintuitive behavior is due to the additional field enhancement near the atoms that form the cap. As the radius increases, E_{appl} (4^{th} line in Table 4) must increase to keep 10nA in the hemisphereonapost, because of the lower field enhancement. However, in the BS model the atoms are the ultimate emitters and increasing E_{appl} in the corresponding BS will largely increase the emission from the atoms. In capped CNTs the BS emission diverges continuously from the SG emission as the thickness of the CNT increases. Open CNTs and Graphene, on the other hand, have emissions which depart around tenfold between SG and BS models and do not scale with the CNT radius or the graphene length.  
The 6^{th} line in Table 4 contains the estimated values in case we had the bonds with radius 0.065nm. This estimate was obtained dividing the values of the 5^{th} line by the factor 2.2, as we just discussed. For the case of the CNT capped with C20 (6^{th} line, 1^{st} column) we managed to run a simulation with bonds. Then, we could check that the simulated emission is just 10% larger than the estimated emission.  
Limitations of the BS model  
Field emission is influenced by many parameters. Any experimental or theoretical analysis can cover only a small range of all possible cases. In this work we assumed φ, I_{SG} and aspect ratio as fixed. Also, our analyses are for short, perfectly conducting single walled CNTs and no quantum mechanics approach; so we could focus on determining the ratio I_{BS}/I_{SG}. Despite the limits in our analyses, we can illustrate the variation of φ and the electrical resistance effects on I_{BS}/I_{SG} in the open cap (7,5) CNT as follows.  
Variation of φ in I_{BS}/I_{SG}  
The φ is seldom reported to be out of the range from 4.5 to 5.5eV. Simulating φ = 4.5, 5 and 5.5eV yields I_{SG}=10nA at E_{appl}=1.029, 1.173 and 1.330V/nm respectively. The ratios I_{BS}/I_{SG} are 7.66, 7.87 and 8.45 respectively. This analysis points out that φ has little influence on the current ratio (<10%); on the other hand, experimental data can be interpreted as having significant lower work function if SG is used instead of BS. The implications of interpreting experimental data with SGs and BS models require extended analyses and will be presented in a forthcoming article.  
Effect of the conductivity  
Assuming a resistivity of 10^{−3}Ωm, which is a typical resistivity of a semiconductor, the I_{SG}=10nA is obtained with 1.4083V/nm. In this case the ratio I_{BS}/I_{SG}=25.8, which is three times larger than that for a perfectly conducting SWNT as shown in Table 1. To keep 10nA in the SG the applied field must be higher due to the voltage drop along the CNT. Because of the rapidly increasing behavior of equation.(1), E_{appl} overcompensates the voltage drop in the BS model. The finite conductivity of the CNTs increases de discrepancy between SG and BS even more.  
Conclusion 

In this paper we have described an atomic simulation model for the field emission from CNTs and graphene sheets. Atoms and bonds are considered as balls and sticks. The results of our simulations show that this atomic model yields 515 larger emission currents as compared to the standard hemisphereonapost model at the same conditions. These conditions are essentially the same aspect ratio and radius that yields 515 more current in the atomic model. However, if instead, an IE experimental curve is given while the aspect ratio and the radius are unknown, then, fitting the curve considering the CNT as a simplified geometry will result in higher aspect ratio and larger radius compared to the same fitting using an atomic BS model. The aspect ratio and radius must be larger in the simplified geometries to provide larger field enhancement factor and larger effective area of emission in order to be at par with the BS model.  
In the BS model for open CNTs, the emission can vary by ~300% among the various possibilities for the chirality for a given radius, aspect ratio and spacing. In capped CNTs the BS current can be larger than the emission calculated with the SGmodel: the thicker the capped CNT, the larger will be the ratio I_{BS}/I_{SG}. The ratio I_{BS}/I_{SG} is increasing with the resistivity of the CNT.  
We think that the atomistic model will enable more detailed and realistic fittings to experimental results in the future.  
Acknowledgement 

This work was supported by the National Counsel of Scientific and Technologic Development (CNPq) of Brazil, grant number 382634/20144.  
^{1}Read the online version of this article showing colors.  
References 

