Abstract—Carbon Nanotubes (CNT) and graphene as a whole have been of interest since their discovery due to their unique properties. These include, but are not limited to: high electron mobility in room temperatures, relatively high thermal conductivity and possibly the highest resistance to mechanical stress. Hence, it is important to properly model structures and systems that might use graphene in future. This paper presents the comparison of several simulation methods in different programs. Special attention has been paid to calculating Radar Cross Section (RCS) of modelled structures.
Index Terms—Graphene, CNT, computer modelling, simulation.
I. INTRODUCTION
When it comes to modelling graphene or any variation of it (ribbons, single- and multiwalled nanotubes) the basic problem is that electromagnetic (EM) solvers do not possess any data regarding graphene. Programs like CST Studio Suite, HFSS, FEKO, IE3D may be accurate and effective solvers when it comes to simulating antennas and/or electrically large objects, however they do not provide user with data about graphene in their libraries. On the other hand, programs like Atomistix Toolkit may accurately simulate current flows in nanoscale, but are generally not fit to solve EM problems or to calculate mono-static RCS. In this paper structures with graphene were simulated in two EM field solvers: CST Studio Suite and FEKO. This approach is correct, provided reader wants to focus more on macroscale effects associated with electromagnetic fields rather than on micro- or nanoscale effects that are linked to quantum mechanics.
Mathematical model used in this scenario also neglects quantum mechanics effects between nanotubes. One of goals of this work was to analyse Radar Cross Section (and/or possibly Absorption Cross Section) dependancy on presence (or lack thereof) of graphene in modelled structures. The general idea was to use a single plane wave at either one designated location or to illuminate structure at many angles. The main goal of latter approach was to determine whether any angles of transparency are present, as they would significantly reduce performance of structure. Frequency used ranges from hundreds of megahertzes (MHz) to several gigahertzes (GHz). The main reason for that is limited computional power available for simulations as they were run on laptop and a
single personal computer (PC). The higher the frequency is, the smaller mesh size gets and thus more time is required to perform simulations.
II. SUMMARY OF USED EM SOLVERS
While both field solvers share many traits, like e. g. modelling can be done with extensive use of parameters, allowing to dynamically change geometry or being able to simulate finite arrays of elements, they have a number of differences that may cause readers to lean to one over the other. If we consider accuracy of mathematical models used in these environments, we immediately notice that CST Studio Suite does not allow electrical conductivity to be a function of frequency, while in FEKO it's possible to use different conductivity values for certain frequencies. On the other hand, CST allows loss tangent's values to be defined in a frequency band. Both programs, though, will not allow reader to enter negative values of either permittivity or permeability (or conductivity for that matter) which limits its possibilities to simulate metamaterials.
When it comes to methods of simulation, these two environments use roughly the same set of numerical methods. These include but are not limited to: Method of moments (MoM), Multilevel fast multipole method (MLFMM) and Finite elements method (FEM). The main difference is FEKO allows user to manually select (or force) a method for simulation, while CST uses methods associated with selected solver. It is not directly stated that they are used but one can with relative ease come to conclusion which method was performed. Additionally, there is a number of differences related to modelling itself. Firstly, default mesh in CST is a tetrahedral mesh (in certain setups it's hexahedral, but it's not important from the perspective of sheer difference). Mesh used by default in FEKO is a triangle mesh with three levels of detail (fine, standard, coarse). Only by specifying e. g. FDTD method can user force a tetrahedral mesh. All in all, one cannot state that one program is strictly better than the other, simply because they can be used in conjuction for different tasks. One example might include comparing 'flat' triangle
mesh size of FEKO with default 'volume' mesh size of CST. There are many more differences, which are not going to be covered by this paper, such as boundary conditions, signal waveforms or abilities to represent conducting ground plane. To sum it up, comparison given in this paper will be performed in simulation methods department in order to possibly confirm that results are identical for the same methods.
III. PROPERTIES OF GRAPHENE EDGES
There are two types of graphene edges, as shown in Figure1. Vertexes represent carbon atoms and lines between them are covalent carbon-carbon bonds. For zigzag type edge, each carbon atom on the edge has an unpaired electron, which causes the entire edge to be more active chemically. On the other hand, covalent bonds on the edges of armchair type allow it to be more stable in a chemical meaning. Nevertheless, both compositions are knows as “closed-edge” configurations, as their setup is entirely determined. When edges are defective, whether it is intentional or not, they are refered to as “open-edge” configuration.
There is a lot of controversy when it comes to properties of graphene edges. Many papers [3], [4], [5] present mutually exclusive results and as such it is rather hard to determine which approach is correct in terms of prolonged use. Differences may often result from various boundary conditions, external fields or even inadvertent user influence. Reference [1] states that graphene nanoribbons (GNR) that possess armchair edges are likely to be semiconducting or entirely metallic in nature. As one may suspect, electrical attributes of graphene nanoribbons are connected to their edge configuration. What is more, electrical conductivity depends on the amount of sp3-like bonds and passivation of the carbon bonds through chemical functionalization. On the other hand, zigzag GNR(ZGNR) are metallic regardless of their widths based on tight-binding approach. It is also noted that nanoribbons with zigzag edges can be transformed into semiconducting form by controlling defective atoms and their distance to edges. Authors indicate that doping edges of ZGNR with boron or nitrogen affects the conductivity due to various effects connected to localized edge states. This, however, is not a reliable method to tune electrical properties as there is currently no method that allows to consistently introduce single dopant atoms(heteroatoms) into carbon rings. It is also worth noting that oxidation of edges will result in changes in stability and electronic properties. Graphene
oxide's (GO) band gap creates a barrier at graphene/GO junctions.
IV. MATHEMATICAL MODEL OF GRAPHENE NANOTUBES
Reference [2] was the main paper which supported the idea of using Boltzmann's equation to model electrical conductivity of graphene nanotubes. Furthermore, Figure 2 shows results of calculating electrical conductivity of carbon nanotubes. Index m is strictly related to radius of these nanotubes by equation:
a=3bm2π
(1)
where : a – radius of a nanotube
b = 0.142 nm – distance between atoms in graphene
m – index related to cylinder axis in armchair nanotubes [2]
Equation (1) represents a simplified case, when armchair nanotubes are considered. Figure 3 shows a typical graphene sheet which can be rolled via any axis. Position vector is a linear sum of basis vectors multiplied by m and n indexes. Armchair nanotubes are acquired through rolling via axis η, in which case n = m and equation (1) is directly derived from equation:
a=√3b2π
√m2+nm+n2 (2)
Rolling via axis ξ allows one to obtain zigzag nanotubes, which are characterized by (m, 0) pair of indexes. If a nanotube is created by means of rolling via any axis that is not ξ or η, such nanotube is called a chiral nanotube and has a default (m. n) pair of indexes. Armchair nanotube are an area of interest in this paper, due to their relative chemical stability, which is a result of them lacking any unpaired electrons on their edges. While zigzag edges may exhibit superior electrical
Figure 1: Two types of graphene edges: a) zigzag edge, b) armchair edge [1]
Figure 2: Electrical conductivity of carbon nanotubes [2]
properties, due to possibilites of introducing dopants like nitrogen and boron, they are not going to be covered. Unpaired electrons which exist on their edges for every carbon ring are also a source of defects and chemical unstability. [2]
As stated before, Boltzmann's equation is the initial stage of deriving formula for electrical conductivity. This equation is as follows [2]:
∂ f∂ t +cE z
∂ f∂ pz
+v z∂ f∂ z =ν[ f 0( p)− f ( p , z , t)] (3)
It should be stated that this form expresses the case, when Ez is directed onto z axis and no other electric field components are present. In equation (3) we have:
f – electron distribution function,vz – electron velocity,e – electron charge,ν – relaxation frequency, which is equal to τ-1, given τ is
relaxation time,p – two-dimensional electron momentum,f0 – Fermi-Dirac distribution.
Before presenting the final formula for electrical conductivity, one should be aware that this model neglects quantum mechanics effects, like tunneling effects, and describes electron transition only within one band. Such movement is called intraband transition, while electron movement within two or more bands is called interband transition. In this paper, however, intraband transition are considered, due to their abundance and influence on electrical conductivity of graphene nanotubes with small diameters.
In general, Boltzmann's equation can be applied to any structure with infinitely small thickness. To narrow down our case, we are going to assume that Fermi energy EF = 0 for armchair nanotubes. Furthermore, it is imperative that we assume energy dependance on two-dimensional (in our case
only z-component shall be considered, two-dimensional case is a general statement) momentum given by equation [2] :
E ( pz , s)=±γ0√1+4cos(ξbz )cos(ξs)+4cos2
(ξb
z ) (4)
where we have
γ0 = 2,7 – 3 eV, constant,ξb
z = √3 πbpz/h ,ξs = πs/m, while s = 1, 2, …, m and represents quantitized
circumferential electron momentum.
After aforementioned and some other derivations, one can obtain a formula for computing electrical conductivity for armchair nanotubes with small diameter (m < 50). This formula is given by equation [2] :
σ zz≃− j4 e2 vF
πh a(ω− j ν) (5)
Given the fact that EM field solvers may not support frequency dependant electrical conductivity, approximtions had to be made.
Figure 4 presents results of computing electrical conductivity for nanotubes with the smallest diameter. The most striking difference between these results and the ones presented in literature is that our calculations are made in regard to absolute value of this conductivity. This is a result of the fact that used EM field solvers do not support real and imaginary values for this parameter. To meet the requirements given by aforementioned software, frequency range was narrowed down to values from 1 to 15 GHz and then results were averaged. This allows to define a structure with finite value of electrical conductivity. There is, however, a major disadvantage of such approach. Equation (5) provides readers with electrical conductivity in units of siemenses (S), but software only accepts units of siemenses by meter (S/m). Analytical model assumes that these nanotubes are infinitely
Figure 3 Graphene sheet with indicated axes, basis vectors and position vectors. Circles represent carbon
atoms [2]
Figure 4 Electrical conductivity vs frequency for nanotubes with the smallest diameter (m = 1)
thin and therefore length is neglected in calculations. This is yet a subject to be considered changed, should results be wrong.
V.SIMULATION RESULTS AND DISCUSSION
Figure 5 depicts a structure consisting of three components: a perfect electric conductor (PEC) layer, at the bottom, that represents a metal surface, silicon layer on top of this PEC and graphene nanotube stripes placed at the very top. Structure itself is small as it's size is 6 mm by 6 mm and it's thickness is 150 μm. Boundary conditions were set to standard open boundaries with space added between the structure and free space. This specific case introduces a medium similar to free space but with a small reflection coefficient to represent air and possibly diffirentiate one from another.
Figure 6 and Figure 7 both show results of calculating cross sections via finite elements method in CST Studio Suite. Most notable feature of these graphs is presence of peaks every 500 MHz. These peaks may resemble resonant peaks but in fact additional simulations were run and it is clear that expanding structure to 180% of its size does not get rid of these peaks. They are 5-7 MHz wide in frequency range and
can obtain values both higher and lower than the surrounding ones. It is also confirmed that it is not an effect coming from graphene nanotube stripes as replacing them with metalic sheets produce different RCS values but peaks still remain every 500 MHz. Analysis of them, however, is not the main aim of this paper.
Figure 8 presents results of calculating Radar Cross Section via method of moments in FEKO. Graph itself is nearly identical in terms of shape, values and number of frequency samples. The main difference is absence of RCS peaks, which makes these results much easier to read. Secondly, frequency samples (discrete points) in the first case were linearly spaced between each other. In the latter case, however, they were spaced logarithmically to account for possible interpolation errors. Reader will notice that spacing does little, if anything, in terms of accuracy. It should also be pointed out that high order basis functions (HOBF) were not used for two reasons. Primarily, this structure has virtually no curvatures and as such, curvilinear mesh was not required. Secondly, simulations were conducted on a single personal computer, which had limited computing power. Further research should extend to not only HOBF but also finite time difference method. This is a priority for two reasons. Firstly, FDTD is present in both solvers, so results would be comparable without much thought. Secondly, modelled structure is ideal for FDTD and its cubical (hexahedral) mesh. It requires singificant time and power efforts however.
On top of everything, such approach has one major flaw. Both EM field solvers view graphene nanotube stripes as dielectric materials, which is a very substantial misunderstanding. In terms of modelling, however, this approach is more correct than strictly introducing graphene as metal sheets. This is because such software requires metal sheets to have a predefined thickness to possibly account for skin depth and skin effect. Even after neglecting quantum mechanics effects, skin effect cannot be expected or possibly accounted for in nanotubes as they are infinitely thin when modelled by Boltzmann's equation.
Figure 5 Example of a modelled structure in CST Studio Suite: stripes of nanotubes placed on top of PEC and
silicon layers
Figure 6 Radar Cross Section vs frequency for aforementioned structure
Figure 7 Absorption Cross Section vs frequency for aforementioned structure
Figure 8 Radar Cross Section vs frequency calculated in FEKO solver
Moreover, there is another important matter to discuss. Analytical model presented in this paper completely neglects positioning of nanotubes in respect to each other, which may be a serious concern. This is because currents induced by electromagnetic wave in nanotubes are sources of reflected waves. These waves can interfere with each other in different phases, causing both amplification and cancellation of reflected waves. Flaring effects, in particular, should be considered, because they describe an ideal case, where both constructive and destrictive interferences may occur.
VI.CONCLUSION
This paper outlines the background of modelling carbon nanotubes in any field solver program. Worth mentioning is the fact that it is entirely possible to ignore quantum mechanics effects in nanoscale and focus completely on macroscale conductivity. Of course, it's not an ideal approach as real value could potentially differ from the simulated results. This holds especially true for this case, as modelled structures are treated like dielectric sheets by both software programs. Neglecting quantum mechanics effects bears the burden of not characterizing nanotubes to their fullest extent. It is, however, an optimal approach in regard to EM field solvers requirements.
It is also important to notice that while CST Studio Suite and FEKO both use different methods of calculating farfields, they were able to come up with nearly identical results (apart from aforementioned peaks, which origins are yet to be determined). This is a promising conclusion, as it allows users to use both softwares without sacrificing too much in results' accuracy.
In regard to acquiring carbon nanotubes samples with demanded properties, there is still a lot of controversy in literature. Studies indicate that there is no method which allows to consistently introduce dopants improving electrical
conductivity. What is more, literature lacks information concerning aligning nanotubes. There are studies, which describe edge shaping, edge cutting and edge reconstruction but they do not approach the topic of nanotube alignment, which is an important matter. Quite possibly chemical methods would be an area of interest as alignment of graphene via atomic forces seems reasonable, given the fact that the same forces are exploited to maintain oxidation and edge passivation. However, stimulating nanotube alignment via electric or magnetic fields should also be considered, because nanotubes can be treated like dipoles. All in all, from the perspective of modelling carbon nanotubes in EM field solvers, methods of obtaining them are a secondary issue. The primary concern is assuming an analytical model that both characterizes nanotubes well and meets requirements given by software.
VII.REFERENCES
[1] Muge Acik, Yves J. Chabal, “Nature of Graphene Edges: A Review”, Japanese Journal of Applied Physics 50 (2011) 070101, DOI: 10.1143/JJAP.50.070101
[2] G. W. Hanson, “Fundamental Transmitting Properties of Carbon Nanotube Antennas” , IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 53, NO. 11, NOVEMBER 2005
[3] S. Saxena and T. A. Tyson, “Ab initio density functional studies of the restructuring of graphene nanoribbons to form tailored single walled carbon nanotubes”, Carbon, Volume 48, Issue 4, April 2010, Pages 1153–1158
[4] A. Incze, A. Pasturel, and C. Chatillon, “Ab initio study of graphite prismatic surfaces”, Applied Surface Science, Volume 177, Issue 4, 15 June 2001, Pages 221–225
[5] P. Koskinen, S. Malola, and H. Ha´kkinen, Self-Passivating Edge Reconstructions of Graphene, Physical Review Letters, Volume 101, Issue 11-12, September 2008
