Investigation on Carbon Nanotube Array Behavior

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314–334, 2006. [4] Chris Rutherglen, Dheeraj Jain, and Peter Burke, “rf resistance and inductance of massively parallel single walled carbon nanotubes: Direct,.
Investigation on Carbon Nanotube Array Behavior K. Louertani1 , H. Talleb1 , C. Tripon-Canseliet1 , D. Lautru1 , D. Decoster2 , JP. Martinaud3 1

UPMC Univ Paris 06, UR2, L2E, BC 252, 4 place Jussieu, 75252 Paris, France 2 IEMN, Avenue Poincar´e, BP 60069, 59652 Villeneuve d’Ascq cedex, France 3 Thales Syst`emes A´eroport´ees, Avenue Gay-Lussac 78990 Elancourt, France Email: [email protected]

Abstract—In this paper, an investigation on the electromagnetic behavior of a carbon nanotube (CNT) array is proposed. The nanotubes are modeled using a RF circuit previously defined in the literature. The studied configuration corresponds to an array of metallic CNTs vertically aligned on a metallic ground plane. The dispersion diagram is calculated and plotted using the classical irreductible Brillouin zone.

I. I NTRODUCTION Carbon nanotubes (CNTs) have drawn a lot of attention these past few years due to their particular electromagnetic and mechanical characteristics. Previous studies have led to RF circuit model definition of a metallic CNT [1] [2] [3] and have highlighted their exceptional electrical and electromagnetic characteristics such as high impedance and slow wave propagation. Indeed, the characteristic impedance of a CNT above a ground plane is close to 12.5 GHz and the wave velocity is a hundred times slower than the speed of light (v = 0.01c). Issues due to the high impedance has been overcome by assuming CNT bundles [4] [5] [6] that lower the impedance and makes easier the measurements of such structures [7]. There are different configurations of bundles i. e. planar, cylindrical. The behavior of the bundle is different for each case depending on the interactions between the surrounding CNTs. In this paper, we propose to study the electromagnetic behavior of an array of CNTs vertically grown on a ground plane by using RF circuit model and transmission line theory [8].

Fig. 1.

Array of SWCNT on a ground plane

nanotubes. Modeling could be improved by adding a contact resistance between CNT. Due to the geometry of the 2D array, each CNT is coupled with four others. Hence, the RF circuit can be defined as in Fig. 2.

II. A RRAY CONFIGURATION The studied configuration showed in Fig. 1 represents an array of CNT equally spaced, by a step d, on a ground plane. In this paper, we consider metallic Single Wall Carbon Nanotube (SWCNT). Here, the system is considered as homogeneous along the z-axis. Previous investigations led by P. Burke have defined an RF circuit model [2] for a SWCNT with a value for the kinetic inductance (Lk = 16 nH/µm) and for the quantum capacitance (CQ = 400 aF/µm). Depending on the surrounding configuration i.e. a single or double CNTs above a ground plane, the RF circuit model may be different in order to take into account the mutual electrical coupling. In this configuration, all the nanotube are vertical and connected at the base of the tubes. Here, we assume a perfect connection between the base of the

Fig. 2.

RF circuit of a SWCNT in 2D configuration

Formulas for the definition of the impedance Z and admittance Y are given in equ. 1 where Y expression represents the CNT and Z expression models the coupling between neighbor nanotubes.

Lk Y = j ω = jLT ot ω 4 1 1 Z= = j2(Ces + 4Cq)ω j2CT ot ω

(1)

Coupling is assumed by a electrostatic capacitance Ces term and quantum capacitance CQ term. A quantum resistance RQ could be introduced in Y expression in order to add losses along the nanotube. III. D ISCUSSION Dispersion diagram of the array can be determined from the RF circuit. Hence, if we assume a propagating wave on the perpendicular plane of the CNT, this diagram gives the evolution of the wave propagation constant over a frequency band. The equation of the dispersion diagram according to the RF circuit is given in equ.2 [8]. In order to represent the two main directions of propagation, the dispersion diagram is plotted over the first Brillouin zone [9]. βy d 1 βx d ) + sin2 ( ) = A.B sin2 ( 2 2 2 h i 1 kd A= 2sin( kd ) − Z0 ωC cos( ) 2 2 T ot h i Z0 kd B= 2sin( kd ) − cos( ) 2 ωLk 2

(2)

Where k is the propagation constant of the transmission line and Z0 is the characteristic impedance. Brillouin zone is defined by the smallest pattern of the array with periodic 2D conditions in order to achieve the entire array. Hence, the Brillouin zone can be represented for three directions as shown in Fig. 3.

Fig. 3.

The results are given in Fig. 4 and 5 for an array of CNT equally spaced by a step d = 1 µm and d = 100 µm. A 3D view of the dispersion diagram is also shown in Fig. 6. Due to the nanotube characteristics, the frequency band reaches several terahertz. In both cases, the dispersion diagram corresponds to a 2-D periodic structure. In the first case, the propagation is not modified by the array and no band gap is observed. However, in the second case, a lattice period effect is produced which creates a frequency band gap. Moreover, a High Pass Stop band is shown at the lower frequencies. This is characteristic of the left handed structures which is confirmed by the RF circuit.

Fig. 4.

Dispersion diagram of the array (d=1 µm)

Brillouin zone

The first one on the axis Γ − X represents the evolution of the propagation constant on x axis and is given by introducing βx d : 0 → π and βy d = 0 in equ. 2. The second one on X − M represents the propagation along y axis and is given by βx d = π and βy d : 0 → π. The last one, M − Γ, is given by βx d : π → 0 and βy d : π → 0.

Fig. 5.

Dispersion diagram of the array (d=100 µm)

Fig. 6.

3D view of the dispersion diagram (d=100 µm)

IV. C ONCLUSION A carbon nanotube array has been studied using the transmission line theory. The dispersion diagram has been determined from the RF circuit and plotted over the first Brillouin zone. The array of CNTs exhibits a 2-D periodic structure dispersion diagram with a frequency band gap at the lower frequencies which corresponds to the behavior of a left hand structure. ACKNOWLEDGMENT The authors would like to thank the DGA for financial support under the contract REI-DGA: 2010 34.0004. R EFERENCES [1] P. J. Burke, “Luttinger liquid theory as a model of the gigahertz electrical properties of carbon nanotubes,” vol. 1, no. 3, pp. 129–144, 2002. [2] P. J. Burke, “An rf circuit model for carbon nanotubes,” vol. 2, no. 1, pp. 55–58, 2003. [3] P. J. Burke, Shengdong Li, and Zhen Yu, “Quantitative theory of nanowire and nanotube antenna performance,” vol. 5, no. 4, pp. 314–334, 2006. [4] Chris Rutherglen, Dheeraj Jain, and Peter Burke, “rf resistance and inductance of massively parallel single walled carbon nanotubes: Direct, broadband measurements and near perfect 50 ohms impedance matching,” Applied Physics Letters, vol. 93, no. 8, pp. 083119, 2008. [5] Sangjo Choi and K. Sarabandi, “Design of efficient terahertz antennas: Carbon nanotube versus gold,” in Proc. IEEE Antennas and Propagation Society Int. Symp. (APSURSI), 2010, pp. 1–4. [6] Yue Wang, Yu Ming Wu, Lei Lei Zhuang, Shao Qing Zhang, Le Wei Li, and Qun Wu, “Electromagnetic performance of single walled carbon nanotube bundles,” in Proc. Asia Pacific Microwave Conf. APMC 2009, 2009, pp. 190–193. [7] Yijun Zhou, Y. Bayram, Feng Du, Liming Dai, and J. L. Volakis, “Polymer-carbon nanotube sheets for conformal load bearing antennas,” vol. 58, no. 7, pp. 2169–2175, 2010. [8] A. Grbic and G. V. Eleftheriades, “Dispersion analysis of a microstripbased negative refractive index periodic structure,” vol. 13, no. 4, pp. 155–157, 2003. [9] C. Kittel, “Introduction to solid state physics,” 1995.