Effective permittivity of single-walled carbon nanotube composites: Two-fluid model Afshin Moradi, Hamid Reza Zangeneh, and Firoozeh Karimi Moghadam Citation: Physics of Plasmas 22, 122104 (2015); View online: https://doi.org/10.1063/1.4936872 View Table of Contents: http://aip.scitation.org/toc/php/22/12 Published by the American Institute of Physics Articles you may be interested in Optical reflection and absorption of carbon nanotube forest films on substrates Journal of Applied Physics 118, 013106 (2015); 10.1063/1.4923390 Determination of complex refractive index of thin metal films from terahertz time-domain spectroscopy Journal of Applied Physics 104, 053110 (2008); 10.1063/1.2970161 Dielectric properties of single-walled carbon nanotubes in the terahertz frequency range Applied Physics Letters 91, 011108 (2007); 10.1063/1.2753747 Optical and electrical properties of preferentially anisotropic single-walled carbon-nanotube films in terahertz region Journal of Applied Physics 95, 5736 (2004); 10.1063/1.1699498 Terahertz conductivity of anisotropic single walled carbon nanotube films Applied Physics Letters 80, 3403 (2002); 10.1063/1.1476713 Anisotropic high-field terahertz response of free-standing carbon nanotubes Applied Physics Letters 108, 241111 (2016); 10.1063/1.4954222 PHYSICS OF PLASMAS 22, 122104 (2015) Effective permittivity of single-walled carbon nanotube composites: Two-fluid model Afshin Moradi,1,2,a) Hamid Reza Zangeneh,3 and Firoozeh Karimi Moghadam3 1 Department of Engineering Physics, Kermanshah University of Technology, Kermanshah, Iran Department of Nano Sciences, Institute for Studies in Theoretical Physics and Mathematics (IPM), Tehran, Iran 3 Department of Photonics, Faculty of Physics, University of Kashan, Kashan, Iran 2 (Received 10 September 2015; accepted 18 November 2015; published online 7 December 2015) We develop an effective medium theory to obtain effective permittivity of a composite of two-dimensional (2D) aligned single-walled carbon nanotubes. Electronic excitations on each nanotube surface are modeled by an infinitesimally thin layer of a 2D electron gas represented by two interacting fluids, which takes into account different nature of the r and p electrons. Calculations of both real and imaginary parts of the effective dielectric function of the system are presented, for C 2015 AIP Publishing LLC. different values of the filling factor and radius of carbon nanotubes. V [http://dx.doi.org/10.1063/1.4936872] I. INTRODUCTION II. BASIC EQUATIONS Since the discovery of carbon nanotube (CNT) by Iijima in 1991,1 this material has been the subject of intensive research efforts throughout the world due to its outstanding mechanical, thermal, electrical, magnetic, and optical properties. Also, advanced composites, combined with aligned CNTs are now fabricated by many research groups using different techniques.2–5 For example, aligned CNTs can be synthesized easily right now, using plasma enhanced hot filament chemical vapor deposition (PE-HF-CVD) method, and arrays of aligned CNTs can be synthesized with controlled diameter, length, and inter-tube distance.4 Recently, Kumar and Tripathi,6 studied the linear and nonlinear interaction of electromagnetic waves with an array of CNTs, using an effective medium theory based on the fluid theory. On the other hand, we investigated the optical properties of an isolated single-walled CNT,7 by means of the vector wave function method and the two-fluid hydrodynamic model.8,9 We found that strong interaction between the r and p fluids gives rise to the splitting of the extinction spectra into two peaks in quantitative agreement with the p and r þ p plasmon energies.10 In present work, inspired by the idea given in Ref. 6, we extend our previous results7 to study the optical characteristics of a composite of 2D aligned single-walled CNTs. For this purpose, we develop the standard effective medium theory proposed by Wu et al.11 Huang and Gao.12,13 Then, using the results obtained in Ref. 7, we derive the effective permittivity of the system. The paper is organized as follows: In Section II, we set the basic equations concerning the problem. Numerical results are discussed in Section III, and finally, Section IV contains our conclusions. Let us consider a composite consisting of infinitely long single-walled CNTs (shown by 1) of radius a and volume fraction f parallel to each other and randomly embedded in a background medium (shown by 2) with dielectric permittivity 0 (i.e., in vacuum) as sketched in Fig. 1(a). To study the effective permittivity eff of the whole composite, we invoke the standard effective medium theory,11–13 which deals with the scattering problem of a single-walled CNT with radius a that is enclosed by a dielectric cylinder of radius b within an effective medium with permittivity eff, as seen in Fig. 1(b). The volume between a and b represents the average envelope per CNT and f ¼ a3/b3. The effective dielectric permittivity of the system eff may be determined by requiring that this single-walled CNT causes no perturbation of the electromagnetic waves in the surrounding medium.11–15 We take an electromagnetic wave normally incident on the structure. For this propagation direction there are two different values of eff corresponding to different polarizations. In the case of transverse magnetic (TM) waves, i.e., when the electric field is parallel to the cylinders axis, no plasmon resonance peak is found in the far-field extinction spectra of an isolated single-walled CNT7 because the polarization direction along the axis of CNTs cannot induce collective motions of the conduction electrons. Therefore, we consider only the case of transverse electric (TE) waves. For TE waves, the magnetic field is parallel to the nanotubes axis and the solutions in the effective medium can be written as Hz ¼ þ1 X ½am ðeff ÞJm ðke rÞ þ bm ðeff ÞHm ðke rÞeim/ ; (1) m¼1 a) Electronic mail: a.moradi@kut.ac.ir 1070-664X/2015/22(12)/122104/4/$30.00 pﬃﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃ where ke ¼ x ef f l0 and x is the frequency of the electromagnetic wave. Here Jm(x) and Hm(x) are the Bessel 22, 122104-1 C 2015 AIP Publishing LLC V 122104-2 Moradi, Zangeneh, and Moghadam Phys. Plasmas 22, 122104 (2015) Hz ¼ þ1 X ½am ð0ÞJm ðk0 rÞ þ bm ð0ÞHm ðk0 rÞeim/ ; (2) m¼1 where k0 ¼ x/c and c is the light speed in vacuum. The usual two boundary conditions at the dielectricdielectric interface require the continuity of the tangential components of the electric and magnetic fields across the interface. By matching the aforementioned boundary conditions on the interface of the background and effective medium, we obtain am ð0Þ A11 A12 am ðeff Þ ¼F ; (3) bm ðeff Þ bm ð0Þ A21 A22 where F ¼ ð0 ke Þ1 ½Jm ðke bÞHm0 ðke bÞ Jm0 ðke bÞHm ðke bÞ1 ; A11 ¼ 0 ke Jm ðk0 bÞH 0m ðke bÞ ef f k0 J 0m ðk0 bÞHm ðke bÞ; A12 ¼ 0 ke Hm ðk0 bÞH 0m ðke bÞ ef f k0 H 0m ðk0 bÞHm ðke bÞ; A21 ¼ ef f k0 Jm ðke bÞJ 0m ðk0 bÞ 0 ke Jm ðk0 bÞJ 0m ðke bÞ; A22 ¼ ef f k0 Jm ðke bÞH 0m ðk0 bÞ 0 ke Hm ðk0 bÞJ 0m ðke bÞ: By using the condition b1(eff) ¼ 011–13 for the present system, from Eq. (3) we obtain A21 b1 ð0Þ ¼ ¼ D1 ðxÞ; A22 a1 ð0Þ (4) where D1(x) represent the Mie scattering coefficients of the single-walled CNT and have the form of7 2 rr// þ rp// J10 ðk0 aÞ D 1 ðx Þ ¼ : (5) 2=ðpxal0 Þ þ rr// þ rp// J 01 ðk0 aÞH 01 ðk0 aÞ FIG. 1. (a) The single-walled CNTs in a dielectric host. (b) A single-walled CNT plus surrounding dielectric shell embedded in the effective medium. The single-walled CNTs are infinitely long in the z-direction. function and Hankel function of the first kind, respectively. Furthermore, we assume a harmonic dependence for all quantities as a function of time [ expðixtÞ, where x is the frequency of electromagnetic wave]. Also, we take cylindrical coordinate ðr; /; zÞ for an arbitrary point in space and use the two-fluid theory. In this way, we consider each CNT to be composed of p and r fluids superimposed at r ¼ a with equilibrium densities (per unit area) n0p ¼ 38 nm2 and n0r ¼ 3n0p ¼ 3 38 nm2 , respectively.3 We should note that according to the Longe and Bose results,16 such classical models give results in agreement with random phase approxpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ imation models when 1 kFa, where kF ¼ 2pðn0p þ n0r Þ is the Fermi momentum. For the single-walled CNT composites that will be considered, we find 1 kFa for each CNT, which justifies our use of the hydrodynamic model. Similarly, the magnetic field in the background medium of the coated single-walled CNT takes the form We note that rk// is the conductivity of the electron fluid k (the index k takes values r and p) along the / direction and may be written as7 rk// ¼ i0 ax2pk x ; xðx þ ick Þ ak =a2 (6) where xpk ¼ ðn0k e2 =0 mk aÞ1=2 . Also, mk and ck are the effective electron mass and the effective collision frequency of the electron fluid k, respectively. By considering the limit of keb 1, the Bessel and Hankel functions of the first kind in A21 and A22 can be approximated as J1 ðxÞ ﬃ x=2; J 01 ðxÞ ﬃ 1=2; H1 ðxÞ ﬃ ðx=2Þ ið2=pxÞ, and H 01 ðxÞ ﬃ ð1=2Þ þ ið2px2 Þ with x ¼ keb. Therefore, using these approximations, Eq. (4) reduces to J1 ðk0 bÞ D 1 ðx Þ k0 bJ 01 ðk0 bÞ Y 01 ðk0 bÞ ; ¼ 0 Y1 ðk0 bÞ iJ 1 ðk0 bÞ 1 þ D1 ðxÞ ef f ðxÞ 0 k0 bY 01 ðk0 bÞ ef f ðxÞ 0 (7) where Ym(x) is the Neumamn function and Hm(x) ¼ Jm(x) þ iYm(x). The above equation is the main result of the 122104-3 Moradi, Zangeneh, and Moghadam Phys. Plasmas 22, 122104 (2015) FIG. 2. Variation of Im[eff] and Re[eff] vs. the frequency x [in the frequency range 1 eV < x < 10eV]. In panels (a1) and (a2) f ¼ 0.11, and a ¼ 4aB, 6aB, and 8aB, while in panels (b1) and (b2) a ¼ 4aB, and f ¼ 0.081, 0.11, and 0.16. present work, namely, the (complex) effective dielectric function of a composite of aligned single-walled CNTs. III. NUMERICAL RESULT AND DISCUSSION Equation (7) can be used to obtain both real and imaginary parts of the effective dielectric function of the system, as a function of the frequency x, volume fraction f, and CNTs radius a. In calculations, we assume that r and p electrons have equal effective masses, mr ¼ mp ¼ me , where me is the electron mass. Also, we choose the friction coefficients to be small but finite parameters cr ¼ 0.01xpr and cp ¼ 0.01xpp. We divide the frequency regime into two regions: (1) 1 eV < x < 10 eV and (2) 10 eV < x < 25 eV. In Fig. 2, panels (a1) and (a2), we show both imaginary and real parts of the effective dielectric function of the system as a function of the frequency x, choosing volume fraction f ¼ 0.11 and different values of a ¼ 4aB, 6aB, and 8aB [aB is the Bohr radius], respectively. Here, the sharp resonant peak in panel (a1) is due to p plasmon excitations.7 One can see that by increasing a, the resonant peak position of p plasmons exhibits a redshift, as can be observed from results of panels (b1) and (b2) of Fig. 2 in Ref. 7. The similar results of resonant peak of r þ p plasmons, in the frequency range 10 eV < x < 25 eV, are presented in Fig. 3, panels (a1) and (a2). To illustrate volume fraction effects, we plot in panels (b1) and (b2) of Fig. 2 and 3, both the imaginary and real parts of the effective permittivity of the system, respectively, versus the frequency x, for the CNTs characterized by a ¼ 4aB and different values of f ¼ 0.081, 0.11, and 0.16. We FIG. 3. The corresponding results of Fig. 2, but in the frequency range of 10 eV < x < 25 eV. 122104-4 Moradi, Zangeneh, and Moghadam see as parameter f increases, the interaction among the CNTs leads to a resonance at lower frequencies for resonant peak of r þ p plasmons [this result corresponds to the result given in Ref. 17 for the optical response of composite materials with metallic nanoparticles], while the resonant peak of p plasmons remains stable. IV. CONCLUSION To summarize, we have developed a standard effective medium theory to obtain the effective permittivity of a composite of 2D aligned single-walled CNTs. We have presented the calculations of both real and imaginary parts of the effective dielectric function of the system, for various values of the filling factor and radius of CNTs. 1 2 S. Iijima, Nature 354, 56 (1991). R. R. Schlitter, J. W. Seo, J. K. Gimzevski, C. Durkan, M. S. M. Saifullah, and M. E. Welland, Science 292, 1136 (2001). Phys. Plasmas 22, 122104 (2015) 3 M. 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