Variation of Plasma Frequency with Applied Magnetic Fields in a Single Walled Carbon Nanotube

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1 International Journal of Physics and Applications. ISSN Volue 2, Nuber 1 (2010), pp International Research Publication House Variation of Plasa Frequency with Applied Magnetic Fields in a Single Walled Carbon Nanotube K.A.Vijayalakshi, Nafeesa Baby TP and Binuol P Kuriakose Research and Developent Centre, Bharathiar University, India E-ail: nafeesababy@gail.co Abstract Using linearised quantu hydrodynaic odel along with Maxwell s Equation the effect of static unifor agnetic field on the frequency of plasa oscillations in a single walled carbon nanotube (SWCNT) is discussed here. In presence of a static transverse agnetic field, in short wavelength liit plasa dispersion relation is found varied in accordance with the applied agnetic field. Keywords: Plasa Oscillations, Magnetic Field, Dispersion Relation, Long Wavelength Liit. Introduction Plasa consists of electrons and ions with total charge zero. Plasa is usually considered as an electron gas oving in the background of positive ions. Plasa is a quasineutral gas of ions, electrons and neutral particles [1].The ter plasa was first used by Languir [2] in The study of carbon nanotubes (CNTs) is now an active area of research, which could lead to the developent of advanced technology devices. One of the ost fascinating aspects about CNTs is their surface ode excitations. During the past years, a lot of experiental and theoretical works have been done to study the highfrequency excitations (electron oscillations) in these systes. Also, it is well-known that in such systes both the positive ions and the electrons oscillate under lowfrequency disturbance. Fetter [3] used a siple hydrodynaic odel to study the electrodynaics of the electron ion plasa in a periodic array and obtained an acoustic branch in addition to the optical branch. Wei and Wang [4] studied the dispersion relation of quantu ion acoustic wave (QIAW) oscillations in single-walled carbon nanotubes (SWCNTs) with the quantu hydrodynaic (QHD) odel which was developed by Haas et al [5,

2 40 K.A.Vijayalakshi et al 6]. In the presence of a static agnetic field, we ay expect a new excitation in carbon nanotubes (CNTs), i.e. quantu agnetosonic wave (QMSW) oscillations. Let us note that the effects of a static agnetic field on the plason oscillations of an electron gas in CNTs have been investigated by several authors using various ethods. Shyu et al studied the agnetoplason of SWCNTs within the tight-binding odel [7]. The low-frequency single-particle and collective excitations of SWCNTs were studied in the presence of a agnetic field by Chiu et al [8, 9]. Vedernikov et al [10] studied the collective oscillations of two-diensional electrons in nanotubes in the presence of a agnetic field parallel to the tube axis. The energies of neutral and charged excitations and plason frequencies in nanotubes as functions of the agnetic field were analyzed by Chaplik [11]. Gub[12], calculated the dispersion relation of the collective agnetoplason excitations for an electron gas confined to the surface of a nanotube when a agnetic field is perpendicular to its axis. In particular, by using the hydrodynaic odel and Maxwell s equations, Kobayashi [13] studied the agneto static plasa wave oscillations of a SWCNT in the voigt configuration. Afshin Moradi [14, 15, and 16] has studied the effect of a static external agnetic field in a SWCNT. Here we are interested in the application of transverse agnetic waves which propagate parallel to the surface of a SWCNT and concentrate on the excitations of the electron ion syste as two fluids confined to its surface. There is assued to be a static agnetic field B0 that is noral to the cylindrical surface (Voigt configuration). Theory and Discussions Let us consider an infinitely long and infinitesially thin SWCNT with a radius a and take the cylindrical polar coordinate x = (r, φ, z) for an arbitrary point in space. Let us consider the CNT to consist of electron and ion fluids superiposed at r = a with charges e and Ze, respectively. The equilibriu densities (per unit area) of electrons are n 0e and ions n 0 i satisfy n 0 i = n 0 and n e 0 Z n 0 i= Zn 0 (x,t) + Zn o. u e (x, t) = 0, (1) And the equation of linearised oentu + n o. u i (x, t) = 0, (2) = e/ [E (x,t) + u e (x, t) x B o ] n e (x, t) + [ 2 n e (x, t)] (3) = [E (x,t) + u i (x, t) x B o ] (4)

3 Variation of Plasa Frequency with Applied Magnetic Fields 41 Where n e (n i ) is the velocity of electron (ion) and = eˆz (ə/əz) +a -1 eˆφ (ə/əφ) (5) The three ters on the RHS of equation (3)are respectively force due to electric field and agnetic field, force due to internal interaction or can be considered as classical pressure of electron fluid and the third ter coes fro the quantu diffraction effect which can be considered as quantu pressure. The electric current density flowing on the surface of the cylinder is given by J e (x,t) =σ e E (x,t) (6) J i (x,t) =σ i E (x,t) (7) Where σ eˆ( σ iˆ) is the conductivity tensor of the electron (ion) Now we can define Fourier-Bessel transfor A (q) of an arbitrary function A(φ,z,t) by A(φ,z,t)= A (q) exp [i (φ + qz- ωt)] (8) Using equations (1) to (7) by eliinating ters n e and n i we get equations for σ e and σ I in ters of cyclotron frequency ω. ie ω ce = eb o / e (ω ci =ZeB 0 / i ) of electron (ion). In the space above and below electron ion cylinder, the transverse electric wave satisfies E E z z () r = E0z I ( κa) K ( κr ) () r = E K ( κa) I ( κr ) 0z r > a r < a Where I (x) and K (x) are the odified Bessel functions and K 2 = q 2 - ω 2 /c 2 where c is the speed of light. For the case speed of light can be taken to be infinitely large we have ω 4 - ω 2 (9) + x g

4 42 K.A.Vijayalakshi et al Which deterines the noral electrostatic odes. The roots of the above equation give (in the liit Ze /i<<1) equations for ω + and ω - where ω + for high frequency dispersion and ω- for low frequency (QMSWquantu agneto sonic wave) dispersion. In short wave length liit i.e. qa we ay use the asyptotic expression of the Bessel functions I (x) = e x / and K (x) = e -x, so that the dispersion relation (10) ω - 2 (q) (11) Using the above equation the variation of the diensionless frequency ω -/ ω s with respect to the variable qab for different values of fields applied for carbon nanotubes are studied. Here ω s = (Z e ω 2 ce/ i ) 1/2.The dispersion curves obtained is as shown in figure 1. qa B Figure 1 : Dispersion relation ω -/ ω s versus the diensionless variable qa B Fro equation (11) for a single walled carbon nanotubes Conclusion In conclusion we can see that the frequency is varied in accordance with the B0, for long wave length liit. The dispersion relation for different values of Bo are grouped in figure 1.This was interpreted as the applied agnetic field is increased the plasa dispersion exhibits stronger dispersions.

5 Variation of Plasa Frequency with Applied Magnetic Fields 43 References [1] Chen F F 1984 Introduction to Plasa Physics and Controlled Fusion (New York: Plenu) [2] Languir I 1928 Proc. Natl Acad. Sci. USA [3] Fetter A L 1974 Ann. Phys [4] Wei L and Wang Y N 2007 Phys. Rev. B [5] Haas F, Manfredi G and Feix M 2000 Phys. Rev. E [6] Haas F, Garcia L G, Goedert J and Manfredi G 2003 Phys.Plasas [7] Shyu F L, Chang C P, Chen R B, Chiu C W and Lin M F 2003 Phys. Rev. B [8] Chiu C W, Chang C P, Shyu F L, Chen R B and Lin M F 2003 Phys. Rev. B [9] Chiu C W, Shyu F L, Chang C P, Chen R B and Lin M F 2004 Physica E [10] Vedernikov A I, Govorov A O and Chaplik A V 2001 JETP [11] Chaplik A V 2002 JETP Lett [12] Gubs G 2005 Phys. Scr. T J. Phys.: Condens. Matter 21 (2009) AMoradi [13] Kobayashi M 1999 Phys. Status Solidi b [14] Moradi A and Khosravi H 2007 Phys. Rev. B [15] Moradi A and Khosravi H 2007 Phys. Lett. A [16] Afshin Moradi. 2009J.Phys.Condens.Matter

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