The quantum ion-acoustic waves in single-wall carbon nanotubes are studied with the quantum hydrodynamic model, in which the electron and ion components of the nanotubes are regarded as a two-species quantum plasma system. An analytical expression of the dispersion relation is obtained for the linear disturbance. Numerical results show that the frequency of the ion-acoustic wave strongly depends on the nanotube’s radius in the long-wavelength cases

Wang, You-Nian

Wei, Li

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Quantum Ion-Acoustic Waves in Single-Walled Carbon Nanotubes Studied with a Quantum Hydrodynamic Model

Wilfrid Laurier University

Wilfrid Laurier University Scholars Commons @ Laurier Physics and Computer Science Faculty Publications Physics and Computer Science 2007 Quantum Ion-Acoustic Waves in Single-Walled Carbon Nanotubes Studied with a Quantum Hydrodynamic Model Li Wei Wilfrid Laurier University, lwei@wlu.ca You-Nian Wang Dalian University of Technology Follow this and additional works at: https://scholars.wlu.ca/phys_faculty Recommended Citation Wei, Li and Wang, You-Nian, "Quantum Ion-Acoustic Waves in Single-Walled Carbon Nanotubes Studied with a Quantum Hydrodynamic Model" (2007). Physics and Computer Science Faculty Publications. 82. https://scholars.wlu.ca/phys_faculty/82 This Article is brought to you for free and open access by the Physics and Computer Science at Scholars Commons @ Laurier. It has been accepted for inclusion in Physics and Computer Science Faculty Publications by an authorized administrator of Scholars Commons @ Laurier. For more information, please contact scholarscommons@wlu.ca. Quantum ion-acoustic waves in single-walled carbon nanotubes studied with a quantumhydrodynamic modelLi Wei*Department of Physics and Computer Science, Wilfrid Laurier University, Waterloo, Ontario N2L 3C5, CanadaYou-Nian WangDepartment of Physics, Dalian University of Technology, Dalian 116023, People’s Republic of ChinaReceived 19 December 2006; revised manuscript received 3 March 2007; published 18 May 2007The quantum ion-acoustic waves in single-wall carbon nanotubes are studied with the quantum hydrody-namic model, in which the electron and ion components of the nanotubes are regarded as a two-speciesquantum plasma system. An analytical expression of the dispersion relation is obtained for the linear distur-bance. Numerical results show that the frequency of the ion-acoustic wave strongly depends on the nanotube’sradius in the long-wavelength cases.DOI: 10.1103/PhysRevB.75.193407 PACS numbers: 73.22.Lp, 52.35.Fp, 71.10.CaOne of the most fascinating aspects about carbon nano-tubes is their collective electronic excitations which may bevery important in understanding the electron interactions incarbon nanotubes as well as the characteristics of their elec-tronic structures. During the past years, the collective exci-tations have been widely studied with different theoreticalmodels, such as the classical hydrodynamic model1,2 and thequantum dielectric-response model with the random-phaseapproximation.3–5 The studies correspond to high-frequencyexcitations which come from electron oscillations in carbonnanotubes.It is well known that a carbon nanotube can be metallic orsemiconducting, which depends on its radius and the geo-metric angle, and their electron and ion components can beregarded as a two-species quantum plasma system. Both theelectrons and ions in the carbon nanotubes oscillate under thelow-frequency disturbances. In contrast to the high-frequency excitations, we may expect that the low-frequencydisturbances will lead to a new excitation in the carbon nano-tube, i.e., quantum ion-acoustic wave mode. A powerful the-oretical tool for studying the quantum plasma is the quantumhydrodynamic QHD model which was developed by Haaset al.6 The QHD has been recently used to the quantum dustacoustic wave7 and instabilities of electromagnetic waves inquantum plasma.8 In particular, Haas studied ion-acousticwaves9 in the one-dimensional quantum plasmas.The aim of this paper is to study the dispersion relation ofthe quantum ion-acoustic wave in the carbon nanotube withthe QHD model, in which we model a single-wall carbonnanotube as an infinitesimally thin and infinitely long cylin-drical shell with a radius a, and consider that electrons andions are distributed uniformly over the cylindrical surface,with the equilibrium density n0. In the cylindrical coordi-nates rs= a , ,z, the electron ion density ne ni and fluidvelocity ue ui can be determined by the following QHDequations:6,9net+  · neue = 0, 1nit+  · niui = 0, 2me uet + ue · ue = e − We + 22me 1ne2ne ,3mi uit + ui · ui = − e . 4where me mi are the electron ion mass, e is the charge,and  is the scaled Planck’s constant; = eˆzz + eˆa−1  , andrs , t is the self-consistent potential. The second term onthe right-hand side RHS of Eq. 3 is the force due to theinternal interaction in the electron species, with We=2menebeing the Fermi energy of the two-dimensional 2D electrongas, and the last term on the RHS of Eq. 3 represents thequantum pressure which comes from the quantum diffractioneffects.In Eqs. 2 and 4, the self-consistent potential in theCGS units is given byrs,t = e	 dsnirs,t − ners,trs − rs . 5where ds=addz. By using the expansion of Coulombpotential, 1 / rs−rs, in the cylindrical coordinates, the poten-tial can be expressed by10rs,t = e m=− 	− dk22ga,k,m 	 dsnirs,t− ners,teikz−z+im−, 6where ga ,k ,m=4ImkaKmka, while Im and Km are thecylindrical Bessel functions of order m.Taking into account the small electron inertial force, i.e.,me /mi1, the left-hand side of Eq. 3 can be neglectedunder the low-frequency disturbance. Therefore, by integrat-ing over Eq. 3 once and considering, the boundary condi-tions ne=n0 and =0 at z=, we can obtain the relationbetween the electron density and the potential as follows:PHYSICAL REVIEW B 75, 193407 20071098-0121/2007/7519/1934073 ©2007 The American Physical Society193407-1e = We − We0 −22mene2ne, 7where We0=2men0.The system of Eqs. 2, 4, 6, and 7 constitutes a setof nonlinear self-consistent equations for determining fourunknown quantities, ne, ni, ui, and , which can only besolved numerically, in general. As usually done in the lineardisturbance theory, we linearize the above equations by as-suming ne=n0+ne1, ni=n0+ni1, ui=ui1, and =1, wherene1, ni1, ui1, and 1 are the perturbed quantities. As a result,we obtain the following linearized equations:ni1t= − n0 · ui1, 8ui1t= −emi1, 9e1 =2mene1 −24men02ne1, 101rs,t = e m=− 	− dk22ga,k,m 	 dsni1rs,t− ne1rs,teikz−z+im−. 11Here, k is the lognitudinal wave number and m is the discreteazimuthal quantum number.We further apply the Fourier transformArs,t = m=− 	− dk22	− d2Aa,k,m,eikz−z+im−−it, 12where Ars , t stands for any of the above-listed perturbedquantities. With this transform, one may express the Fouriertransforms of the perturbed electron density, ion density, andpotential as follows:ne1a,k,m =eme21a,k,m1 + km2 /2kF2 , 13ni1a,k,m =en0km2mi2 1a,k,m , 14and1a,k,m = eaga,k,mni1a,k,m − ne1a,k,m ,15where kF= 2n01/2 is the Fermi wave number of the 2Delectron gas and km2=k2+m2 /a2. By substituting Eqs. 13and 14 into Eq. 15, we can get the following dispersionrelation for the linear quantum ion-acoustic wave:2/s2= kmaB2ga,k,maB/a + 2kF2ga,k,m/km2 + 2kF2, 16where s=cs /aB, aB=2 / e2me is the Bohr radius, cs=TF /mi is the ion-acoustic velocity, and TF= 2men0 is theFermi temperature of the 2D electron gas.In the long-wavelength limit, i.e., ka→0, we can use thebehavior of the Imx and Kmx functions when x→0,namely, Imx→amxm, K0x→ ln1.123/x, and Kmx→bmx−mm0, where am=2−m /	m+1 and bm=2m−1	m, then we can get, for m=0, = 0, 17and for m0,2/s2=2m2aB/a2maB/a + 4kF2/2kF2 + m2/a2, 18which depends strongly on the radius of the nanotube.In contrast, we may use the asymptotic expressions of theBessel functions Imx=ex /2x and Kmx= 2xe−x whenx→. Thus, the dispersion relation can be written approxi-mately as2/s2=kmaB2kaB/2 + 2kF2/km2 + 2kF2. 19Especially, when the nanotube radius becomes infinity, i.e.,a→, Eq. 19 can be reduced to2/s2=kaB2kaB/2 + 2kF2/k2 + 2kF2, 20which is the dispersion relation of the ion-acoustic wave in a2D planar quantum plasma.It can be seen from Eq. 16 that the dispersion relationdepends on the tubule’s radius a and the surface electrondensity n0. Generally, radii of the single-wall carbon nano-tubes range from 1 nm up to almost 15 nm. Using the atomicdensity of a graphite sheet 38 nm−2, the surface electron den-sity of a single-walled carbon nanotube can be approximatedby n0=438 nm−2.11 To further reveal the characteristics ofthe dispersion relation in the long-wavelength limit, we plot-ted the dependence of the normalized frequency  /s on thevariable kaB for different nanotube’s radii a with fixed m=2 in Fig. 1. One can see that for long-wavelength regioni.e., kaB→0, as increasing the nanotube’s radius the nor-malized frequency  /s decreases, while the normalized fre-quency approaches to each other for short-wavelength regioni.e., kaB→. To see the effect of the discrete azimuthalquantum number m on the normalized frequency, we plottedthe normalized frequency as a function of kaB for differentvalues of m with fixed a=5aB in Fig. 2. It can be seen fromFig. 2 that for long-wavelength region, as increasing thenumber of m, the normalized frequency increases, and forshort-wavelength region, the same asymptotic behavior asthat in Fig. 1 is observed.In summary, we have used the QHD model to describe thepropagation of the quantum ion-acoustic wave in the single-wall carbon nanotues. An analytical expression of the disper-BRIEF REPORTS PHYSICAL REVIEW B 75, 193407 2007193407-2sion relations has been derived in the case of the linear dis-turbance. Simulation results show that for a fixed surfaceelectron density of a single-walled carbon nanotube, the fre-quencies of the ion-acoustic waves strongly depend on thenanotube’s radius and the azimuthal quantum number in thelong-wavelength cases.This work was supported by the Natural Sciences andEngineering Research Council of Canada.*Electronic address: lwei@wlu.ca1 C. Yannouleas, E. N. Bogachek, and U. Landman, Phys. Rev. B53, 10225 1996.2 X. Jiang, Phys. Rev. B 54, 13487 1996.3 O. Sato, Y. Tanaka, M. Kobayashi, and A. Hasegawa, Phys. Rev.B 48, 1947, 1993.4 M. F. Lin and Kenneth W.-K. Shung, Phys. Rev. B 47, 66171993.5 M. F. Lin, D. S. Chuu, C. S. Huang, Y. K. Lin, and K. W.-K.Shung, Phys. Rev. B 53, 15493 1996.6 F. Haas, G. Manfredi, and M. Feix, Phys. Rev. E 62, 2763 2000.7 P. K. Shukla and S. Ali, Phys. Plasmas 12, 114502 2005.8 P. K. Shukla and L. S. Stenflo, Phys. Plasmas 13, 044505 2006.9 F. Haas, L. G. Garcia, J. Goedert, and G. Manfredi, Phys. Plasmas10, 3858 2003.10 J. D. Jackson, Classical Electrodynamics Wiley, New York,1975.11 D. Ostling, D. Tomanek, and A. Rosen, Phys. Rev. B 55, 139801997.FIG. 1. The dispersion relation  /s of the quantum ion-acoustic wave for the discrete azimuthal quantum numbers m=2and different nanotube’s radii: a=2aB, a=5aB, a=8aB, anda=10aB.FIG. 2. The dispersion relation  /s of the quantum ion-acoustic wave for the nanotube’s radius a=5aB and different azi-muthal quantum numbers m=0, 1, 2, 3, and 4.BRIEF REPORTS PHYSICAL REVIEW B 75, 193407 2007193407-3

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Quantum Ion-Acoustic Waves in Single-Walled Carbon Nanotubes Studied with a Quantum Hydrodynamic Model

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