INTERNATIONAL JOURNAL OF PURE AND APPLIED RESEARCH IN ENGINEERING AND TECHNOLOGY

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1 INTERNATIONAL JOURNAL OF PURE AND APPLIED RESEARCH IN ENGINEERING AND TECHNOLOGY A PATH FOR HORIZING YOUR INNOVATIVE WORK AN EVALUATION OF DISPERSION RELATION OF PROPAGATION OF ELECTROMAGNETIC WAVE WITH TM MODES IN CARBON NANOTUBE WITH AND WITHOUT ELECTRON ENERGY-BAND EFFECTS RAKESH KUMAR, N.P SINGH, SHEO KUMAR YADAV 3, L. K. MISHRA 3. s/o Sri Rajana Prasad, Vill- Badra, P.O-Bhore, Dist-Gaya (Bihar). Departent of Physics, Gaya College, Gaya-8300(Bihar) 3. Departent of Physics, Magadh University, Bodh-Gaya-8434(Bihar) Accepted Date: 7/0/06; Published Date: 0/03/06 Abstract: - Using the theoretical foralis of Li Wei et al (Phys. LettA 333 (004)) and Afshin Moradi (J. Electroagnetic Analysis & Applications (00), we have studied the dispersion relation of TM-ode with and without electron energy band effects. Without including electron energy band effects, the dispersion relation can be studied with linearized hydrodynaic theory with Maxwell equations. This indicates that the TM-ode is very different fro TE-ode. Here, the dispersion relation does not approach to well-known dispersion relation of D electron-gas. Besides, it also indicates that internal interaction forces play an iportant role on the dispersion relation of TM-ode. The dispersion relation of TM-ode including electron energy band effects can be studied by eans of the sei classical kinectic theory of the electron dynaics. The effect of energy band structure is taken into account for surface plason oscillations in the zigzag and archair nanotube of etallic character. Our theoretical results also indicate that plason waves are not sensitive to the types of etallic nanotube with sae radius. Our theoretically evaluated results are in good agreeent with those of the other theoretical workers. \ Keywords: Single wall carbon nanotubes, lineaized hydrodynaic theory, Nano waveguide, hydrodynaic equations, dispersion relation for surface waves, Transverse electric ode (TE-ode), Transverse agnetic ode (TM-ode), Low frequency electroagnetic wave, Electrostatic collective excitations, Single electron excitation effects, surface plason wave, internal interaction forces, Zig-zag and archair nanotubes. Corresponding Author: MR. RAKESH KUMAR Access Online On: How to Cite This Article: PAPER-QR CODE 07

2 . INTRODUCTION In an earlier paper, we have studied the propagation of electroagnetic wave propagation in carbon nanotubes and presented a ethod of evaluation of dispersion relation of TE-ode. In this paper, we have presented a ethod of evaluation of TM-ode. The evaluation has been perfored by theoretical foralis of Li Wei etal.. In this foralis, they have oduled the nanotube surface as infinitesially thin layer of free-electron gas which is described by eans of linearzied hydrodynaical theory. General expression of dispersion relation are obtained for the low frequency electroagnetic wave with transverse agnetic ode. This has been achieved by solving Maxweell and hydrodynaic equations with appropriate boundary conditions. Nuerical ethods were perfored and it was observed that TM ode is quite different fro TE-ode. These two odes are different for large wave nubers and also for large nanotube radius. We first obtained the dispersion relation for TM ode using hydrodynaical odel which does not consider the electron energy- band effects. These effects are valid only for describing dispersion relation for chiral carbon nanotube. We have studied the energy band effects on the dispersion relation of the surface plason wave in single walled carbon nanotube (SWCNTs) of etallic behavior by the classical kinetic theory of the electron dynaics. Earlier 3,4, one focussed on plason wave oscillations in a cylindrical electron gas as a siple odel of etallic tube. In this case, we have taken the ore exact analysis of geoetrical effect including the radius and the chiral angle of the nanotube. Fro the discovery by Iiji 5 of carbon nanotubes (CNTs), there has been a growing interest in electroagnetic wave propagation in single-wall carbon nanotubes. Since CN can be etals or seiconductors depending upon its radius and the geoetrical angle, soe iportant inforation about the structural and electronic properties can be obtained using electroagnetic probe techniques 6,7 or electron probe techniques 8. In particular, being very long, a CN can be regarded as a nano waveguide to guide electroagnetic waves. During the past years, different theoretical odels have been used to describe the physical properties of CNs. With a classical hydrodynaic odel, Yanouleas 9 and Jiang 0 studied the collective excitation behavior of σ and π electrons in single or ulti-wall CNs. The collective excitation properties of CNs are quite different fro those of well-known graphite sheets. The difference is that the collective excitations in CNs have traditional one-diensional (D) characters for sall wave nuber, while exhibiting two-diensional (D) behavior for large wave nubers. The quantu dielectric-response theory, taking into account the electron energy band structures in CNs have used -3 to describe low frequency electronic excitations in CNs. Beyond the electrostatic excitations, Slepyan etal. 4-6 studied electroagnetic processes in CNs. With 08

3 the classical electrodynaics and a sei-classical kinetic theory, they derived the dispersion relation of surface wave in CN s. They found out that the CNs can be used as a waveguide for controlling electroagnetic wave propagation in specified frequency ranges (exaples are infrared and optical). They also presented a general quantu echanical theory of the conductivity of a single wall carbon nanotubes with interband transitions. Now days, there is uch interest to electroagnetic high-frequency properties of carbon nanotubes. This is because of their potential applications in nanoelectronics 7, nanoantennas 8-, polarizers, free electron lasers 3, devices for THz sensing and iaging 4. As we know that carbon nanotubes (CNs) possess etallic properties therefore special interest has been paid due to their high conductivity at THz frequencies and copared to etal nanowires 5. By this reason, their applications see to be proising in THz and infrared ranges due to noticeable lower losses copared to other conductive aterials. One of the ost iportant electroagnetic property of etallic CNTs is a capability to support propagation of strongly delayed surface waves 6,7. It is caused by a very high kinetic inductance of thin single wall CNTs 8. It akes electroagnetic (EM) wave propagation in CNTs strongly different copared to transission lines, ade of usual bulk etals. For description of electroagnetic properties of etallic CNTs, very often the odel of ipedance cylinder and effective boundary condition is used 9. The odel of ipedance cylinder takes into account quantu properties of CNTs via the coplex surface frequency-dependent conductivity. This odel was applied for theoretical study of CNT transission lines and interconnects. These structures are coposed of closely packed bundles of parallel identical etallic CNTs. It was applied for studying two-diensional periodic arrays of single wall etallic CNTs.. MATERIALS AND METHODS: One odels a single wall carbon nanotube as an infinitesially thin and infinitely long cylindrical shell with a radius a. One assues that the valence electrons can be considered as free electron gas distributed uniforly over the cylindrical surface. Let the density per unit area be n0. One uses cylindrical coordinate r=(ρ,φ,z). Consider an electroagnetic wave with frequency ω propagating along the nanotube z-axis. The hoogeneous electron gas will be perturbed by the electroagnetic wave and can be regarded as a charged fluid with velocity field u(rs,t) and the perturbed density (per unit area) n(rs,t). rs=(φ,z) is the coordinate of a point at the cylindrical surface of the nanotube. Velocity field u has only tangential coponents to the nanotube surface. Based on the linearized hydrodynaic odel 30, the electronic excitations on the cylindrical surface can be described to the continuity equation 09

4 n ( r S, t) n0. u ( rs, t) 0 t () And the oentu balance equation is given by u ( rs, t) e E ( r, t ) n ( r, t ) ( n ( r, t )) t n n e S S S 0 0 () E Eze z E e Where n is the tangential coponent of the electroagnetic field. is the charge density polarization of the electron gas, e is the electronic charge and e is the electron e z e ass. Here z only differentiates tangentially to the nanotube surface. The first ter on the right hand side of equation () is the force on electrons on the nanotube due to the tangential coponent of the electric field the second and third ters ay be regarded as parts of the internal interaction force in the electron gas. Here α=(vf /) is the ( n0a ) v speed of propagation of density disturbances in the electron gas with VF= the Feri velocity of the D electron gas and ( av ) / 4 being describes single electron a excitations in the electron gas. Here v and are the Bohr radius and Bohr velocity respectively. We have neglected the second and third ters in the calculation as it was neglected in the works of Yannouleas etal 5. The electric field vector E(r,t) and the agnetic field vector B(r,t) can be expanded in the following Fourier fors E(r,φ,z,t)= dqe ( r, q) e i qz t ( ) (3) B(r,φ,z,t) dqb ( r, q) e i qz t ( ) (4) Using Maxwell s equations, one can obtain the following Helholtz equations for the z- coponents Ez and Bz of the expanding coefficients Ez and Bz 0

5 d Ez dez ( ) Ez 0 dr r dr r (5) And d Bz dbz ( ) Bz 0 dr r dr r (6) Where q k, k c 7(a) k is wave nuber and c is velocity of light. We have assued that the propagation of electroagnetic waves are in the infrared regie so that k q. By eliinating the velocity field u(r,t), one can obtain the following equations fro equation() and () n ( r, t) en. E ( r, t) n ( r, t) ( n ( r, t)) s 0 t e s s s (8) Upon solving equation (8) by eans of space-tie Fourier transfors for the induced density n(rs,t) on the cylindrical surface, one finds n (,, ) ( ) i qz t z t dqn q e ( ) 9(a) Where With en N i q E 0. e W q qe z ( ) e a 9(b) 9(c) W q q 4 9(d) To solve equations (5) and (6), one has to provide appropriate boundary conditions. With the induced density, these boundary conditions can be written as E ( a) E ( a) r ra r ra en 0 0 0(a)

6 And ( E ) ( E ) 0 ra ra B ( a) B ( a) 0 r ra r ra 0(b) 0(c) where ε0 is the perittivity of free space. Equation 0(a) indicates that due to the polarization of the electron gas on the nanotube surface, the radial coponent of the electric field is discontinuous at the cylinder at r=a. Dispersion relation of TM-ode without including electron energy-band effects For the TM ode, the longitudinal agnetic field is zero, ie. BZ=0. Fro equation (5), the longitudinal electric field can be expressed by E ( r) C I ( r),( r a) z () And E ( r) D K ( r),( r a) z () Siilarly, with Maxwell equations, we find the expressions of E E r, E and as follows () q de () z r Er r i dr q E ( r) E ( ) z r r (3) (4) And q B ( r) ( )( ) E ( r) r r z (5) In this case, the Fourier Coefficent N of the induced density is written as en0 N i ( qez E ) W a e (6) Cobining the boundary conditions (Equations 0(a), 0(b) & 0(c)) with equations ()-(6), the dispersion relation of TM ode can be written as c a c a ( ) ( )

7 ( a) p ( a ) I ( ) ( ) a K a (7) With the diensionless variable frequency case can be reduced to y p and x a, the dispersion relation in the low y ( x ) ( x ) ( x ) I ( ) ( ) x K x x (8) The results are shown in table T, T and T3 in this paper. Dispersion relation of TM-ode including electron energy-band effects Here, one considers both zig-zag (, 0) and archair (, ) nanotubes as infinitesially thin and infinitely long cylindrical shells of radius rc with its axis along the z-direction. CNT consists of π-electrons superiposed with equlibriu densities (per unit area) n0. One assues that in equlibriu the π-electron fluid has no velocity and n is the perturbed density (per unit area) of fluid, produced by the π-electrons theselves under the action of the electric field generated by the fixed positive ions of the lattice. Here, one considers the surface plason waves with TM odes. The tube radius of the CNT is given by a 0 rc n n (9) a Where and n are integers, 0 3b0is the lattice constant of the graphite sheet and b0.4a 0 is the distance between the nearest-neighboring carbon atos. A SWCNTs is n 3q etallic if, where q=0,,, The archair nanotubes are always etallic, whereas zig-zag nanotubes are etallic only if =3q with q=,--- The dispersion equation for TM ode with electron energy-band effects can be written as ( i ) ( ) r ( ) I ( r ) K ( r ) p c c c r c r c (0) Where I(x) and K(x) are odified Bessel s functions, α=vf / is the speed of propagation of density disturbances in a unifor D hoogeneous electron fluid. γ is friction coefficent. 3

8 c region q. q en p [ ] r 0 c 0 eff c, c is light speed. The paraeter κ is the real quantity in the is the eigen-frequency of the π-electron gas layer in etallic SWCNTs. The solution of equation (0) yield coplex frequencies r ii. It ay be i r i observed that the iaginary part will be. Now by writing, the solution for finite daping will be of the for [ ( ) r ( ) I ( r ) K ( r ) ] i 4 p c c c r c r c () The friction coefficent 3 is the inverse of the relaxation tie τ=3x0 - s. The dispersion characteristics of the surface waves in the syste depend upon nanotube geoetry. This includes the radius and chiral angle of the nanotube. It also depends on the wave nuber, angular oentu and the friction coefficent. It is observed that by increasing friction coefficent, the dispersion curves shift to lower frequencies, then one set γ=0. At this stage, fro equation (), one can see for investigation the dispersion characteristics of the zig-zag and archair SWCNTs. One has to give the values of n0/eff. The paraeter n0/eff takes into account the influence of the atoic crystal field. By using the seiclassical odel of the π-electron dynaics, one obtains the following estiation n 0 eff v F rc () Where vf is the velocity of the electrons at the Feri level v F 3 b 0 0 (3) Here γ0 is the characteristic energy of the graphene lattice (γ0 =.7-3eV), b0 is the distance3 between the nearst-neighbouring carbon atos b0=.4a 0. The value of vf coes out = (0.9-) x0 6 /s. The equation (3) holds for zig-zag nanotubes with =3q<60, for archair nanotubes with <50 and for chairl nanotubes with n+=3q. In the range of validity of equation (3) the paraeter decreases as the nanotube radius increases. 4

9 To see clearly the nergy band effects on the dispersion relation of the surface waves in SWCNTs, one takes the long and short wavelength liits of the equation (). For r c, using the well known asyptotic expressions 3, we have I ( x) e x x 4(a) x K( x) e x 4(b) (with finite ) The dispersion relation can be written approxiately as evf 0 r c (5) This shows that earliar the dispersion relation is independent of the geoetrical effects of the tube, the right hand side of equation (5) the dispersion relation depends strongly on the radius of the tube. It appears as the nanotube radius rc increases the value of ω decreases. In the r 0 opposite liit c, where the phase velocity of the surface plason is coparable to the velocity of light, surface plason oscillations couple with the electroagnetic wave and retardation effects are present. Retardation effects in low-diensional plasons were investigated 33. If one neglects the retardation effects by using well known expressions of Bessel functions I x ( x) ( ) ( ) 6(a) And K ( ) ( x) ( ) x ln.3 K0( x) x (for 0 ) 6(b) (for =0) 6 (c ) Then, one obtains for =0 4evF.3 0 rc [ 0, 0] [ ln( )] (8) 5

10 This is a quasiacoustic ode and for 0, one gets evf r c 0 r c (9) 3. RESULTS AND DISCUSSION: In this paper, using the theoretical foralis of Li Wei etal. and Afshin Moradi 34, we have evaluated the diepersion relation (ω/ωp) for T odes as a function of variable κa without including electron energy band effects and also including electron energy band effects respectively. We have evaluated dispersion relation of diensionless frequency (ω/ωp) for TM ode as a function of diensionless paraeter κa for nanotube diaeter a=5n and different values of. The evaluated results are shown in table T. Fro our evaluate results, it appears that the character of dispersion relation of the TM-ode is quite different fro that of TEode. Here, the values of (ω/ωp) decreases as a function of κa for all values of. At κa=0, its value is large for all values of starting fro =0 to =4. Afterwards, it decreases and decreases very sharply. After κa=6 decrease is slower for all values of. In another calculation, we have evaluated (ω/ωp) as a function of κa for different values of nanotube diaeter a for fixed values of =0. The results are shown in table T. Fro our evaluated results, we observe that the value of (ω/ωp) first decreases and then increases and increases very sharply. The value decreases upto value of κa=5 and then increases. The increase is large for a=n and sall for a=5n. It is also noticed that unlike the TE-ode, the dispersion relation of TM- ode does not approach to the well-known dispersion relation fot the D electron gas in the liit κa. In table T3, we have evaluated (ω/ωp) as a function of κa for fixed nanotube radius a=5n and =0 for three cases (a) 0, 0 (b) 0, 0 (c) 0 using equation (8).Fro our evaluated results, it appears that (ω/ωp) is large in the case of (a) for all values of κa starting fro 0 to 60 and very sall in the case of (c) for all values of κa. These results give the influence of the internal interaction forces of the electron gas on the dispersion relation of TM-odes. On coparing this calculation with TE-odes, it can be seen that the internal interaction forces play an iportant role on the dispersion relation of TM-odes. If these forces are not included, the frequency will decrease rapidly as the wave nuber increases 35. The above three calculations were perfored without including the electron energy band effects. We have included electron energy band effects for zig-zag (, o) and archair (, ) nanotubes. These are considered as infinitely thin and infinitely long cylindical shells of radius rc with its axis along the z-direction.the dispersion relation including electron energy band effects 6

11 were calculated using equations (5), (8) and (9). The results are shown in table T4, T5, T6, T7 and T8. In table T4, wehave shown the evaluated results of dispersion curve ω (ev) as a function of q (A 0- ) of surface waves for different nanotube geoetries for =0 and γ=0. This gives the electron energy band effects on the dispersion relation of plason waves in the syste. These results were obtained for zig-zag (, 0), (7, 0) and archair (9, 9), (5, 5) nanotubes respectively. Our theoretically evaluated results indicate that ω (ev) increases with q (A 0- ) for both varities of nanotubes. The value is large foe zig-zag (, 0) and sall for archair (5, 5) nanotubes. In table T5, we repeated the above calculation for = and γ=0 and siilar results were obtained. The value is large or zig-zag (, 0) and sall for archair (5, 5) nanotubes. The radius of zig-zag nanotube is rc=.056n and for archair rc=.07n. This indicates that behavior of plason wave is not sensitive to the types of etallic nanotubes with sae radius. In table T6, we have shown the results of ω (ev) as a function of q (A 0- ) for three values of electron bea velocities by considering the expression ω=vq. We have taken zig-zag nanotube (7, 0) and =0, γ=0. The three electron bea velocities are v=4x0 6 /s, v=.5x0 6 /s and v=0.93x0 6 /s. Our theoretical results indicate that ω (ev) as a function of q (A 0- ) increases very fast for v=4x0 6 /s and slow for v=0.93x0 6 /s. The velocity of the electron bea is equal to phase velocity of the surface plason odes. The electron bea is in synchronization with the surface wave and they interact with each other and instability occurs between the. This also indicates that surface waves in the syste can only be excited by applying soe relativistic electron bea which speed is about 0 6 /s. In table T7, we have shown the evaluated results of Ez/Eoz as a function of (r/rc) for =0, = and = for vacuu (r<<rc). This gives the surface TM ode of a nanotube as a function of radial coordinate r. Our theoretically evaluated results indicate that (Ez/Eoz) increases with (r/rc). The value is large for =0 and sall for =. The value becoes axiu at (r/rc)=.0 for each value of. In table T8, we have repeated the calculation for vacuu(r>>rc). Here, (Ez/Eoz) decreases with (r/rc) for each value of. The value decreases fro (r/rc) =.0 to.0 for each. Soe ore works 36-5 in this field also reveals the siilar behavior. 4. CONCLUSION: Fro the above theoretical investigation and analysis, we coe across the following conclusion Dispersion relation of TM ode is quite different fro TE-ode. TM-ode does not approach to well-known dispersion relation of the D-electron gas in the liit a. The internal interaction forces play an iportant role on the dispersion relation for the TMode. If these forces are not included the frequency will decrease rapidly as the wave nuber increases. 7

12 If one does not include electron energy band effects then linearzied hydrodynaic odel along with Maxwell equations can be used to study the dispersion relation of TM-ode. The electron energy band effects can be studied with the help of zig-zag (, 0) and archair (, ) nanotubes. The conduction electron of the syste are odelled by an infinitesinally thin layer of free electron gas which is described by eans of the seiclassical kinetic theory of the electron dynaics. The propagation of surface plason wave in etallic single-walled carbon nanotube is odelled within the fraework of classical electrodynaics. The results obtained in this paper ake one s to believe that the hydrodynaic theory with seiclassical odel is appropriate for studies of plason oscillations in CNTs for different nanotubes geoetries. This also indicates that it can be a good theoretical foralis of carbon nanotube as optical nano waveguides. ( ) Table T: An evaluated result of the dispersion relation p of TM-ode for carbon nanotube with radius a=5n and =0,,, 3 and 4 as a function of κa. κa ( ) < p =0 = = =3 =

13 ( ) TableT: An evaluated result of the dispersion relation p of TM-ode for carbon nanotube with radius a=, 5, 0 and 5n for = as a function of κa. κa ( ) p (=) a=n a =5n a =0n a=5n

14 TableT3: An evaluated result of the dispersion relation ( ) p of TM-ode for a=5n and =0 for (a) 0 (b) 0, 0 (c) 0 as a function of κa. κa ( ) p ( a=5n and =0) (a) 0 (b) 0, 0 (c) 0 0,

15 Table T4: An evaluated results of dispersion relation ω (ev) as a function of wave nuber q(a - 0 ) for different nanotube geoetries for =0 and γ=0. This shows the electron energ band effects on the dispersion relation of plason waves in the syste q(a -0 ) < ω (ev) ( =0 and γ=0) Zig-Zag(,0) Zig-Zag(7,0) Archair(9,9) Archair(5,5) ,

16 Table T5: An evaluated results of dispersion relation ω (ev) as a function of wave nuber q(a - 0 ) for different nanotube geoetries for = and γ=0. This shows the electron energ band effects on the dispersion relation of plason waves in the syste q(a -0 ) < ω (ev) ( =0 and γ=0) Zig-Zag(,0) Zig-Zag(7,0) Archair(9,9) Archair(5,5) ,

17 Table T6: An evaluated results of dispersion curve ω (ev) as a function of wave nuber q(a -0 ) for different values of electron bea velocities v keeping =0 and γ=0 fpr zig-zag nanotube (7,0) writing ω=vq q(a -0 ) <--ω (ev) ( =0 and γ=0) (Zig-zag(7,0)) V=4x0 6 /s V=.5x0 6 /s V=0.93x0 6 /s

18 Table T7: An evaluated results of Ez/E0z as a function of r/rc for =0, = and = for vacuu (r<<rc). This gives the surface TM odes of a nanotube as a function of radial coordinate r. (r/rc) Ez/E0z (Vacuu r<<rc) =0 = =

19 Table T8: An evaluated results of Ez/E0z as a function of r/rc for =0, = and = for vacuu (r>>rc). This gives the surface TM odes of a nanotube as a function of radial coordinate r. (r/rc) Ez/E0z (Vacuu r>>rc) =0 = =

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