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Model calculations of electric forces acting on carbon nanotube tip in DC-plasma sheath
Diamond and Related Materials 13 (2004) 503–506
Model calculations of electric forces acting on carbon nanotube tip in DC-plasma sheath a,b,c a ˇ ´ a, Ch. Taschner ¨ ˇ a,b,*, P. Spatenka J. Blazek , F. Pacal , A. Leonhardta a Institute of Solid State and Materials Research Dresden, D-01171 Dresden, Postfach 270016, Germany ˇ ´ 31, CZ-370 05 Ceske´ Budejovice, Czech Republic University of South Bohemia, Group of Plasma Physics and Chemistry, Branisovska c ´ Technical University Liberec, Department of Material Sciences, Halkova 6, CZ-461 17 Liberec, Czech Republic
b
Received 7 January 2003; received in revised form 5 August 2003; accepted 5 December 2003
Abstract Numerous Øexperimental investigations indicate the necessity of negative bias for low-pressure CVD of aligned carbon nanotubes. Based on the experimentally determined electron density in the dual hot filamentyDC plasma deposition system the electrical field close to the substrate was calculated. Taking into account the field enhancement in the vicinity of the CNTs the force acting on their tips is determined. The calculated force has been found to exceed the weight of the droplet by four orders of magnitude. The computations have shown only a weak dependence between the electrical forces and the droplet shape. 䊚 2003 Elsevier B.V. All rights reserved. Keywords: Nanotubes; Bias growth; Electric field
Carbon nanotubes (CNTs) have attracted a great deal of research effort since their discovery in 1991 w1x. Their unique structural, chemical, electrical and mechanical properties potentially make them to be an appropriate material for a variety of applications, such as field emission sources, nanoelectronic devices or mechanical reinforcement in composite materials w2,3x. Hot filament chemical vapor deposition (HFCVD), originally established for diamond film deposition, has also been found to be an appropriate tool for deposition of CNTs, even for industrial applications w4x. The combination of HFCVD with additional DC plasma is the favored technique for deposition of highly aligned CNTs w5– 10x. Although many authors have reported that biasing the substrate is necessary for the aligned growth of CNTs in low-pressure CVD systems, the knowledge concerning the role of the electric field on the aligned growth is incomplete. Only a few attempts have been made to explain the mechanism of the oriented growth andyor to estimate the field close to the nanotube tip. Han et al. w11x reported the oriented growth, when the bias *Corresponding author. Tel.: q420387773052; 420387312194. ˇ E-mail address: [email protected] (J. Blazek).
fax:
q
exceeds 550 V. Yu et al. w12x assumed that the charged particles form bonds along the electric field direction. Chen et al. w13x hypothesized that the catalytic particles are pulled in the direction of the electric field. Bower et al. w14x tried to estimate the electric field from the length of the nanotubes and the self-bias in the microwave discharge. They assessed the magnitude of the field to be 0.1 Vymm. Similar values of electric field strength in a DC PECVD system, estimated from the DC voltage of the substrate and collisionless sheath thickness, have been reported by Chhowalla et al. w15x. Assuming a uniform potential drop in the sheath Maiti et al. w16,17x calculated the electric field acting on the nanotube tip in the arc discharge. Tanemura et al. w18x calculated the electric force on a cylindrical tip with a diameter of 40 nm assuming the electrical field used for CNT emitter tips. To obtain a more realistic view of the electric conditions close to the nanotube tip during their growth under a DC bias, we performed a detailed model calculation of the electric field in the low-pressure collisional plasma sheath above the DC-biased substrate holder, including the field enhancement in the vicinity of the CNTs. The force acting on their tips is also calculated.
0925-9635/04/$ - see front matter 䊚 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.diamond.2003.12.009
ˇ et al. / Diamond and Related Materials 13 (2004) 503–506 J. Blazek
504
Assuming a homogeneous large substrate area we confine our calculation in the first approximation to the one-dimensional planar geometry. The dependence of the electric field on the distance x from the substrate is described by the Poisson and continuity equations dE en s dx ´0
B
CEsy D
dw E F dx G
(1)
envsyJ0 with boundary conditions
wŽ0.syUs wŽs.s0, EŽs.s0
(2)
where E and w are the electric intensity and potential, respectively, n and v are the ion number density and ion velocity, respectively, and s is the sheath thickness. The x-axis is oriented outward from the cathode. The discharge current density J0 and the sheath bias Us are the values to be determined experimentally. The system of equations is completed by an equation connecting the ion motion with the electric field. For the typical pressure of 10 mbar in our experiments the collisional ion drift motion must be considered. Supposing that the ion thermal velocity is negligible in comparison with the drift velocity, according to Refs. w19,20x the ion mobility is taken in the form 2 e mis Ø )v) p M li
(3)
where the ion mean free path li is approximately constant. The computations were carried out using experimental data typically used for the deposition of aligned nanotubes. The nanotubes were grown in a dual hot filamenty DC plasma reactor at a pressure of 10 mbar from a hydrogenymethane mixture. The electron number density and electron temperature determined from the probe measurements in the plasma bulk were nbs2=1017 my3 and Tes0.8 eV, respectively. The sheath bias of Uss650 V was approximately equal to the voltage between the electrodes. The ion mean free path was estimated from comparing the experimentally determined electron current density J0 with the theoretical expression involving the ion density and ion velocity at
Fig. 1. The electric field strength in the planar cathode sheath region for the bias of y650 V.
the sheath edge. All experimental details are described elsewhere w21x. Fig. 1 illustrates the computed electric field inside the sheath. The values of sheath thickness and electric field strength close to the cathode surface computed for our experimental conditions were 1.5 mm and 7.2=105 Vy m, respectively. The true value of the field strength at the surface is approximately two times higher than the field strength calculated from the sheath thickness and the DC bias, assuming uniform potential drop across the sheath. Close to the nanotube tip the electric intensity is distorted and stronger in comparison with the intensity calculated for purely planar geometry because of the presence of the metallic droplet. The Poisson equation in the vicinity of the nanotubes was solved for the ion charge density taken from the planar model and under the assumption that the nanotubes as well as the substrate are conductive and that, therefore their surface is equipotential with the cathode. At relatively large distances from the nanotubes, we assumed an undisturbed planar intensity. The computations were performed at various radii and distances between the neighboring nanotubes. It has been found, that the electric field is amplified at their tips and is fully screened in the space among them. For the spherical tip with the radius of 20 nm the factor of maximum field enhancement gsEy E0 is approximately 2.5 (Fig. 2). This value is substantially lower than the value obtained by Maiti et al. w16,17x or Edgcombe and Valdre` w22x. As follows from our preliminary calculations, the enhancement factor strongly depends on the distance between the individual nanotubes. If the nanotubes are close to each other, they start to form a virtual cathode resulting in a strong
ˇ et al. / Diamond and Related Materials 13 (2004) 503–506 J. Blazek
505
enhancement of the local electrical field at the tip but the computations showed, that the force acting on the tip of the nanotube depends only weakly on the radius and shape of the catalyst. The reason is in integrating effects and could be in particular cases found analytically. It has been shown in Ref. w23x, that the electrical forces acting on an isolated needle placed normally to the conductive plane are in the ‘slender body approximation’ independent on a detailed shape of its ends. The more detailed analysis will be published in our next article. Acknowledgments This work has been undertaken in cooperation with IFW Dresden and University of South Bohemia under the project DFG, contract No. 4-7531.50-04-823-00y18. Two of us (P.S and J.B.) acknowledge partial support by the projects GA CR No. 201y00y1592, OC 527.60 and project MSM CR No.124100004. Fig. 2. Enhancement factor gsEyE0 of the electric field close to the nanotube tip of the radius of 20 nm. CNTs are placed at distances 70 nm each from other, the ‘planar’ value of intensity E0s7.2=105 Vym.
decrease of both the enhancement factor and its dependence on the nanotube height. Our estimation of the force acting on an 80 nm high nanotube indicates an enlargement by a factor of 20 with an increase in distance from 70 nm to 1 mm, when this nanotube may be treated as separate. Higher values of the enhancement factor obtained by the above authors are due to the neglecting the ‘screening effect’ of neighboring nanotubes. This screening effect is more pronounced for higher nanotubes. In connection with the hypothesis that the CNTs alignment is caused by the electric field we also estimated the magnitude of the electric force acting on the nanotube tip in a direction perpendicular to the surface. This force is Fels
´0 2
| E cosu dS 2
(4)
S
where E is the intensity just above the surface, S is the outer surface of the catalyst tip and u is the angle between the normals of curved and planar surfaces. The resulting value of the force Felf1=10y14 N, acting on the CNTs of 20-nm radius placed at distances of 70 nm from each other is approximately four order of magnitude higher than the weight of the cobalt droplet and is equal to the weight of the amorphous carbon cylinder of the same radius and height of 350 mm. Due to its relatively huge value this force could be responsible for CNTs alignment. The calculations were also carried out for elliptically shaped droplets. Sharpening of the tube tip results in an
References w1x S. Iijima, Nature 354 (1991) 56. w2x T.W. Ebbesen, Carbon Nanotubes: Preparation and Properties, Chemical Rupper Corp, Boca Raton, FL, 1997. w3x S. Subramoney, Adv. Mater. 10 (1998) 1157. w4x L. Marty, V. Bouchiat, A.M. Bonnot, M. Chaumont, T. Fournier, S. Decossas, et al., Microelectron. Eng. 61–62 (2002) 485. w5x Z.F. Ren, Z.P. Huang, J.W. Xu, J.H. Wang, P. Bush, M.P. Siegal, et al., Science 282 (1998) 1105. w6x J.H. Han, B.S. Moon, W.S. Yang, J.B. Yoo, C.Y. Park, Surf. Coat. Technol. 131 (2000) 93. w7x Z.P. Huang, J.W. Xu, Z.F. Ren, J.H. Wang, M.P. Siegal, P.N. Provencio, Appl. Phys. Lett. 73 (1998) 3845. w8x Y. Chen, L.P. Guo, S. Patel, D.T. Shaw, J. Mater. Sci. 35 (2000) 5517. w9x Y. Liao, C.H. Li, Z.Y. Ye, C. Chang, G.Z. Wang, R.C. Fang, Diamond Relat. Mater. 9 (2000) 1716. w10x X.D. Zhu, M. Hu, R.J. Zhan, X.H. Wen, H.Y. Zhou, J. Zuo, et al., J. Phys. D-Appl. Phys. 31 (1998) 3327. w11x J. Han, W.S. Yang, J.B. Yoo, C.Y. Park, J. Appl. Phys. 88 (2000) 7363. w12x J. Yu, X.D. Bai, J. Ahn, S.F. Yoon, E.G. Wang, Chem. Phys. Lett. 323 (2000) 529. w13x Y. Chen, L. Guo, S. Patel, D.T. Shaw, J. Mater. Sci. 35 (2000) 5517. w14x C. Bower, W. Zhu, S. Jin, O. Zhou, Appl. Phys. Lett. 77 (2000). w15x M. Chhowalla, K.B.K. Teo, C. Ducati, N.L. Rupesinghe, G.A.J. Amaratunga, A.C. Ferrari, et al., J. Appl. Phys. 90 (2001) 5308. w16x A. Maiti, C.J. Brabec, C.M. Roland, J. Bernholc, Phys. Rev. Lett. 73 (1994) 2468. w17x A. Maiti, C.J. Brabec, C.M. Roland, J. Bernholc, Phys. Rev. B 52 (1995) 14 850. w18x M. Tanemura, K. Iwata, K. Takahashi, Y. Fujimoto, F. Okuyama, H. Sugie, et al., J. Appl. Phys. 90 (2001) 1529. w19x B.M. Smirnov, Physics of weakly ionized gases, Mir, Moscow 1981. w20x V.A. Godyak, N. Sternberg, IEEE Trans. Plasma. Sci. 18 (1990) 159.
506
ˇ et al. / Diamond and Related Materials 13 (2004) 503–506 J. Blazek
ˇ w21x Ch. Taschner, ¨ ´ F. Pacal, A. Leonhardt, P. Spatenka, R. Kaltofen, A. Graff, Synthesis of aligned carbon nanotubes by DC plasmaenhanced hot filament CVD, Plasma Source Science and Technol. – accepted.
w22x C.J. Edgcombe, U. Valdre, ` Solid State Electron. 46 (2001) 857. w23x G. Taylor, Proc. Roy. Soc. Lond. A 313 (1969) 453, Appendix by M.D. Van Dyke.
Model calculations of electric forces acting on carbon nanotube tip in DC-plasma sheath a,b,c a ˇ ´ a, Ch. Taschner ¨ ˇ a,b,*, P. Spatenka J. Blazek , F. Pacal , A. Leonhardta a Institute of Solid State and Materials Research Dresden, D-01171 Dresden, Postfach 270016, Germany ˇ ´ 31, CZ-370 05 Ceske´ Budejovice, Czech Republic University of South Bohemia, Group of Plasma Physics and Chemistry, Branisovska c ´ Technical University Liberec, Department of Material Sciences, Halkova 6, CZ-461 17 Liberec, Czech Republic
b
Received 7 January 2003; received in revised form 5 August 2003; accepted 5 December 2003
Abstract Numerous Øexperimental investigations indicate the necessity of negative bias for low-pressure CVD of aligned carbon nanotubes. Based on the experimentally determined electron density in the dual hot filamentyDC plasma deposition system the electrical field close to the substrate was calculated. Taking into account the field enhancement in the vicinity of the CNTs the force acting on their tips is determined. The calculated force has been found to exceed the weight of the droplet by four orders of magnitude. The computations have shown only a weak dependence between the electrical forces and the droplet shape. 䊚 2003 Elsevier B.V. All rights reserved. Keywords: Nanotubes; Bias growth; Electric field
Carbon nanotubes (CNTs) have attracted a great deal of research effort since their discovery in 1991 w1x. Their unique structural, chemical, electrical and mechanical properties potentially make them to be an appropriate material for a variety of applications, such as field emission sources, nanoelectronic devices or mechanical reinforcement in composite materials w2,3x. Hot filament chemical vapor deposition (HFCVD), originally established for diamond film deposition, has also been found to be an appropriate tool for deposition of CNTs, even for industrial applications w4x. The combination of HFCVD with additional DC plasma is the favored technique for deposition of highly aligned CNTs w5– 10x. Although many authors have reported that biasing the substrate is necessary for the aligned growth of CNTs in low-pressure CVD systems, the knowledge concerning the role of the electric field on the aligned growth is incomplete. Only a few attempts have been made to explain the mechanism of the oriented growth andyor to estimate the field close to the nanotube tip. Han et al. w11x reported the oriented growth, when the bias *Corresponding author. Tel.: q420387773052; 420387312194. ˇ E-mail address: [email protected] (J. Blazek).
fax:
q
exceeds 550 V. Yu et al. w12x assumed that the charged particles form bonds along the electric field direction. Chen et al. w13x hypothesized that the catalytic particles are pulled in the direction of the electric field. Bower et al. w14x tried to estimate the electric field from the length of the nanotubes and the self-bias in the microwave discharge. They assessed the magnitude of the field to be 0.1 Vymm. Similar values of electric field strength in a DC PECVD system, estimated from the DC voltage of the substrate and collisionless sheath thickness, have been reported by Chhowalla et al. w15x. Assuming a uniform potential drop in the sheath Maiti et al. w16,17x calculated the electric field acting on the nanotube tip in the arc discharge. Tanemura et al. w18x calculated the electric force on a cylindrical tip with a diameter of 40 nm assuming the electrical field used for CNT emitter tips. To obtain a more realistic view of the electric conditions close to the nanotube tip during their growth under a DC bias, we performed a detailed model calculation of the electric field in the low-pressure collisional plasma sheath above the DC-biased substrate holder, including the field enhancement in the vicinity of the CNTs. The force acting on their tips is also calculated.
0925-9635/04/$ - see front matter 䊚 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.diamond.2003.12.009
ˇ et al. / Diamond and Related Materials 13 (2004) 503–506 J. Blazek
504
Assuming a homogeneous large substrate area we confine our calculation in the first approximation to the one-dimensional planar geometry. The dependence of the electric field on the distance x from the substrate is described by the Poisson and continuity equations dE en s dx ´0
B
CEsy D
dw E F dx G
(1)
envsyJ0 with boundary conditions
wŽ0.syUs wŽs.s0, EŽs.s0
(2)
where E and w are the electric intensity and potential, respectively, n and v are the ion number density and ion velocity, respectively, and s is the sheath thickness. The x-axis is oriented outward from the cathode. The discharge current density J0 and the sheath bias Us are the values to be determined experimentally. The system of equations is completed by an equation connecting the ion motion with the electric field. For the typical pressure of 10 mbar in our experiments the collisional ion drift motion must be considered. Supposing that the ion thermal velocity is negligible in comparison with the drift velocity, according to Refs. w19,20x the ion mobility is taken in the form 2 e mis Ø )v) p M li
(3)
where the ion mean free path li is approximately constant. The computations were carried out using experimental data typically used for the deposition of aligned nanotubes. The nanotubes were grown in a dual hot filamenty DC plasma reactor at a pressure of 10 mbar from a hydrogenymethane mixture. The electron number density and electron temperature determined from the probe measurements in the plasma bulk were nbs2=1017 my3 and Tes0.8 eV, respectively. The sheath bias of Uss650 V was approximately equal to the voltage between the electrodes. The ion mean free path was estimated from comparing the experimentally determined electron current density J0 with the theoretical expression involving the ion density and ion velocity at
Fig. 1. The electric field strength in the planar cathode sheath region for the bias of y650 V.
the sheath edge. All experimental details are described elsewhere w21x. Fig. 1 illustrates the computed electric field inside the sheath. The values of sheath thickness and electric field strength close to the cathode surface computed for our experimental conditions were 1.5 mm and 7.2=105 Vy m, respectively. The true value of the field strength at the surface is approximately two times higher than the field strength calculated from the sheath thickness and the DC bias, assuming uniform potential drop across the sheath. Close to the nanotube tip the electric intensity is distorted and stronger in comparison with the intensity calculated for purely planar geometry because of the presence of the metallic droplet. The Poisson equation in the vicinity of the nanotubes was solved for the ion charge density taken from the planar model and under the assumption that the nanotubes as well as the substrate are conductive and that, therefore their surface is equipotential with the cathode. At relatively large distances from the nanotubes, we assumed an undisturbed planar intensity. The computations were performed at various radii and distances between the neighboring nanotubes. It has been found, that the electric field is amplified at their tips and is fully screened in the space among them. For the spherical tip with the radius of 20 nm the factor of maximum field enhancement gsEy E0 is approximately 2.5 (Fig. 2). This value is substantially lower than the value obtained by Maiti et al. w16,17x or Edgcombe and Valdre` w22x. As follows from our preliminary calculations, the enhancement factor strongly depends on the distance between the individual nanotubes. If the nanotubes are close to each other, they start to form a virtual cathode resulting in a strong
ˇ et al. / Diamond and Related Materials 13 (2004) 503–506 J. Blazek
505
enhancement of the local electrical field at the tip but the computations showed, that the force acting on the tip of the nanotube depends only weakly on the radius and shape of the catalyst. The reason is in integrating effects and could be in particular cases found analytically. It has been shown in Ref. w23x, that the electrical forces acting on an isolated needle placed normally to the conductive plane are in the ‘slender body approximation’ independent on a detailed shape of its ends. The more detailed analysis will be published in our next article. Acknowledgments This work has been undertaken in cooperation with IFW Dresden and University of South Bohemia under the project DFG, contract No. 4-7531.50-04-823-00y18. Two of us (P.S and J.B.) acknowledge partial support by the projects GA CR No. 201y00y1592, OC 527.60 and project MSM CR No.124100004. Fig. 2. Enhancement factor gsEyE0 of the electric field close to the nanotube tip of the radius of 20 nm. CNTs are placed at distances 70 nm each from other, the ‘planar’ value of intensity E0s7.2=105 Vym.
decrease of both the enhancement factor and its dependence on the nanotube height. Our estimation of the force acting on an 80 nm high nanotube indicates an enlargement by a factor of 20 with an increase in distance from 70 nm to 1 mm, when this nanotube may be treated as separate. Higher values of the enhancement factor obtained by the above authors are due to the neglecting the ‘screening effect’ of neighboring nanotubes. This screening effect is more pronounced for higher nanotubes. In connection with the hypothesis that the CNTs alignment is caused by the electric field we also estimated the magnitude of the electric force acting on the nanotube tip in a direction perpendicular to the surface. This force is Fels
´0 2
| E cosu dS 2
(4)
S
where E is the intensity just above the surface, S is the outer surface of the catalyst tip and u is the angle between the normals of curved and planar surfaces. The resulting value of the force Felf1=10y14 N, acting on the CNTs of 20-nm radius placed at distances of 70 nm from each other is approximately four order of magnitude higher than the weight of the cobalt droplet and is equal to the weight of the amorphous carbon cylinder of the same radius and height of 350 mm. Due to its relatively huge value this force could be responsible for CNTs alignment. The calculations were also carried out for elliptically shaped droplets. Sharpening of the tube tip results in an
References w1x S. Iijima, Nature 354 (1991) 56. w2x T.W. Ebbesen, Carbon Nanotubes: Preparation and Properties, Chemical Rupper Corp, Boca Raton, FL, 1997. w3x S. Subramoney, Adv. Mater. 10 (1998) 1157. w4x L. Marty, V. Bouchiat, A.M. Bonnot, M. Chaumont, T. Fournier, S. Decossas, et al., Microelectron. Eng. 61–62 (2002) 485. w5x Z.F. Ren, Z.P. Huang, J.W. Xu, J.H. Wang, P. Bush, M.P. Siegal, et al., Science 282 (1998) 1105. w6x J.H. Han, B.S. Moon, W.S. Yang, J.B. Yoo, C.Y. Park, Surf. Coat. Technol. 131 (2000) 93. w7x Z.P. Huang, J.W. Xu, Z.F. Ren, J.H. Wang, M.P. Siegal, P.N. Provencio, Appl. Phys. Lett. 73 (1998) 3845. w8x Y. Chen, L.P. Guo, S. Patel, D.T. Shaw, J. Mater. Sci. 35 (2000) 5517. w9x Y. Liao, C.H. Li, Z.Y. Ye, C. Chang, G.Z. Wang, R.C. Fang, Diamond Relat. Mater. 9 (2000) 1716. w10x X.D. Zhu, M. Hu, R.J. Zhan, X.H. Wen, H.Y. Zhou, J. Zuo, et al., J. Phys. D-Appl. Phys. 31 (1998) 3327. w11x J. Han, W.S. Yang, J.B. Yoo, C.Y. Park, J. Appl. Phys. 88 (2000) 7363. w12x J. Yu, X.D. Bai, J. Ahn, S.F. Yoon, E.G. Wang, Chem. Phys. Lett. 323 (2000) 529. w13x Y. Chen, L. Guo, S. Patel, D.T. Shaw, J. Mater. Sci. 35 (2000) 5517. w14x C. Bower, W. Zhu, S. Jin, O. Zhou, Appl. Phys. Lett. 77 (2000). w15x M. Chhowalla, K.B.K. Teo, C. Ducati, N.L. Rupesinghe, G.A.J. Amaratunga, A.C. Ferrari, et al., J. Appl. Phys. 90 (2001) 5308. w16x A. Maiti, C.J. Brabec, C.M. Roland, J. Bernholc, Phys. Rev. Lett. 73 (1994) 2468. w17x A. Maiti, C.J. Brabec, C.M. Roland, J. Bernholc, Phys. Rev. B 52 (1995) 14 850. w18x M. Tanemura, K. Iwata, K. Takahashi, Y. Fujimoto, F. Okuyama, H. Sugie, et al., J. Appl. Phys. 90 (2001) 1529. w19x B.M. Smirnov, Physics of weakly ionized gases, Mir, Moscow 1981. w20x V.A. Godyak, N. Sternberg, IEEE Trans. Plasma. Sci. 18 (1990) 159.
506
ˇ et al. / Diamond and Related Materials 13 (2004) 503–506 J. Blazek
ˇ w21x Ch. Taschner, ¨ ´ F. Pacal, A. Leonhardt, P. Spatenka, R. Kaltofen, A. Graff, Synthesis of aligned carbon nanotubes by DC plasmaenhanced hot filament CVD, Plasma Source Science and Technol. – accepted.
w22x C.J. Edgcombe, U. Valdre, ` Solid State Electron. 46 (2001) 857. w23x G. Taylor, Proc. Roy. Soc. Lond. A 313 (1969) 453, Appendix by M.D. Van Dyke.