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Computations of local electric field and electric forces acting on carbon nanotubes in a direct current plasma sheath
Thin Solid Films 489 (2005) 291 – 295 www.elsevier.com/locate/tsf
Computations of local electric field and electric forces acting on carbon nanotubes in a direct current plasma sheath J. Bla”eka,b,*, P. Sˇpatenkaa,c, F. Pa´cala, Ch. Ta¨schnera, A. Leonhardta b
a Institute of Solid State and Materials Research Dresden, D-01171 Dresden, Postfach 270016, Germany University of South Bohemia, Department of Physics, Jerony´mova 10, CZ-371 15 Cˇeske´ Budeˇjovice, Czech Republic c Technical University Liberec, Department of Material Sciences, Ha´lkova 6, CZ-461 17 Liberec, Czech Republic
Received 15 June 2004; received in revised form 25 February 2005; accepted 22 April 2005 Available online 31 May 2005
Abstract The direct current bias has been reported as a necessary condition for aligned growth of carbon nanotubes. To clarify the mechanisms of nanotube alignment we performed numerical calculations of the electrical field in the collisional sheath and the resulting electric force acting on the metallic droplet on the nanotube tip. Based on the model calculation, dependences of the force on various parameters, e.g. distance between nanotubes, height of nanotubes and form of their tips, is discussed in detail. The theoretical model also predicts how this force depends on the direct current bias and the resistivity of the biased substrate. D 2005 Elsevier B.V. All rights reserved. Keywords: Carbon nanotubes; Plasma processing and deposition; Electrical properties and measurements
1. Introduction Carbon nanotubes (CNTs) have attracted much research effort since their discovery in 1991 [1]. Their unique structural, chemical, electrical and mechanical properties promise to make nanotubes an appropriate material for a variety of applications, such as field emission sources, nanoelectronics devices and mechanical reinforcement in composite materials [2,3]. Hot filament chemical vapor deposition, originally established for diamond film deposition, has also been found to be a suitable tool for deposition of CNTs even for industrial applications [4]. Combination of hot filament chemical vapor deposition with additional direct current (DC) plasma is a favorable technique for deposition of highly aligned CNTs [5 –10]. Although biasing the substrate as a necessity for aligned growth of CNTs has been reported by many authors, * Corresponding author. University of South Bohemia, Department of Physics, Jerony´mova 10, CZ-371 15 Cˇeske´ Budeˇjovice, Czech Republic. Tel.: +42 0387 773 052. E-mail address: [email protected] (J. Bla”ek). 0040-6090/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2005.04.084
knowledge concerning the role of the electric field on the aligned growth is still incomplete. Only a few attempts have been undertaken to explain the mechanism of the oriented growth and/or to estimate the field close to the nanotube tip. Han et al. [11] reported oriented growth when the bias exceeds 550 V. Yu et al. [12] supposed that the charged particles form bonds along the electric field direction, Chen et al. [13] expressed a hypothesis that the catalytic particles on the CNTs tips are pulled in the direction of the electric field and Tanemura et al. [14] suggested that the alignment could be caused by the electrostatic field selectively supplying positive ions to the CNTs tips. Bower et al. [15] explained the CNTs alignment in the microwave discharge as a result of electrostatic forces established in the self-bias on the substrate. They showed that the nanotubes always grow in a direction perpendicular to the local substrate surface regardless of the tilt and curvature of this surface and that switching off the plasma source leads to thermal unaligned growth. Similar conclusions were made by Chhowalla et al. [16]. They investigated the dependence of the nanotubes’ alignment on different bias voltages in a DC plasma enhanced chemical vapor deposition system.
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J. Blazˇek et al. / Thin Solid Films 489 (2005) 291 – 295
In the above-mentioned articles the electric forces acting on CNTs tips were not evaluated and only rough approximate formulae were employed to estimate the field strength on the tips. In Ref. [14] field strength was assessed from a simple approximate formula for long cylindrical objects, in Refs. [15] and [16] the magnitude of the field was estimated from the bias and thickness of a collisionless sheath. More work has been done in connection with investigation of the field emission of CNTs. Edgcombe and Valdre` [17] computed the field enhancement factor E tip / E numerically by the finite element method. Kokkorakis et al. [18] modeled the nanotube as a cylindrical array of touching spheres with the resulting potential obtained as a linear combination of the potentials produced by each of the spheres. Bonard et al. [19] modeled the shape of the nanotube with an equipotential surface of a suitably chosen geometry object with known exact solution of Laplace’s equation. But none of the methods developed in connection with the nanotube field emitters suppose the presence of plasma and some even omit the screening effect arising from the surrounding CNTs. To clarify the alignment mechanism of the CNT growth we performed model calculation of the electric field assuming a collisional plasma sheath above the DC-biased substrate holder. Based on this model we computed the induced electric forces acting on the CNT tips. The real plasma conditions as well as the electrostatic screening from the neighboring nanotubes were taken into consideration.
3-D Eqs. (1) – (3) with corresponding boundary conditions represent five scalar equations for five scalar Y unknowns U ; ni ; vi . As the location of the sheath boundary is not known, it is impossible to solve the whole system (1) – (3) directly. Fortunately, as it will be shown later, the electric field is distorted only in the close vicinity of CNT tips and the influence of the CNTs on the ion current is negligible. This fact enables us to evaluate the electric field near the nanotube tips in two steps. At first the presence of the CNTs is ignored and the 3-D system (1) –(3) is replaced by the 1-D equations describing the collisional 1D planar sheath ([21], p. 170). Secondly, the values of the ion density and electric field at the substrate obtained from the 1-D model are applied to the 3-D Poisson Eq. (1) in the vicinity of CNTs. The value of the electric field at the planar substrate is used as the boundary condition at the distance of several (e.g. 10) radii above the CNT tips. This set of equations is completed by additional assumptions and boundary conditions. Supposing conductive connection of the nanotubes with the cathode, we set the potential on the nanotubes’ surface equal to the cathode potential. For uniformly distributed nanotubes the existence of symmetry planes between neighboring nanotubes can be taken into account, thus only one nanotube is considered for the calculation. The starting values of the ion number density and ion velocity at the sheath – presheath boundary were calculated from the electron density and electron temperature measured in the plasma bulk [21,22]. The field intensity E tip at the nanotube tip determines the induced electric force
2. Basic equations Due to a high negative voltage of the cathode the electron density in the sheath is neglected and only the ion density n i is considered. The electric potential U then satisfies the Poisson equation eni l2 U ¼ : ð1Þ e0
Fel ¼
e0 2
Z S
2 Etip cosh dS
ð4Þ
In this equation, S is the surface of the nanotube tip and h is the angle between the nanotube axis and the normal to the surface.
Neglecting ionization in the sheath, the continuity equation
3. Numerical results and discussion
Y lI ni vi Þ,0
Eq. (1) has been solved with a commercial program FEMLAB (COMSOL AB, www.comsol.com), based on the finite element method. To verify the reliability of the implemented algorithm we calculated the electrical field in the vicinity of a conductive sphere placed in an originally homogeneous electric field, as the geometry of the sphere is similar to the geometry of a nanotube tip and the exact solution is known. The field around the sphere was numerically established with high accuracy, with the exception of the surface, where the error reached its maximum. Despite it the resulting numeric value of the force F el = 4.9 (computed for unit radius of the sphere and unit permittivity) in comparison with its exact value 6.3 is fully satisfactory for our purposes. The error of about 20% is partly due to the boundary condition E = E 0 defined on a
ð2Þ
holds. The equation system is completed by an equation connecting the ion velocity Y vi with the electric field Y E ¼ lU . Assuming a working pressure of about 10 mbar, the collisional ion drift motion has to be considered. For the description of the ion velocity v i we used the formula according to Ref. [20] Y vi
¼
2 eki Y E p Mi j Y vi j
ð3Þ
where M i and k i are the ion mass and ion mean free path, respectively.
J. Blazˇek et al. / Thin Solid Films 489 (2005) 291 – 295
293
motion, as the relative change of ion velocity in the potential bias DU ¨ 0.1 V is only Dv eDU , ¨1%: v 2Ekin
Fig. 1. Electric force F el versus nanotube height v, evaluated for two different distances d between neighboring cylinders.
finite cubic box surrounding the sphere and replacing the asymptotic behavior E YE 0 for rYV, where E 0 is the undisturbed field. As the geometry involving only one hemisphere is simpler, one can expect even more precise computations for CNTs. The computations of the field around CNTs were carried out for our experimental conditions described in detail elsewhere [23]. The CNTs were grown in a dual hot filament/DC plasma reactor. Hydrogen/methane mixture was used as the working gas. The working pressure was 10 mbar. The electron density and electron temperature above the substrate were determined by the Langmuir probe method. The typical values were 2 1017 m3 and 0.8 eV, respectively. The voltage of 650 V between the electrodes was equal to the sheath bias with high accuracy. The ion mean free path was estimated from a formula similar to that in [21] (Eq. (3.5.7)). The radius of 20 nm and the average distance between particular nanotubes of 70 nm were estimated from images made by scanning electron microscope (Philips XL 30). For the sake of simplicity a spherical shape for the catalytic particle on the nanotube tip is supposed. As the computations proved only slight dependence of the electric field on the ion species, we present here only the values calculated for CH3+ ions. The values of the ion density and their energy close to the cathode surface computed for the above mentioned experimental conditions are 1.8 1016 m3 and 4.6 eV, respectively. The field strength at the planar cathode surface without CNTs is 7.3 105 V/m. This value is approximately two times higher than the field strength estimated as the ratio of the DC bias and sheath thickness, assuming uniform potential drop across the sheath. Close to the nanotube tip the electric intensity is distorted and for the chosen geometry approximately three times stronger in comparison with the intensity computed for purely planar geometry. This field distortion reaches up only to several radii above the CNTs tips, i.e. 107 m, and the enhanced field is of the order 106 V/m. The distortion of the field in the presence of the nanotubes does not significantly influences the ion
ð5Þ
Actually, the change of the ion velocity together with the ion density will be smaller as the more noticeable values of the field enhancement are concentrated very closely to the CNT tips. Thus, the enhanced field does not affect the ions striking the nanotube tips. The formula (4) for the force acting on the nanotube tip in the axial direction gives the resulting value F el = 1.0 1014 N. Neglecting the field enhancement, one obtains approximately a three times lower value. The electric force is relatively large. Just for comparison, it is equal to the weight of a 360 Am high amorphous carbon cylinder of 20 nm in radius, and is four orders of magnitude higher than the weight of the cobalt droplet ending the nanotube tip. The force strongly depends on the geometrical configuration of deposited nanotubes. The dependence of the electric force on the nanotube height between particular nanotubes is shown in Fig. 1. The force reaches its maximum for the heights of the order of magnitude equal to the distance between neighboring cylinders. For lower heights the influence of the bottom predominates whereas for higher heights the bottom is screened out by neighboring nanotubes and the field remains confined above their tips. From the opposite point of view, the force strongly depends on the distances between neighboring nanotubes (Fig. 2). When the distance between the nanotubes is smaller than their height, they screen each other and the force is predominately determined by the undisturbed electrical field. As the distance increases the field strength rises together with the electrical force. The value of the force is saturated when the distance between cylinders is approximately 5– 6 times longer than their height. In such cases the value of the force is the same as for an isolated nanotube.
Fig. 2. Electric force F el versus distance d between two neighboring nanotubes (radius 20 nm, spherical tip); the limited value of the force strongly depends on the height v of the nanotube cylinder.
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Fig. 3 demonstrates the dependence of the force on the CNT’s radius. To depict the force as well as the tip area and field enhancement factor in one graph, these quantities were normalized to reference values taken at r = 20 nm. The force does not rise with the square of the radius but is constant. This is due to the compensation of the tip area increase by the decrease of the field enhancement factor. The relationship between the force and the shape of the catalytic droplet on the tip is shown in Fig. 4. The droplet was modeled as a rotational ellipsoid with defined ratio k of the longitudinal semiaxis to the CNT’s radius. The sensitivity of the force to the shape of the CNT’s tip depends on the screening effect. The obtained result suggests that for densely distributed nanotubes the force does not notably depend on other forms of the shape, including tiny protrusions on the CNT’s tip. The nanotubes in our experiment were grown on a SiOx film. Observations in a high resolution transmission electron microscope (type FEI Tecnai F30) showed this film through-etched during the pretreatment, thus we assumed the nanotube surface to be conductively connected with the cathode. The nonzero surface resistance may decrease the sheath bias and consequently the electrical force. To estimate the influence of the nonzero resistance, we expressed the undisturbed field strength from the planar model in the form E ” (U s J i)2/5, where U s and J i are the actual sheath bias and ion current density in the presence of the resistive sheath, respectively. The field intensity at the CNT’s tip is proportional to the planar value, E tip ” E, with the coefficient of proportionality given by the CNT’s geometry alone. The ion current density is a function of the electron density and electron temperature. Supposing only slight dependence of these parameters on the sheath voltage, we obtain the estimation F el ” U s4/5, or in more details RJi 4=5 Fel ,Fel0 I 1 ð6Þ Us0 where R is the electric resistance of the film per square unit and R J i is the corresponding voltage drop. The
Fig. 4. Electric force F el as a function of the ratio k = c / r, where c is the longitudinal semiaxis of the rotational ellipsoidal droplet on the CNT’s tip and r = 20 nm is its radius; the curves are computed for two different distances d between the nanotubes and two different heights v.
quantities subscripted by zero are related to the perfectly conducting substrate. The nonzero resistance between the nanotubes and the cathode reduces only the electric field and the absolute value of the force acting on the nanotube tip, but the dependences discussed above remain unchanged.
4. Conclusion The electric field in the vicinity of carbon nanotubes grown in a DC collisional plasma sheath was computed from a complex model involving real plasma parameters. Enhancement of the electrical field close to the nanotube tip does not influence the energy of ions bombarding the substrate, but strongly influences the electrical force acting on the growing nanotube. The force calculated for CNTs of 20 nm in diameter is 1.0 1014 N, which is, for instance, four orders of magnitude greater than the weight of a metal droplet embedded on the nanotube tip. Neither the radii nor the shapes of the CNT tips significantly affect the magnitude of the force, which instead strongly depends on the nanotube height and the distance between the nanotubes. The force increases with growing distance. The value of the force becomes saturated when the distance between the cylinders is approximately 5 – 6 times longer than their height. In such cases the force approaches the value as that for an insulated nanotube. The resistive film on the cathode decreases the magnitude of the electrical forces but does not influence their qualitative behavior.
Acknowledgments
Fig. 3. Electric force F el as a function of the CNT’s radius r. The distance between neighboring nanotubes is kept constant. All depicted quantities are related to their values at r = 20 nm.
This research was supported by Deutsche Forschungsgemeinschaft (DFG), project no. LE 863/9-1 and by the Ministry of Education of the Czech Republic, project no. MSM 124100004 and OC 527.60.
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References [1] S. Iijima, Nature (Lond.) 354 (1991) 56. [2] T.W. Ebbesen, Carbon Nanotubes: Preparation and Properties, Chemical Rupper Corp., Boca Raton, FL, 1997. [3] S. Subramoney, Adv. Mater. 10 (1998) 1157. [4] L. Marty, V. Bouchiat, A.M. Bonnot, M. Chaumont, T. Fournier, S. Decossas, S. Roche, Microelectron. Eng. 61 – 62 (2002) 485. [5] Z.F. Ren, Z.P. Huang, J.W. Xu, J.H. Wang, P. Bush, M.P. Siegal, P.N. Provencio, Science 282 (1998) 1105. [6] J.H. Han, B.S. Moon, W.S. Yang, J.B. Yoo, C.Y. Park, Surf. Coat. Technol. 131 (2000) 93. [7] Z.P. Huang, J.W. Xu, Z.F. Ren, J.H. Wang, M.P. Siegal, P.N. Provencio, Appl. Phys. Lett. 73 (1998) 3845. [8] Y. Chen, L.P. Guo, S. Patel, D.T. Shaw, J. Mater. Sci. 35 (2000) 5517. [9] Y. Liao, C.H. Li, Z.Y. Ye, C. Chang, G.Z. Wang, R.C. Fang, Diamond Relat. Mater. 9 (2000) 1716. [10] X.D. Zhu, M. Hu, R.J. Zhan, X.H. Wen, H.Y. Zhou, J. Zuo, C.Y. Xu, J. Phys. D: Appl. Phys. 31 (1998) 3327. [11] J. Han, W.S. Yang, J.B. Yoo, C.Y. Park, J. Appl. Phys. 88 (2000) 7363. [12] J. Yu, X.D. Bai, J. Ahn, S.F. Yoon, E.G. Wang, Chem. Phys. Lett. 323 (2000) 529.
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[13] Y. Chen, L. Guo, S. Patel, D.T. Shaw, J. Mater. Sci. 35 (2000) 5517. [14] M. Tanemura, K. Iwata, K. Takahashi, Y. Fujimoto, F. Okuyama, H. Sugie, V. Filip, J. Appl. Phys. 90 (2001) 1529. [15] C. Bower, W. Zhu, S. Jin, O. Zhou, Appl. Phys. Lett. 77 (2000). [16] M. Chhowalla, K.B.K. Teo, C. Ducati, N.L. Rupesinghe, G.A.J. Amaratunga, A.C. Ferrari, D. Roy, J. Robertson, W.I. Milne, J. Appl. Phys. 90 (2001) 5308. [17] C.J. Edgcombe, U. Valdre`, Solid-State Electron. 45 (2001) 857. [18] G.C. Kokkorakis, A. Modinos, J.P. Xanthakis, J. Appl. Phys. 91 (2002) 4580. [19] J.-M. Bonard, M. Croci, I. Arfaoui, O. Noury, D. Sarangi, A. Chaˆtelain, Diamond Relat. Mater. 11 (2002) 763. [20] B.M. Smirnov, Physics of Weakly Ionized Gases, Mir, Moscow, 1981, problem 4.5. [21] M.A. Lieberman, A.J. Lichtenberg, Principles of Plasma Discharges and Materials Processing, John Wiley & Sons, 1994. [22] V.A. Godyak, N. Sternberg, IEEE Trans. Plasma Sci. 18 (1990) 159. [23] Ch. Ta¨schner, F. Pacal, A. Leonhardt, P. Spatenka, K. Bartsch, A. Graff, R. Kaltofen, Surf. Coat. Technol. 174 – 175 (2003) 81.
Computations of local electric field and electric forces acting on carbon nanotubes in a direct current plasma sheath J. Bla”eka,b,*, P. Sˇpatenkaa,c, F. Pa´cala, Ch. Ta¨schnera, A. Leonhardta b
a Institute of Solid State and Materials Research Dresden, D-01171 Dresden, Postfach 270016, Germany University of South Bohemia, Department of Physics, Jerony´mova 10, CZ-371 15 Cˇeske´ Budeˇjovice, Czech Republic c Technical University Liberec, Department of Material Sciences, Ha´lkova 6, CZ-461 17 Liberec, Czech Republic
Received 15 June 2004; received in revised form 25 February 2005; accepted 22 April 2005 Available online 31 May 2005
Abstract The direct current bias has been reported as a necessary condition for aligned growth of carbon nanotubes. To clarify the mechanisms of nanotube alignment we performed numerical calculations of the electrical field in the collisional sheath and the resulting electric force acting on the metallic droplet on the nanotube tip. Based on the model calculation, dependences of the force on various parameters, e.g. distance between nanotubes, height of nanotubes and form of their tips, is discussed in detail. The theoretical model also predicts how this force depends on the direct current bias and the resistivity of the biased substrate. D 2005 Elsevier B.V. All rights reserved. Keywords: Carbon nanotubes; Plasma processing and deposition; Electrical properties and measurements
1. Introduction Carbon nanotubes (CNTs) have attracted much research effort since their discovery in 1991 [1]. Their unique structural, chemical, electrical and mechanical properties promise to make nanotubes an appropriate material for a variety of applications, such as field emission sources, nanoelectronics devices and mechanical reinforcement in composite materials [2,3]. Hot filament chemical vapor deposition, originally established for diamond film deposition, has also been found to be a suitable tool for deposition of CNTs even for industrial applications [4]. Combination of hot filament chemical vapor deposition with additional direct current (DC) plasma is a favorable technique for deposition of highly aligned CNTs [5 –10]. Although biasing the substrate as a necessity for aligned growth of CNTs has been reported by many authors, * Corresponding author. University of South Bohemia, Department of Physics, Jerony´mova 10, CZ-371 15 Cˇeske´ Budeˇjovice, Czech Republic. Tel.: +42 0387 773 052. E-mail address: [email protected] (J. Bla”ek). 0040-6090/$ - see front matter D 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.tsf.2005.04.084
knowledge concerning the role of the electric field on the aligned growth is still incomplete. Only a few attempts have been undertaken to explain the mechanism of the oriented growth and/or to estimate the field close to the nanotube tip. Han et al. [11] reported oriented growth when the bias exceeds 550 V. Yu et al. [12] supposed that the charged particles form bonds along the electric field direction, Chen et al. [13] expressed a hypothesis that the catalytic particles on the CNTs tips are pulled in the direction of the electric field and Tanemura et al. [14] suggested that the alignment could be caused by the electrostatic field selectively supplying positive ions to the CNTs tips. Bower et al. [15] explained the CNTs alignment in the microwave discharge as a result of electrostatic forces established in the self-bias on the substrate. They showed that the nanotubes always grow in a direction perpendicular to the local substrate surface regardless of the tilt and curvature of this surface and that switching off the plasma source leads to thermal unaligned growth. Similar conclusions were made by Chhowalla et al. [16]. They investigated the dependence of the nanotubes’ alignment on different bias voltages in a DC plasma enhanced chemical vapor deposition system.
292
J. Blazˇek et al. / Thin Solid Films 489 (2005) 291 – 295
In the above-mentioned articles the electric forces acting on CNTs tips were not evaluated and only rough approximate formulae were employed to estimate the field strength on the tips. In Ref. [14] field strength was assessed from a simple approximate formula for long cylindrical objects, in Refs. [15] and [16] the magnitude of the field was estimated from the bias and thickness of a collisionless sheath. More work has been done in connection with investigation of the field emission of CNTs. Edgcombe and Valdre` [17] computed the field enhancement factor E tip / E numerically by the finite element method. Kokkorakis et al. [18] modeled the nanotube as a cylindrical array of touching spheres with the resulting potential obtained as a linear combination of the potentials produced by each of the spheres. Bonard et al. [19] modeled the shape of the nanotube with an equipotential surface of a suitably chosen geometry object with known exact solution of Laplace’s equation. But none of the methods developed in connection with the nanotube field emitters suppose the presence of plasma and some even omit the screening effect arising from the surrounding CNTs. To clarify the alignment mechanism of the CNT growth we performed model calculation of the electric field assuming a collisional plasma sheath above the DC-biased substrate holder. Based on this model we computed the induced electric forces acting on the CNT tips. The real plasma conditions as well as the electrostatic screening from the neighboring nanotubes were taken into consideration.
3-D Eqs. (1) – (3) with corresponding boundary conditions represent five scalar equations for five scalar Y unknowns U ; ni ; vi . As the location of the sheath boundary is not known, it is impossible to solve the whole system (1) – (3) directly. Fortunately, as it will be shown later, the electric field is distorted only in the close vicinity of CNT tips and the influence of the CNTs on the ion current is negligible. This fact enables us to evaluate the electric field near the nanotube tips in two steps. At first the presence of the CNTs is ignored and the 3-D system (1) –(3) is replaced by the 1-D equations describing the collisional 1D planar sheath ([21], p. 170). Secondly, the values of the ion density and electric field at the substrate obtained from the 1-D model are applied to the 3-D Poisson Eq. (1) in the vicinity of CNTs. The value of the electric field at the planar substrate is used as the boundary condition at the distance of several (e.g. 10) radii above the CNT tips. This set of equations is completed by additional assumptions and boundary conditions. Supposing conductive connection of the nanotubes with the cathode, we set the potential on the nanotubes’ surface equal to the cathode potential. For uniformly distributed nanotubes the existence of symmetry planes between neighboring nanotubes can be taken into account, thus only one nanotube is considered for the calculation. The starting values of the ion number density and ion velocity at the sheath – presheath boundary were calculated from the electron density and electron temperature measured in the plasma bulk [21,22]. The field intensity E tip at the nanotube tip determines the induced electric force
2. Basic equations Due to a high negative voltage of the cathode the electron density in the sheath is neglected and only the ion density n i is considered. The electric potential U then satisfies the Poisson equation eni l2 U ¼ : ð1Þ e0
Fel ¼
e0 2
Z S
2 Etip cosh dS
ð4Þ
In this equation, S is the surface of the nanotube tip and h is the angle between the nanotube axis and the normal to the surface.
Neglecting ionization in the sheath, the continuity equation
3. Numerical results and discussion
Y lI ni vi Þ,0
Eq. (1) has been solved with a commercial program FEMLAB (COMSOL AB, www.comsol.com), based on the finite element method. To verify the reliability of the implemented algorithm we calculated the electrical field in the vicinity of a conductive sphere placed in an originally homogeneous electric field, as the geometry of the sphere is similar to the geometry of a nanotube tip and the exact solution is known. The field around the sphere was numerically established with high accuracy, with the exception of the surface, where the error reached its maximum. Despite it the resulting numeric value of the force F el = 4.9 (computed for unit radius of the sphere and unit permittivity) in comparison with its exact value 6.3 is fully satisfactory for our purposes. The error of about 20% is partly due to the boundary condition E = E 0 defined on a
ð2Þ
holds. The equation system is completed by an equation connecting the ion velocity Y vi with the electric field Y E ¼ lU . Assuming a working pressure of about 10 mbar, the collisional ion drift motion has to be considered. For the description of the ion velocity v i we used the formula according to Ref. [20] Y vi
¼
2 eki Y E p Mi j Y vi j
ð3Þ
where M i and k i are the ion mass and ion mean free path, respectively.
J. Blazˇek et al. / Thin Solid Films 489 (2005) 291 – 295
293
motion, as the relative change of ion velocity in the potential bias DU ¨ 0.1 V is only Dv eDU , ¨1%: v 2Ekin
Fig. 1. Electric force F el versus nanotube height v, evaluated for two different distances d between neighboring cylinders.
finite cubic box surrounding the sphere and replacing the asymptotic behavior E YE 0 for rYV, where E 0 is the undisturbed field. As the geometry involving only one hemisphere is simpler, one can expect even more precise computations for CNTs. The computations of the field around CNTs were carried out for our experimental conditions described in detail elsewhere [23]. The CNTs were grown in a dual hot filament/DC plasma reactor. Hydrogen/methane mixture was used as the working gas. The working pressure was 10 mbar. The electron density and electron temperature above the substrate were determined by the Langmuir probe method. The typical values were 2 1017 m3 and 0.8 eV, respectively. The voltage of 650 V between the electrodes was equal to the sheath bias with high accuracy. The ion mean free path was estimated from a formula similar to that in [21] (Eq. (3.5.7)). The radius of 20 nm and the average distance between particular nanotubes of 70 nm were estimated from images made by scanning electron microscope (Philips XL 30). For the sake of simplicity a spherical shape for the catalytic particle on the nanotube tip is supposed. As the computations proved only slight dependence of the electric field on the ion species, we present here only the values calculated for CH3+ ions. The values of the ion density and their energy close to the cathode surface computed for the above mentioned experimental conditions are 1.8 1016 m3 and 4.6 eV, respectively. The field strength at the planar cathode surface without CNTs is 7.3 105 V/m. This value is approximately two times higher than the field strength estimated as the ratio of the DC bias and sheath thickness, assuming uniform potential drop across the sheath. Close to the nanotube tip the electric intensity is distorted and for the chosen geometry approximately three times stronger in comparison with the intensity computed for purely planar geometry. This field distortion reaches up only to several radii above the CNTs tips, i.e. 107 m, and the enhanced field is of the order 106 V/m. The distortion of the field in the presence of the nanotubes does not significantly influences the ion
ð5Þ
Actually, the change of the ion velocity together with the ion density will be smaller as the more noticeable values of the field enhancement are concentrated very closely to the CNT tips. Thus, the enhanced field does not affect the ions striking the nanotube tips. The formula (4) for the force acting on the nanotube tip in the axial direction gives the resulting value F el = 1.0 1014 N. Neglecting the field enhancement, one obtains approximately a three times lower value. The electric force is relatively large. Just for comparison, it is equal to the weight of a 360 Am high amorphous carbon cylinder of 20 nm in radius, and is four orders of magnitude higher than the weight of the cobalt droplet ending the nanotube tip. The force strongly depends on the geometrical configuration of deposited nanotubes. The dependence of the electric force on the nanotube height between particular nanotubes is shown in Fig. 1. The force reaches its maximum for the heights of the order of magnitude equal to the distance between neighboring cylinders. For lower heights the influence of the bottom predominates whereas for higher heights the bottom is screened out by neighboring nanotubes and the field remains confined above their tips. From the opposite point of view, the force strongly depends on the distances between neighboring nanotubes (Fig. 2). When the distance between the nanotubes is smaller than their height, they screen each other and the force is predominately determined by the undisturbed electrical field. As the distance increases the field strength rises together with the electrical force. The value of the force is saturated when the distance between cylinders is approximately 5– 6 times longer than their height. In such cases the value of the force is the same as for an isolated nanotube.
Fig. 2. Electric force F el versus distance d between two neighboring nanotubes (radius 20 nm, spherical tip); the limited value of the force strongly depends on the height v of the nanotube cylinder.
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Fig. 3 demonstrates the dependence of the force on the CNT’s radius. To depict the force as well as the tip area and field enhancement factor in one graph, these quantities were normalized to reference values taken at r = 20 nm. The force does not rise with the square of the radius but is constant. This is due to the compensation of the tip area increase by the decrease of the field enhancement factor. The relationship between the force and the shape of the catalytic droplet on the tip is shown in Fig. 4. The droplet was modeled as a rotational ellipsoid with defined ratio k of the longitudinal semiaxis to the CNT’s radius. The sensitivity of the force to the shape of the CNT’s tip depends on the screening effect. The obtained result suggests that for densely distributed nanotubes the force does not notably depend on other forms of the shape, including tiny protrusions on the CNT’s tip. The nanotubes in our experiment were grown on a SiOx film. Observations in a high resolution transmission electron microscope (type FEI Tecnai F30) showed this film through-etched during the pretreatment, thus we assumed the nanotube surface to be conductively connected with the cathode. The nonzero surface resistance may decrease the sheath bias and consequently the electrical force. To estimate the influence of the nonzero resistance, we expressed the undisturbed field strength from the planar model in the form E ” (U s J i)2/5, where U s and J i are the actual sheath bias and ion current density in the presence of the resistive sheath, respectively. The field intensity at the CNT’s tip is proportional to the planar value, E tip ” E, with the coefficient of proportionality given by the CNT’s geometry alone. The ion current density is a function of the electron density and electron temperature. Supposing only slight dependence of these parameters on the sheath voltage, we obtain the estimation F el ” U s4/5, or in more details RJi 4=5 Fel ,Fel0 I 1 ð6Þ Us0 where R is the electric resistance of the film per square unit and R J i is the corresponding voltage drop. The
Fig. 4. Electric force F el as a function of the ratio k = c / r, where c is the longitudinal semiaxis of the rotational ellipsoidal droplet on the CNT’s tip and r = 20 nm is its radius; the curves are computed for two different distances d between the nanotubes and two different heights v.
quantities subscripted by zero are related to the perfectly conducting substrate. The nonzero resistance between the nanotubes and the cathode reduces only the electric field and the absolute value of the force acting on the nanotube tip, but the dependences discussed above remain unchanged.
4. Conclusion The electric field in the vicinity of carbon nanotubes grown in a DC collisional plasma sheath was computed from a complex model involving real plasma parameters. Enhancement of the electrical field close to the nanotube tip does not influence the energy of ions bombarding the substrate, but strongly influences the electrical force acting on the growing nanotube. The force calculated for CNTs of 20 nm in diameter is 1.0 1014 N, which is, for instance, four orders of magnitude greater than the weight of a metal droplet embedded on the nanotube tip. Neither the radii nor the shapes of the CNT tips significantly affect the magnitude of the force, which instead strongly depends on the nanotube height and the distance between the nanotubes. The force increases with growing distance. The value of the force becomes saturated when the distance between the cylinders is approximately 5 – 6 times longer than their height. In such cases the force approaches the value as that for an insulated nanotube. The resistive film on the cathode decreases the magnitude of the electrical forces but does not influence their qualitative behavior.
Acknowledgments
Fig. 3. Electric force F el as a function of the CNT’s radius r. The distance between neighboring nanotubes is kept constant. All depicted quantities are related to their values at r = 20 nm.
This research was supported by Deutsche Forschungsgemeinschaft (DFG), project no. LE 863/9-1 and by the Ministry of Education of the Czech Republic, project no. MSM 124100004 and OC 527.60.
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