- Email: [email protected]
Channeling of protons in double-walled carbon nanotubes in kinetic model
Nuclear Instruments and Methods in Physics Research B 267 (2009) 3133–3136
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Channeling of protons in double-walled carbon nanotubes in kinetic model Shu-Yan You, Yuan-Hong Song, You-Nian Wang * School of Physics and Optoelectronic Technology, Dalian University of Technology, Dalian 116023, China
a r t i c l e
i n f o
Article history: Available online 16 June 2009 PACS: 73.63.-b 34.50.Bw Keywords: Carbon nanotube Self-energy Stopping power
a b s t r a c t A kinetic model combined with the dielectric response theory is employed to study the electronic excitation on the nanotube walls during the channeling of protons through double-walled carbon nanotubes. Analytical expressions of the self-energy and stopping power are obtained with protons moving along the axis of the double-walled nanotubes. Calculation results show us interesting double-peak curves of the self-energy and stopping power, under strong influence of the damping factor and the special doublewalled nanotube geometry. Relatively increasing the damping factor and the chiral parameter of the outer wall can reduce the interference effects between the two walls and weaken the double-peak to one-peak shapes. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Ever since the discovery of carbon nanotubes by Iijima [1], many efforts have been made both theoretically and experimentally to investigate various aspects of nanotubes owing to their exceptional properties and great expectations for their applications. Carbon nanotubes are classified by the dual index (n, l). Two integers n and l represent the vector characterizing the pattern of rolling a planar sheet to a cylindrical cavity, with l = 0 for zigzag carbon nanotubes, l = n for armchair carbon nanotubes, and n – l for chiral carbon nanotubes, showing either metallic or semiconducting properties. The channeling of charged particles through nanotubes obtains us not only information about the surface electronic excitation, but also a new application for particles channeling and steering. In the theoretical studies, the dielectric theory has been taken as a valid theoretical tool to investigate interactions of charged particles with cylindrical structures, including fast ions and clusters moving paraxially through microcapillaries and nanocapillaries in solids [2–4] or fast electrons moving perpendicularly through metallic and semiconductor cylindrical nanowires [5]. Besides, the dielectric properties of carbon nanotubes have been investigated by means of the hydrodynamic theory, in which plasmon excitation in a thin layer of free electron gas have been deduced [6]. Also by using the two-dimensional fluid model, the self-energy and the energy loss of charged particles moving parallel to the axis of a single-walled (SWNT) or a double-walled nanotube (2WNT) have
* Corresponding author. Tel.: +86 411 84707307; fax: +86 411 84709304. E-mail address: [email protected] (Y.-N. Wang). 0168-583X/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2009.06.037
been calculated [7–9]. Moreover, two-fluid hydrodynamic model has been adopted to describe the dielectric response of r and p electron systems on SWNTs and multi-walled carbon nanotubes (MWNTs) [10,11]. Meanwhile, considering the energy band structure of the electrons on the nanotube surface, semi-classical kinetic model [12] combined with the dielectric response theory has been introduced to study the collective excitation, self-energy and energy loss, with charged particles moving inside the SWNTs and parallel with axes [13]. However, as MWNTs have shown rich plasmon spectra properties [14], the effects of the band structure of the electrons during the channeling have to be considered. In the present paper, we try to study the collective excitation in 2WNTs while charged particles move along the axis, based on the semi-classical kinetic theory. Our attention will also be focused on the self-energy and the stopping power for charged particles channeling. The cylindrical coordinates r = (q, u, z) for carbon nanotubes will be adopted, with the orientation of z along the axis of nanotubes. Atomic units (a.u.) will be used throughout this paper, unless otherwise indicated. 2. Theoretical model The present work is an extension of our previous study of SWNT [13] to the case of 2WNT. In this paper, we only take p electrons as our aim for their important effects on electronic properties. The 2WNT is regarded as infinitesimally thin and infinitely long cylindrical tubes. Consider a 2WNT consists of 2 coaxial walls with the inner radius a1 and the outer radius a2. When a charged particle goes through inside the nanotubes, we use cylindrical coordination r0 = (q0, /0, vt) for its instantaneous position, involving the velocity
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of the particle v. We employ U2 to express the potential between two walls a1 < q < a2, U3 to express the potential outside the outer wall q > a2,while the values of the total potential U1 = U0 + Uind inside the inner wall q < a1 may include the potential U0 from the moving charged particle itself and the induced potential Uind from the charge polarization on the inner surface induced by charged particles. Conforming to the work in [13], these potentials could be expanded in terms of Bessel functions as
U0 ðr; tÞ ¼
þ1 Z Q X
Q ¼ jr r0 j
þ1
p m¼1 1 Im ðjkjq< ÞK m ðjkjq> Þ;
Uind ðr; tÞ ¼
þ1 Z Q X
p
þ1
dke
ikðzv tÞþimð//0 Þ
dke
ð1Þ
ikðzv tÞþimð//0 Þ
m¼1
U1 ðr; tÞq¼a1 ¼ U2 ðr; tÞq¼a1
ð2Þ
1
Im ðjkjq0 Þ½Im ðjkjqÞBm ðkÞ þ K m ðjkjqÞC m ðkÞ
and the radial component of the displacement field keeps discontinuous at the boundary,
@ U2 @ U1 ðr; tÞq¼a1 ðr; tÞq¼a1 ¼ 4pn1 ða1 ; /; z; tÞ; @q @q @ U3 @ U2 ðr; tÞq¼a2 ðr; tÞq¼a2 ¼ 4pn2 ða2 ; /; z; tÞ; @q @q
þ1 Z Q X
p
m¼1
ð3Þ
nj ðaj ; /; z; tÞ ¼
þ1
ikðzv tÞþimð//0 Þ
dke
1
Im ðjkjq0 ÞK m ðjkjqÞDm ðkÞ;
ð4Þ
where k is the longitudinal wave number, m is the angular momentum. Also in these above equations, Q is the charge of the particle, Im(x) and Km(x) are the modified Bessel functions, and q< and q> are the smaller or larger of q and q0. Moreover, exposed to the intrusion of charged ions, the distribution functions of p electrons on the inner and outer surfaces of the 2WNT are assumed to satisfy the first-order Boltzmann kinetic equation, respectively,
@fj @f0 þ v rsj fj ¼ eEj cfj @t @p
j ¼ 1; 2;
ð5Þ
@ ^ @ ^ e/ þ @z ez ðj ¼ 1; 2Þ is the two-dimensional differenwhere rsj ¼ a1 @/ j tial operator only to each surface of the 2WNT, Ej = rUj is the electric field on the surface, f0 ðpÞ ¼ 1þexpfe1ðpÞ=kB Tg is the Fermi equilibrium distribution function of the electron gas, p is the electron’s two dimensional quasimomentum tangential to the nanotube’s surface, kB is the Boltzmann constant, T is the temperature, and e(p) is the electron energy with respect to Fermi level, which is determined by the nanotube geometry and can be obtained by using the tight-binding model [12,15,16]. Besides, fj is the perturbation of the distribution function, c is the friction coefficient owing to electron scattering on the positive charge background, which is introduced to satisfy the nonconservation of the system. In terms of a Fourier transform, the perturbation part fj can be adopted here,
fj ¼
þ1 Z Q X
p
m¼1
þ1
ikðzv tÞþimð//0 Þ
dke
fjm ðaj ; pÞ:
ð6Þ
1
Combining Eqs. (5) and (6) with Eq. (3) in which U ¼ U2 ðr; tÞq¼a2 , we can obtain
fjm ðaj ; p; k; xÞ ¼
ð9Þ
where n1(a1,/,z,t) and n2(a2,/,z,t) are the individual induced density on the inner and outer surface of the nanotube. By using the Fourier–Bessel transform, the induced density nj(aj,/,z,t) can be expressed as follows, while considering the R expression as njm ðaj ; k; xÞ ¼ ð2p4hÞ2 dpfjm ðaj ; p; k; xÞ, we obtain
and
U3 ðr; tÞ ¼
ð8Þ
U2 ðr; tÞq¼a2 ¼ U3 ðr; tÞq¼a2
1
m¼1
Im ðjkjq0 ÞIm ðjkjqÞAm ðkÞ; þ1 Z þ1 Q X ikðzv tÞþimð//0 Þ U2 ðr; tÞ ¼ dke
p
For the sake of determining the coefficients Am(k), Bm(k), Cm(k) and Dm(k), the following boundary conditions at q = aj are adopted: the potential keeps continuous at the surface of cylinder,
eUjm
m @f0 aj @p/
@f0 þ k @p z
x maj v / kv z þ ic
;
ð7Þ
where Pz and P/ are the projections of p on the axis and the direcphS for armchair, tion of the nanotube, P/ ¼ p2pffiffi3hnbS for zigzag, P/ ¼ 23nb x = kv, Ujm = Im(jkjq0)[Bm(k)Im(jkjaj) + s = 1,2 . . . ,n. And, Cm(k)Km(jkjaj)], respectively. But for v/, with p/ a quantized variable, we approximate the partial derivative with finite difference, as v/ = e[pz, p/(s + 1)] e[pz,p/].
¼
þ1 Z Q X
p
m¼1
m¼1
imðzv tÞþimð//0 Þ
dke
njm ðaj ; k; xÞ
1
þ1 Z Q X
p
þ1
þ1
imðzv tÞþimð//0 Þ
dke
Ujm vðk; m; x; aj Þ;
1
ð10Þ where v(k, m, x, aj) is the response function, which is different for zigzag and armchair nanotube. The detailed expressions of the response function can be found in [13]. Finally, the coefficients in the potential expressions in Eqs. (2)– (4) can be obtained, especially for the induced potential Uind, which allows us to pay further attention on the self-energy, Eself ¼ 12 Uind ðr; tÞjr¼r0 and the stopping power S ¼ Q @Uind@zðr;tÞ jr¼r0 , showing how the electron polarization influences the ion channeling inside 2WNTs. 3. Results and conclusions Consider a proton moving along the axis q0 = 0 of a 2WNT. It is noted that only m = 0 mode can contribute to the energy loss because of axial symmetry. In Fig. 1, we show how the self-energy and stopping power depend on proton speed v in 2WNTs. Four 2WNTs are chosen to be channelled in this figure: both walls are zigzag or armchair, or with one wall zigzag and the other one armchair. It is interesting to note that double peaks occur in the selfenergy and stopping power curves, with the narrow one in low speed region and the wide one in high speed region. In our opinion, it is owing to the double-walled structures, in which electron polarization in both walls and the interference between them contribute to the stoppings for the particle. In order to make comparison between SWNTs and 2WNTs, we also calculate the velocity dependencies of the self-energy and stopping power in SWNTs and a 2WNT, as shown in Fig. 2. Here, we take the 2WNT with inner zigzag (15, 0) and outer zigzag (27, 0) for example. The two SWNTs are zigzag (15, 0) and zigzag (27, 0) nanotubes, respectively. We can observe that, in the stopping power curves, the plasmon excitations for both of the 2WNT and SWNTs start to take place near v 0.6, and the broad peak of the 2WNT locate between the peaks of the two SWNTs. Moreover, owing to the existence of the outer wall, the peak value of the stopping power in the 2WNT reduces and the high speed tail is raised significantly, compared with that of only channeling in zigzag (15, 0). Thus, from Fig. 2, the presence of double-walled structure which gives rise to strong interference has evident effects on the velocity dependencies of both the stopping power and the self energy.
S.-Y. You et al. / Nuclear Instruments and Methods in Physics Research B 267 (2009) 3133–3136
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Fig. 1. Self-energy (a) and stopping power (b) versus the speed v, for a proton moving along axis of four kinds of 2WNTs, including zigzag (15, 0) and (27, 0), armchair (9, 9) and (16, 16), zigzag (15, 0) with armchair (16, 16), and armchair (9, 9) with zigzag (27, 0).
Fig. 2. Self-energy (a) and stopping power (b) versus the speed v, for a proton moving along the axis of the double-walled nanotube zigzag (15, 0) and (27, 0) and two singlewalled nanotubes: zigzag (15, 0) and zigzag (27, 0).
Fig. 3. Self-energy (a) and stopping power (b) versus the speed v, for a proton moving along the axis of the double-walled nanotube zigzag (15, 0) and (27, 0) with different friction coefficient c.
In the following, we will study the influence of the frictionq coefffiffiffiffiffiffiffiffi ficient c on the self-energy and stopping power. Here, Xp ¼ 4pa1n0 , n0 is the surface density of the valence electrons, which can be obtained from the integration of equilibrium distribution functions R n0 ¼ ð2p4hÞ2 dpf0 ða1 ; pÞ. Considering a set of different values of c, the calculation results are shown in Fig. 3. As the increase of the damping factors, we can easily notice that both of the self-energy and stopping power keep decreasing in magnitude at higher speed, while the negative regions of the self-energy and the wide peak of the stopping power move to lower velocities. In the meanwhile, the narrow peaks in the self-energy and stopping power gradually reduce and disappear at last. Thus, significant damping may wash out the interference effects from double-walled structure. Thresholds for plasmon excitation shift to lower speeds with the increas-
ing of c, showing more relaxing effects from damping, as also seen in [13] in single-walled nanotube investigation. Similar multi-peak profiles have been shown in [11], in which a two-fluid model was used for multiwalled carbon nanotube. However, the difference of the maxima position also exists, due to different theoretical modes as discussed in [13]. Finally, in Fig. 4, we show the impact of the chiral parameter n of the outer wall on the self-energy and stopping power for a proton moving along the axis of a 2WNT. Here, both of the two walls are zigzag nanotubes. So, as the chiral parameter n of the outer wall gets larger, the outer radius of the nanotube increases, resulting in weaker electron polarization in this wall and smaller interference effects on the proton channeling. Thus, the two peaks in the stopping power get closer to each other, and superpose to almost
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Fig. 4. Self-energy (a) and stopping power (b) versus the speed v, for a proton moving along axes of 2WNTs with different outer wall nanotube chiral parameter n.
one peak at last. And also, the narrow peaks in the negative region of the self-energy get reduced and move to higher speed region. In conclusion, we have proposed a semi-classical kinetic model combined with the dielectric response theory to describe the plasmon excitation of 2WNTs during channeling of a proton along the axis. Compared with the self-energy and stopping power in SWNTs, double peaks structure is noticed in those in 2WNTs attributing to electron polarizations and their interference from both walls. And, the self-energy and stopping power in 2WNTs also have strong dependencies on the damping and the nanotubes geometry. As the damping factor or the chiral parameter n of the outer wall increase, less interference effects are noticed and responsible for the changes in the curves of the self-energy and the stopping power, including the gradual fade-out of the double-peak profiles. In our future work, we will extend our present work to channeling of charged particles through MWNTs, including channeling not only inside but also between walls of the nanotubes. Acknowledgement This work was supported by the National Natural Science Foundation of China (Y.N.W.), Grant No. 10275009.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
S. Iijima, Nature (London) 354 (1991) 56. N.R. Arista, M.A. Fuentes, Phys. Rev. B 63 (2001) 165401. N.R. Arista, Phys. Rev. A 64 (2001) 032901. N.R. Arista, Nucl. Instr. and Meth. B 182 (2001) 109. N. Zabala, E. Ogando, A. Rivacoba, F.J. Garcíde Abajo, Phys. Rev. B 64 (2001) 205410. T. Stökli, J.M. Bonard, A. Châtelain, Z.L. Wang, P. Stadelmann, Phys. Rev. B 64 (2001) 115424. Y.N. Wang, Z.L. Miškovic´, Phys. Rev. A 69 (2004) 022901. D.J. Mowbray, Z.L. Miškovic´, F.O. Goodman, Y.N. Wang, Phys. Lett. A 329 (2004) 94. D.J. Mowbray, S. Chung, Z.L. Miškovic´, F.O. Goodman, Y.N. Wang, Nucl. Instr. and Meth. B 230 (2005) 142. D.J. Mowbray, Z.L. Miškovic´, F.O. Goodman, Phys. Rev. B 74 (2006) 195435. Sangwoo Chung, D.J. Mowbray, Z.L. Miškovic´, F.O. Goodman, Y.N. Wang, Rad. Phys. Chem. 76 (2007) 524. G.Ya. Slepyan, S.A. Maksimenko, A. Lakhtakia, O. Yevtushenko, A.V. Gusakov, Phys. Rev. B 60 (1999) 17136. Yuan-Hong Song, Dan Zhao, You-Nian Wang, Phys. Rev. A 78 (2008) 012901. Yannouleas, E.N. Bogachek, U. Landman, Phys. Rev. B 53 (1996). R. Saito, M. Fujita, G. Dresselhaus, M.S. Dresselhaus, Phys. Rev. B 46 (1992) 1804. M.F. Lin, W.-K. Kenneth Shung, Phys. Rev. B 52 (1995) 8423.
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Channeling of protons in double-walled carbon nanotubes in kinetic model Shu-Yan You, Yuan-Hong Song, You-Nian Wang * School of Physics and Optoelectronic Technology, Dalian University of Technology, Dalian 116023, China
a r t i c l e
i n f o
Article history: Available online 16 June 2009 PACS: 73.63.-b 34.50.Bw Keywords: Carbon nanotube Self-energy Stopping power
a b s t r a c t A kinetic model combined with the dielectric response theory is employed to study the electronic excitation on the nanotube walls during the channeling of protons through double-walled carbon nanotubes. Analytical expressions of the self-energy and stopping power are obtained with protons moving along the axis of the double-walled nanotubes. Calculation results show us interesting double-peak curves of the self-energy and stopping power, under strong influence of the damping factor and the special doublewalled nanotube geometry. Relatively increasing the damping factor and the chiral parameter of the outer wall can reduce the interference effects between the two walls and weaken the double-peak to one-peak shapes. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Ever since the discovery of carbon nanotubes by Iijima [1], many efforts have been made both theoretically and experimentally to investigate various aspects of nanotubes owing to their exceptional properties and great expectations for their applications. Carbon nanotubes are classified by the dual index (n, l). Two integers n and l represent the vector characterizing the pattern of rolling a planar sheet to a cylindrical cavity, with l = 0 for zigzag carbon nanotubes, l = n for armchair carbon nanotubes, and n – l for chiral carbon nanotubes, showing either metallic or semiconducting properties. The channeling of charged particles through nanotubes obtains us not only information about the surface electronic excitation, but also a new application for particles channeling and steering. In the theoretical studies, the dielectric theory has been taken as a valid theoretical tool to investigate interactions of charged particles with cylindrical structures, including fast ions and clusters moving paraxially through microcapillaries and nanocapillaries in solids [2–4] or fast electrons moving perpendicularly through metallic and semiconductor cylindrical nanowires [5]. Besides, the dielectric properties of carbon nanotubes have been investigated by means of the hydrodynamic theory, in which plasmon excitation in a thin layer of free electron gas have been deduced [6]. Also by using the two-dimensional fluid model, the self-energy and the energy loss of charged particles moving parallel to the axis of a single-walled (SWNT) or a double-walled nanotube (2WNT) have
* Corresponding author. Tel.: +86 411 84707307; fax: +86 411 84709304. E-mail address: [email protected] (Y.-N. Wang). 0168-583X/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2009.06.037
been calculated [7–9]. Moreover, two-fluid hydrodynamic model has been adopted to describe the dielectric response of r and p electron systems on SWNTs and multi-walled carbon nanotubes (MWNTs) [10,11]. Meanwhile, considering the energy band structure of the electrons on the nanotube surface, semi-classical kinetic model [12] combined with the dielectric response theory has been introduced to study the collective excitation, self-energy and energy loss, with charged particles moving inside the SWNTs and parallel with axes [13]. However, as MWNTs have shown rich plasmon spectra properties [14], the effects of the band structure of the electrons during the channeling have to be considered. In the present paper, we try to study the collective excitation in 2WNTs while charged particles move along the axis, based on the semi-classical kinetic theory. Our attention will also be focused on the self-energy and the stopping power for charged particles channeling. The cylindrical coordinates r = (q, u, z) for carbon nanotubes will be adopted, with the orientation of z along the axis of nanotubes. Atomic units (a.u.) will be used throughout this paper, unless otherwise indicated. 2. Theoretical model The present work is an extension of our previous study of SWNT [13] to the case of 2WNT. In this paper, we only take p electrons as our aim for their important effects on electronic properties. The 2WNT is regarded as infinitesimally thin and infinitely long cylindrical tubes. Consider a 2WNT consists of 2 coaxial walls with the inner radius a1 and the outer radius a2. When a charged particle goes through inside the nanotubes, we use cylindrical coordination r0 = (q0, /0, vt) for its instantaneous position, involving the velocity
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of the particle v. We employ U2 to express the potential between two walls a1 < q < a2, U3 to express the potential outside the outer wall q > a2,while the values of the total potential U1 = U0 + Uind inside the inner wall q < a1 may include the potential U0 from the moving charged particle itself and the induced potential Uind from the charge polarization on the inner surface induced by charged particles. Conforming to the work in [13], these potentials could be expanded in terms of Bessel functions as
U0 ðr; tÞ ¼
þ1 Z Q X
Q ¼ jr r0 j
þ1
p m¼1 1 Im ðjkjq< ÞK m ðjkjq> Þ;
Uind ðr; tÞ ¼
þ1 Z Q X
p
þ1
dke
ikðzv tÞþimð//0 Þ
dke
ð1Þ
ikðzv tÞþimð//0 Þ
m¼1
U1 ðr; tÞq¼a1 ¼ U2 ðr; tÞq¼a1
ð2Þ
1
Im ðjkjq0 Þ½Im ðjkjqÞBm ðkÞ þ K m ðjkjqÞC m ðkÞ
and the radial component of the displacement field keeps discontinuous at the boundary,
@ U2 @ U1 ðr; tÞq¼a1 ðr; tÞq¼a1 ¼ 4pn1 ða1 ; /; z; tÞ; @q @q @ U3 @ U2 ðr; tÞq¼a2 ðr; tÞq¼a2 ¼ 4pn2 ða2 ; /; z; tÞ; @q @q
þ1 Z Q X
p
m¼1
ð3Þ
nj ðaj ; /; z; tÞ ¼
þ1
ikðzv tÞþimð//0 Þ
dke
1
Im ðjkjq0 ÞK m ðjkjqÞDm ðkÞ;
ð4Þ
where k is the longitudinal wave number, m is the angular momentum. Also in these above equations, Q is the charge of the particle, Im(x) and Km(x) are the modified Bessel functions, and q< and q> are the smaller or larger of q and q0. Moreover, exposed to the intrusion of charged ions, the distribution functions of p electrons on the inner and outer surfaces of the 2WNT are assumed to satisfy the first-order Boltzmann kinetic equation, respectively,
@fj @f0 þ v rsj fj ¼ eEj cfj @t @p
j ¼ 1; 2;
ð5Þ
@ ^ @ ^ e/ þ @z ez ðj ¼ 1; 2Þ is the two-dimensional differenwhere rsj ¼ a1 @/ j tial operator only to each surface of the 2WNT, Ej = rUj is the electric field on the surface, f0 ðpÞ ¼ 1þexpfe1ðpÞ=kB Tg is the Fermi equilibrium distribution function of the electron gas, p is the electron’s two dimensional quasimomentum tangential to the nanotube’s surface, kB is the Boltzmann constant, T is the temperature, and e(p) is the electron energy with respect to Fermi level, which is determined by the nanotube geometry and can be obtained by using the tight-binding model [12,15,16]. Besides, fj is the perturbation of the distribution function, c is the friction coefficient owing to electron scattering on the positive charge background, which is introduced to satisfy the nonconservation of the system. In terms of a Fourier transform, the perturbation part fj can be adopted here,
fj ¼
þ1 Z Q X
p
m¼1
þ1
ikðzv tÞþimð//0 Þ
dke
fjm ðaj ; pÞ:
ð6Þ
1
Combining Eqs. (5) and (6) with Eq. (3) in which U ¼ U2 ðr; tÞq¼a2 , we can obtain
fjm ðaj ; p; k; xÞ ¼
ð9Þ
where n1(a1,/,z,t) and n2(a2,/,z,t) are the individual induced density on the inner and outer surface of the nanotube. By using the Fourier–Bessel transform, the induced density nj(aj,/,z,t) can be expressed as follows, while considering the R expression as njm ðaj ; k; xÞ ¼ ð2p4hÞ2 dpfjm ðaj ; p; k; xÞ, we obtain
and
U3 ðr; tÞ ¼
ð8Þ
U2 ðr; tÞq¼a2 ¼ U3 ðr; tÞq¼a2
1
m¼1
Im ðjkjq0 ÞIm ðjkjqÞAm ðkÞ; þ1 Z þ1 Q X ikðzv tÞþimð//0 Þ U2 ðr; tÞ ¼ dke
p
For the sake of determining the coefficients Am(k), Bm(k), Cm(k) and Dm(k), the following boundary conditions at q = aj are adopted: the potential keeps continuous at the surface of cylinder,
eUjm
m @f0 aj @p/
@f0 þ k @p z
x maj v / kv z þ ic
;
ð7Þ
where Pz and P/ are the projections of p on the axis and the direcphS for armchair, tion of the nanotube, P/ ¼ p2pffiffi3hnbS for zigzag, P/ ¼ 23nb x = kv, Ujm = Im(jkjq0)[Bm(k)Im(jkjaj) + s = 1,2 . . . ,n. And, Cm(k)Km(jkjaj)], respectively. But for v/, with p/ a quantized variable, we approximate the partial derivative with finite difference, as v/ = e[pz, p/(s + 1)] e[pz,p/].
¼
þ1 Z Q X
p
m¼1
m¼1
imðzv tÞþimð//0 Þ
dke
njm ðaj ; k; xÞ
1
þ1 Z Q X
p
þ1
þ1
imðzv tÞþimð//0 Þ
dke
Ujm vðk; m; x; aj Þ;
1
ð10Þ where v(k, m, x, aj) is the response function, which is different for zigzag and armchair nanotube. The detailed expressions of the response function can be found in [13]. Finally, the coefficients in the potential expressions in Eqs. (2)– (4) can be obtained, especially for the induced potential Uind, which allows us to pay further attention on the self-energy, Eself ¼ 12 Uind ðr; tÞjr¼r0 and the stopping power S ¼ Q @Uind@zðr;tÞ jr¼r0 , showing how the electron polarization influences the ion channeling inside 2WNTs. 3. Results and conclusions Consider a proton moving along the axis q0 = 0 of a 2WNT. It is noted that only m = 0 mode can contribute to the energy loss because of axial symmetry. In Fig. 1, we show how the self-energy and stopping power depend on proton speed v in 2WNTs. Four 2WNTs are chosen to be channelled in this figure: both walls are zigzag or armchair, or with one wall zigzag and the other one armchair. It is interesting to note that double peaks occur in the selfenergy and stopping power curves, with the narrow one in low speed region and the wide one in high speed region. In our opinion, it is owing to the double-walled structures, in which electron polarization in both walls and the interference between them contribute to the stoppings for the particle. In order to make comparison between SWNTs and 2WNTs, we also calculate the velocity dependencies of the self-energy and stopping power in SWNTs and a 2WNT, as shown in Fig. 2. Here, we take the 2WNT with inner zigzag (15, 0) and outer zigzag (27, 0) for example. The two SWNTs are zigzag (15, 0) and zigzag (27, 0) nanotubes, respectively. We can observe that, in the stopping power curves, the plasmon excitations for both of the 2WNT and SWNTs start to take place near v 0.6, and the broad peak of the 2WNT locate between the peaks of the two SWNTs. Moreover, owing to the existence of the outer wall, the peak value of the stopping power in the 2WNT reduces and the high speed tail is raised significantly, compared with that of only channeling in zigzag (15, 0). Thus, from Fig. 2, the presence of double-walled structure which gives rise to strong interference has evident effects on the velocity dependencies of both the stopping power and the self energy.
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Fig. 1. Self-energy (a) and stopping power (b) versus the speed v, for a proton moving along axis of four kinds of 2WNTs, including zigzag (15, 0) and (27, 0), armchair (9, 9) and (16, 16), zigzag (15, 0) with armchair (16, 16), and armchair (9, 9) with zigzag (27, 0).
Fig. 2. Self-energy (a) and stopping power (b) versus the speed v, for a proton moving along the axis of the double-walled nanotube zigzag (15, 0) and (27, 0) and two singlewalled nanotubes: zigzag (15, 0) and zigzag (27, 0).
Fig. 3. Self-energy (a) and stopping power (b) versus the speed v, for a proton moving along the axis of the double-walled nanotube zigzag (15, 0) and (27, 0) with different friction coefficient c.
In the following, we will study the influence of the frictionq coefffiffiffiffiffiffiffiffi ficient c on the self-energy and stopping power. Here, Xp ¼ 4pa1n0 , n0 is the surface density of the valence electrons, which can be obtained from the integration of equilibrium distribution functions R n0 ¼ ð2p4hÞ2 dpf0 ða1 ; pÞ. Considering a set of different values of c, the calculation results are shown in Fig. 3. As the increase of the damping factors, we can easily notice that both of the self-energy and stopping power keep decreasing in magnitude at higher speed, while the negative regions of the self-energy and the wide peak of the stopping power move to lower velocities. In the meanwhile, the narrow peaks in the self-energy and stopping power gradually reduce and disappear at last. Thus, significant damping may wash out the interference effects from double-walled structure. Thresholds for plasmon excitation shift to lower speeds with the increas-
ing of c, showing more relaxing effects from damping, as also seen in [13] in single-walled nanotube investigation. Similar multi-peak profiles have been shown in [11], in which a two-fluid model was used for multiwalled carbon nanotube. However, the difference of the maxima position also exists, due to different theoretical modes as discussed in [13]. Finally, in Fig. 4, we show the impact of the chiral parameter n of the outer wall on the self-energy and stopping power for a proton moving along the axis of a 2WNT. Here, both of the two walls are zigzag nanotubes. So, as the chiral parameter n of the outer wall gets larger, the outer radius of the nanotube increases, resulting in weaker electron polarization in this wall and smaller interference effects on the proton channeling. Thus, the two peaks in the stopping power get closer to each other, and superpose to almost
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Fig. 4. Self-energy (a) and stopping power (b) versus the speed v, for a proton moving along axes of 2WNTs with different outer wall nanotube chiral parameter n.
one peak at last. And also, the narrow peaks in the negative region of the self-energy get reduced and move to higher speed region. In conclusion, we have proposed a semi-classical kinetic model combined with the dielectric response theory to describe the plasmon excitation of 2WNTs during channeling of a proton along the axis. Compared with the self-energy and stopping power in SWNTs, double peaks structure is noticed in those in 2WNTs attributing to electron polarizations and their interference from both walls. And, the self-energy and stopping power in 2WNTs also have strong dependencies on the damping and the nanotubes geometry. As the damping factor or the chiral parameter n of the outer wall increase, less interference effects are noticed and responsible for the changes in the curves of the self-energy and the stopping power, including the gradual fade-out of the double-peak profiles. In our future work, we will extend our present work to channeling of charged particles through MWNTs, including channeling not only inside but also between walls of the nanotubes. Acknowledgement This work was supported by the National Natural Science Foundation of China (Y.N.W.), Grant No. 10275009.
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