- Email: [email protected]
Surface and bulk plasmon excitations in carbon nanotubes. Comparison with the hydrodynamic model
Nuclear Instruments and Methods in Physics Research B 267 (2009) 415–418
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Surface and bulk plasmon excitations in carbon nanotubes. Comparison with the hydrodynamic model Mario Zapata Herrera a,*, Juana L. Gervasoni a,b a b
Instituto Balseiro and Centro Atómico Bariloche, Comisión Nacional de Energía Atómica, 8400 S.C. Bariloche, Argentina Carrera de Investigador Científicoy Tecnológico del CONICET, Argentina
a r t i c l e
i n f o
Available online 6 November 2008 PACS: 71.45 Keywords: Carbon nanotubes Dielectric response model Hydrodynamic model Plasmons in carbon nanostructures
a b s t r a c t In this work, we compare two models describing the interaction of external charged particles with carbon nanotubes. One is the semiclassical dielectric response model (DRM) in the Drude approximation, which approximate the valence electrons of the system by a gas of non interacting classical particles. The other is the hydrodynamic model (HDM) which uses Fluid Dynamics to describe their collective excitations. We found that both models agree for those cases where it is possible to define a dispersion relation which depends on a single frequency xp . We found that in the description of the electronic response of a singlewalled carbon nanotube (SWCNT) with the DRM, the connection between a three- and a two-dimensional system is non trivial and the equivalence is not direct. In spite of this, the DRM can be an important basic tool for the calculation and physical interpretation of the plasmon excitations in a nanodimensions system. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Starting with the seminal works by Endo in the 70’s [1], the discovery of carbon nanotubes (CNT’s) by Iijima [2] and the ulterior synthesis of single-walled carbon nanotubes (SWCNT) achieved two years later by Iijima and Ichihashi [3], the study of these systems has generated great expectations in the scientific community due to their great variety of physical properties and technological applications. Its single dimensional quantum character [4,5], its elasticity modulus of the order of 1 TPa and shear modulus around 1 GPa [6], its unusual band structure [7] that is strongly dependent on its geometrical configuration [5,8,9], are some of the unusual characteristics of this fascinating device. On the other hand, its use for the storage of hydrogen (see [10]), and its integration in logical circuits [11] and nanoelectro-mechanical systems [12] are only two of the still growing number of applications of the CNT’s. The plasmon excitations in nanostructures have been studied using the dielectric response model (DRM) in cylindrical cavities by Arista and Fuentes [13], as well as in nanowires by Gervasoni and Arista [14] and in metallic nanotubes by Seguí et al. [15]. By means of this model, the stopping force acting on an incident particle can also be calculated, as well as its dependence with the impact parameter [15]. From the quantum point of view, the plasmon field can be described by means of a second quantization
formalism. The equivalence between the classic (or semiclassical) and quantum descriptions was demonstrated by Aligia et al. in [16]. The hydrodynamic model (HDM) provides an alternative to the dielectric formalism that allows for a direct physical interpretation through an analogy between the valence electrons and a fluid. Stockli et al. [17,18] applied this model to describe the plasmon excitation in a CNT as a two-dimensional electron gas confined to a cylinder surface. Miskovic et al. [19] introduced a more realistic approach, by considering the p and r electrons as two separate two-dimensional fluids. The corresponding dispersion relations agrees reasonably well with that obtained experimentally [20]. In the present work we use the dielectric and hydrodynamic formalisms to study the excitation of bulk and surface plasmon in carbon nanotubes due to the presence of charged projectiles. We compare different patterns of excitation for different paths of the incident particles, allowing us to obtain different dispersion relations for each formalism used. 2. Dielectric formalism: a short review Within the dielectric response model (DRM), it is assumed that the displacement vector ~ Dð~ r; tÞ and the electric field Eð~ r; tÞ of an isotropic material are linearly related in the reciprocal space ð~ k; xÞ by a constitutive equation
~ Dð~ k; xÞ ¼ ð~ k; xÞ~ Eð~ k; xÞ; * Corresponding author. E-mail addresses: [email protected] (M.Z. Herrera), [email protected]. gov.ar (J.L. Gervasoni). 0168-583X/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2008.10.078
ð1Þ
Similarly, the induced (qind ) and the external (qext ) charge densities are related by
416
M.Z. Herrera, J.L. Gervasoni / Nuclear Instruments and Methods in Physics Research B 267 (2009) 415–418
qind ð~ k; xÞ ¼ qext ð~ k; xÞ
1 1 : ðk; xÞ
ð2Þ
For a charged particle q moving along a classical path ~ RðtÞ, the external charge density reads
qext ð~r; tÞ ¼ qd~r ~ RðtÞ;
ð3Þ
The stopping force acting on the charged particle q due to the inR; tÞ reads duced electric field ~ Eind ð~
~ F ¼ q~ Eind ð~ r; tÞj~r¼~RðtÞ ;
ð4Þ
where ~ Eind ð~ r; tÞ is obtained by Fourier transforming ~ Eind ð~ k; xÞ namely
~ Eind ð~ r; tÞ ¼
Z
1 ð2pÞ
4
d~ k
Z
1
~ Eind ð~ dxeiðk~rxtÞ~ k; xÞ:
ð5Þ
1
3. Plasmon excitations in carbon nanotubes: dielectric response model
with the boundary conditions
Usð1Þ jq¼a ¼ Uð2Þ s jq¼a ; ð1Þ s jq¼a
rq U
ð9Þ ð2Þ s jq¼a ;
¼ ðxÞrq U
Usð2Þ jq¼b ¼ Uð3Þ s jq¼b
ð10Þ ð11Þ
and ð3Þ rq Uð2Þ s jq¼b ¼ ðxÞrq Us jq¼b :
ð12Þ
Let us now employ the Drude approximation,
ðxÞ 1
x2p ; x2
ð13Þ
for describing the dielectric function, where xp is the bulk plasma frequency of the material. By solving the electrostatic problem, as described in [21], we obtain the dispersion relation xm ðkÞ for this system for the case of a metallic nanotube of finite and infinitesimal thickness D ¼ ja bj. Within the limit D ! 0, the dispersion relations xm ðkÞ splits up into two contributions for low ðÞ and high
Now, we study the plasmon excitations in carbon nanotubes (see Fig. 1), where the material between the radii a and b is characterized by a dielectric function ðxÞ. Once the electrostatic problem is solved, the description of a metallic SWCNT is obtained by making its thickness infinitesimal, i.e. a ! b. The electrostatic potential for each space region is given in terms of Bessel functions Im ðkqÞ and K m ðkqÞ by:
Uð1Þ s ðq; u; zÞ ¼
X
Am;k eiðkzþmuÞ Im ðkqÞeixt
ð6Þ
m;k
for q < a (region 1) and
Uð3Þ s ðq; u; zÞ ¼
X
Bm;k eiðkzþmuÞ K m ðkqÞeixt
ð7Þ
m;k
for q > a (region 3). For a < q < b (region 2) we have
Uð2Þ s ðq; u; zÞ ¼
X
C m;k Im ðkqÞ þ Dm;k K m ðkqÞ eiðkzþmuÞ eixt :
ð8Þ
m;k
Fig. 1. Nanotube with internal radius a and external radius b. The region between a < q < b is characterized by a dielectric function ðxÞ. The tube is in vacuum, so that ¼ 1 for q < a and q > b.
Fig. 2. Dispersion relation obtained by the dielectric response model (DRM) (upper figure), which can be compared with the Random Phase Approximation, RPA [22] (figure below).
M.Z. Herrera, J.L. Gervasoni / Nuclear Instruments and Methods in Physics Research B 267 (2009) 415–418
417
ðþÞ energies, similar to those of a cylindrical wire [14] and cavity [13], respectively. This characteristic is shown in Fig. 2, for the modes m ¼ 0; 1 in a carbon nanotube of thickness D ¼ 3 a:u. This dispersion relation is compared with the one obtained by means a Random Phase Approximation, RPA [22]. Actually, the low frequency modes are similar to those of the RPA for a two-dimensional model for the plasmon oscillations frequency. Within this approximation, the media is described as a uniform electron gas characterized by the radius r s of the sphere occupied by one electron. For our calculation in Fig. 2 we adopted r s ¼ 0:86 a:u. 4. Plasmon excitations in carbon nanotubes: hydrodynamic model The hydrodynamic model (HDM) provides an alternative for the description of collective excitations of a free electron gas of density n, by modelling it as a fluid that satisfies the Navier–Stokes equation Fig. 5. Stopping force on a charged particle moving along the symmetry axis of a carbon nanotube of different radii.
oWð~ r; tÞ 1 ~ e 1 ¼ jrWð~ r; tÞ: r; tÞj2 Uð~ r; tÞ þ Ið~ ot 2 m m
ð14Þ
Here Uð~ r; tÞ is the electrostatic potential, and the term Ið~ r; tÞ is 2 given by Ið~ r; tÞ ¼ ð h =2mÞð3p2 Þ2=3 n2=3 . The velocity field of the free r; tÞ as electron gas is readily obtained from potential Wð~ ~ v ð~r; tÞ ¼ rWð~r; tÞ. The socalled Bloch Hydrodynamic System is completed by the Poisson,
r2 U ¼ 4peðn nþ Þ
ð15Þ
and the continuity equation
on ¼ r ðnrWÞ: ot
Fig. 3. Dispersion relation obtained with the DRM (3-D) with xp 60 eV for the low frequencies in Fig. 2.
ð16Þ
Using this model, Miskovic et al. [24] evaluated the dispersion relation for carbon nanotubes by describing the electron gas as composed of two interacting fluids of r and p electrons confined within the same cylindrical nanotube surface. This dispersion relation shows an excellent agreement with the experimental data obtained by Pichler et al. in [20]. On the other hand, our calculation within a three-dimensional DRM model, for xp 60 eV, as shown in Fig. 3 for the low frequencies, also compares favorably with both the HDM calculation and the experimental data. 5. Stopping force: external and internal path In Fig. 4 we show the DRM calculation of the stopping force acting on a charged particle moving along a rectilinear trajectory parallel to the symmetry axis of a carbon nanotube. We observe a maximum in the stopping force for each value of q0 . This maximum diminishes in magnitude for increasing q0 , but its position is weakly dependent on the incident velocity of the particle. Similarly, in Fig. 5 we show the stopping force on a charged particle moving along the symmetry axis of carbon nanotubes for different internal radii. As in all previous cases, these results show a reasonable agreement with those obtained with the HDM model [23]. 6. Conclusions
Fig. 4. Stopping force on a charged particle moving along a rectilinear path parallel to the symmetry axis outside a carbon nanotube, with D ¼ 10 a:u: and different impact parameters q0 .
In this work, we made a comparison between the dielectric response model (DRM) and the hydrodynamic model (HDM) for the plasmon excitation due to the interaction of charged particles with carbon nanotubes. Both models agree for those cases where it is possible to define a dispersion relation dependent on a single
418
M.Z. Herrera, J.L. Gervasoni / Nuclear Instruments and Methods in Physics Research B 267 (2009) 415–418
frequency xp . These results demonstrate that both models represent important basic and complementary tools for the physical interpretation of the collective excitations of the electron gas of a system of nanoscopic dimensions. The dispersion relations obtained in this work by means of the DRM are equivalent to those obtained with the HDM for an infinitesimal thickness, where the separation between the branches of high and low frequencies becomes bigger when D ! 0. For internal trajectories, the agreement between the DRM and the HDM is very good, although the results differ at least in a factor 2, but might provide an initial idea of the energy losses due to plasmons excitation in experimental problems. References [1] M. Endo, Ph.D. Thesis Dissertation, Orleans University, France, 1975. [2] S. Iijima, Nature 354 (1991) 56. [3] S. Iijima, T. Ichihashi, Nature 363 (1993) 603.
[4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
S.J. Tans et al., Nature 386 (1997) 474. N. Hamada et al., Phys. Rev. Lett. 68 (1992) 1579. J.P. Salvetat et al., Phys. Rev. Lett. 82 (1999) 944. J.W. Mintmire et al., Phys. Rev. Lett. 68 (1992) 631. R. Saito et al., Phys. Rev. B 46 (1992) 3. R. Saito et al., Imperial College Press, 1998. A.C. Dillon et al., Nature 386 (1997) 377. P.C. Collins et al., Science 292 (2001) 706. T. Cohen-Karni et al., Nature-nano 1 (2006) 36. N.R. Arista, M.A. Fuentes, Phys. Rev. B 63 (2001) 165401. J.L. Gervasoni, N.R. Arista, Phys. Rev. B 68 (2003) 235302. S. Seguí, J.L. Gervasoni, N.R. Arista, Nucl. Instr. and Meth. B 233 (2005) 227. A. Aligia et al., Phys. Rev. B 70 (2004) 235331. T. Stockli et al., Phys. Rev. B 64 (2001) 115424. T. Stocki et al., Appl. Phys. Lett. 80 (2002) 29820. D.J. Mowbray et al., Phys. Rev. B 70 (2004) 195418. T. Pichler et al., Phys. Rev. Lett. 80 (1998) 4729. S. Seguí. Personal communication. Y.N. Wang, Z.L. Miskovic, Phys. Rev. A 66 (2002) 042904. Y.N. Wang, Z.L. Miskovic, Phys. Rev. A 69 (2004) 022901. D.J. Mowbray et al., Nucl. Instr. and Meth. Phys. Res. B 230 (2005) 142.
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Surface and bulk plasmon excitations in carbon nanotubes. Comparison with the hydrodynamic model Mario Zapata Herrera a,*, Juana L. Gervasoni a,b a b
Instituto Balseiro and Centro Atómico Bariloche, Comisión Nacional de Energía Atómica, 8400 S.C. Bariloche, Argentina Carrera de Investigador Científicoy Tecnológico del CONICET, Argentina
a r t i c l e
i n f o
Available online 6 November 2008 PACS: 71.45 Keywords: Carbon nanotubes Dielectric response model Hydrodynamic model Plasmons in carbon nanostructures
a b s t r a c t In this work, we compare two models describing the interaction of external charged particles with carbon nanotubes. One is the semiclassical dielectric response model (DRM) in the Drude approximation, which approximate the valence electrons of the system by a gas of non interacting classical particles. The other is the hydrodynamic model (HDM) which uses Fluid Dynamics to describe their collective excitations. We found that both models agree for those cases where it is possible to define a dispersion relation which depends on a single frequency xp . We found that in the description of the electronic response of a singlewalled carbon nanotube (SWCNT) with the DRM, the connection between a three- and a two-dimensional system is non trivial and the equivalence is not direct. In spite of this, the DRM can be an important basic tool for the calculation and physical interpretation of the plasmon excitations in a nanodimensions system. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Starting with the seminal works by Endo in the 70’s [1], the discovery of carbon nanotubes (CNT’s) by Iijima [2] and the ulterior synthesis of single-walled carbon nanotubes (SWCNT) achieved two years later by Iijima and Ichihashi [3], the study of these systems has generated great expectations in the scientific community due to their great variety of physical properties and technological applications. Its single dimensional quantum character [4,5], its elasticity modulus of the order of 1 TPa and shear modulus around 1 GPa [6], its unusual band structure [7] that is strongly dependent on its geometrical configuration [5,8,9], are some of the unusual characteristics of this fascinating device. On the other hand, its use for the storage of hydrogen (see [10]), and its integration in logical circuits [11] and nanoelectro-mechanical systems [12] are only two of the still growing number of applications of the CNT’s. The plasmon excitations in nanostructures have been studied using the dielectric response model (DRM) in cylindrical cavities by Arista and Fuentes [13], as well as in nanowires by Gervasoni and Arista [14] and in metallic nanotubes by Seguí et al. [15]. By means of this model, the stopping force acting on an incident particle can also be calculated, as well as its dependence with the impact parameter [15]. From the quantum point of view, the plasmon field can be described by means of a second quantization
formalism. The equivalence between the classic (or semiclassical) and quantum descriptions was demonstrated by Aligia et al. in [16]. The hydrodynamic model (HDM) provides an alternative to the dielectric formalism that allows for a direct physical interpretation through an analogy between the valence electrons and a fluid. Stockli et al. [17,18] applied this model to describe the plasmon excitation in a CNT as a two-dimensional electron gas confined to a cylinder surface. Miskovic et al. [19] introduced a more realistic approach, by considering the p and r electrons as two separate two-dimensional fluids. The corresponding dispersion relations agrees reasonably well with that obtained experimentally [20]. In the present work we use the dielectric and hydrodynamic formalisms to study the excitation of bulk and surface plasmon in carbon nanotubes due to the presence of charged projectiles. We compare different patterns of excitation for different paths of the incident particles, allowing us to obtain different dispersion relations for each formalism used. 2. Dielectric formalism: a short review Within the dielectric response model (DRM), it is assumed that the displacement vector ~ Dð~ r; tÞ and the electric field Eð~ r; tÞ of an isotropic material are linearly related in the reciprocal space ð~ k; xÞ by a constitutive equation
~ Dð~ k; xÞ ¼ ð~ k; xÞ~ Eð~ k; xÞ; * Corresponding author. E-mail addresses: [email protected] (M.Z. Herrera), [email protected]. gov.ar (J.L. Gervasoni). 0168-583X/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2008.10.078
ð1Þ
Similarly, the induced (qind ) and the external (qext ) charge densities are related by
416
M.Z. Herrera, J.L. Gervasoni / Nuclear Instruments and Methods in Physics Research B 267 (2009) 415–418
qind ð~ k; xÞ ¼ qext ð~ k; xÞ
1 1 : ðk; xÞ
ð2Þ
For a charged particle q moving along a classical path ~ RðtÞ, the external charge density reads
qext ð~r; tÞ ¼ qd~r ~ RðtÞ;
ð3Þ
The stopping force acting on the charged particle q due to the inR; tÞ reads duced electric field ~ Eind ð~
~ F ¼ q~ Eind ð~ r; tÞj~r¼~RðtÞ ;
ð4Þ
where ~ Eind ð~ r; tÞ is obtained by Fourier transforming ~ Eind ð~ k; xÞ namely
~ Eind ð~ r; tÞ ¼
Z
1 ð2pÞ
4
d~ k
Z
1
~ Eind ð~ dxeiðk~rxtÞ~ k; xÞ:
ð5Þ
1
3. Plasmon excitations in carbon nanotubes: dielectric response model
with the boundary conditions
Usð1Þ jq¼a ¼ Uð2Þ s jq¼a ; ð1Þ s jq¼a
rq U
ð9Þ ð2Þ s jq¼a ;
¼ ðxÞrq U
Usð2Þ jq¼b ¼ Uð3Þ s jq¼b
ð10Þ ð11Þ
and ð3Þ rq Uð2Þ s jq¼b ¼ ðxÞrq Us jq¼b :
ð12Þ
Let us now employ the Drude approximation,
ðxÞ 1
x2p ; x2
ð13Þ
for describing the dielectric function, where xp is the bulk plasma frequency of the material. By solving the electrostatic problem, as described in [21], we obtain the dispersion relation xm ðkÞ for this system for the case of a metallic nanotube of finite and infinitesimal thickness D ¼ ja bj. Within the limit D ! 0, the dispersion relations xm ðkÞ splits up into two contributions for low ðÞ and high
Now, we study the plasmon excitations in carbon nanotubes (see Fig. 1), where the material between the radii a and b is characterized by a dielectric function ðxÞ. Once the electrostatic problem is solved, the description of a metallic SWCNT is obtained by making its thickness infinitesimal, i.e. a ! b. The electrostatic potential for each space region is given in terms of Bessel functions Im ðkqÞ and K m ðkqÞ by:
Uð1Þ s ðq; u; zÞ ¼
X
Am;k eiðkzþmuÞ Im ðkqÞeixt
ð6Þ
m;k
for q < a (region 1) and
Uð3Þ s ðq; u; zÞ ¼
X
Bm;k eiðkzþmuÞ K m ðkqÞeixt
ð7Þ
m;k
for q > a (region 3). For a < q < b (region 2) we have
Uð2Þ s ðq; u; zÞ ¼
X
C m;k Im ðkqÞ þ Dm;k K m ðkqÞ eiðkzþmuÞ eixt :
ð8Þ
m;k
Fig. 1. Nanotube with internal radius a and external radius b. The region between a < q < b is characterized by a dielectric function ðxÞ. The tube is in vacuum, so that ¼ 1 for q < a and q > b.
Fig. 2. Dispersion relation obtained by the dielectric response model (DRM) (upper figure), which can be compared with the Random Phase Approximation, RPA [22] (figure below).
M.Z. Herrera, J.L. Gervasoni / Nuclear Instruments and Methods in Physics Research B 267 (2009) 415–418
417
ðþÞ energies, similar to those of a cylindrical wire [14] and cavity [13], respectively. This characteristic is shown in Fig. 2, for the modes m ¼ 0; 1 in a carbon nanotube of thickness D ¼ 3 a:u. This dispersion relation is compared with the one obtained by means a Random Phase Approximation, RPA [22]. Actually, the low frequency modes are similar to those of the RPA for a two-dimensional model for the plasmon oscillations frequency. Within this approximation, the media is described as a uniform electron gas characterized by the radius r s of the sphere occupied by one electron. For our calculation in Fig. 2 we adopted r s ¼ 0:86 a:u. 4. Plasmon excitations in carbon nanotubes: hydrodynamic model The hydrodynamic model (HDM) provides an alternative for the description of collective excitations of a free electron gas of density n, by modelling it as a fluid that satisfies the Navier–Stokes equation Fig. 5. Stopping force on a charged particle moving along the symmetry axis of a carbon nanotube of different radii.
oWð~ r; tÞ 1 ~ e 1 ¼ jrWð~ r; tÞ: r; tÞj2 Uð~ r; tÞ þ Ið~ ot 2 m m
ð14Þ
Here Uð~ r; tÞ is the electrostatic potential, and the term Ið~ r; tÞ is 2 given by Ið~ r; tÞ ¼ ð h =2mÞð3p2 Þ2=3 n2=3 . The velocity field of the free r; tÞ as electron gas is readily obtained from potential Wð~ ~ v ð~r; tÞ ¼ rWð~r; tÞ. The socalled Bloch Hydrodynamic System is completed by the Poisson,
r2 U ¼ 4peðn nþ Þ
ð15Þ
and the continuity equation
on ¼ r ðnrWÞ: ot
Fig. 3. Dispersion relation obtained with the DRM (3-D) with xp 60 eV for the low frequencies in Fig. 2.
ð16Þ
Using this model, Miskovic et al. [24] evaluated the dispersion relation for carbon nanotubes by describing the electron gas as composed of two interacting fluids of r and p electrons confined within the same cylindrical nanotube surface. This dispersion relation shows an excellent agreement with the experimental data obtained by Pichler et al. in [20]. On the other hand, our calculation within a three-dimensional DRM model, for xp 60 eV, as shown in Fig. 3 for the low frequencies, also compares favorably with both the HDM calculation and the experimental data. 5. Stopping force: external and internal path In Fig. 4 we show the DRM calculation of the stopping force acting on a charged particle moving along a rectilinear trajectory parallel to the symmetry axis of a carbon nanotube. We observe a maximum in the stopping force for each value of q0 . This maximum diminishes in magnitude for increasing q0 , but its position is weakly dependent on the incident velocity of the particle. Similarly, in Fig. 5 we show the stopping force on a charged particle moving along the symmetry axis of carbon nanotubes for different internal radii. As in all previous cases, these results show a reasonable agreement with those obtained with the HDM model [23]. 6. Conclusions
Fig. 4. Stopping force on a charged particle moving along a rectilinear path parallel to the symmetry axis outside a carbon nanotube, with D ¼ 10 a:u: and different impact parameters q0 .
In this work, we made a comparison between the dielectric response model (DRM) and the hydrodynamic model (HDM) for the plasmon excitation due to the interaction of charged particles with carbon nanotubes. Both models agree for those cases where it is possible to define a dispersion relation dependent on a single
418
M.Z. Herrera, J.L. Gervasoni / Nuclear Instruments and Methods in Physics Research B 267 (2009) 415–418
frequency xp . These results demonstrate that both models represent important basic and complementary tools for the physical interpretation of the collective excitations of the electron gas of a system of nanoscopic dimensions. The dispersion relations obtained in this work by means of the DRM are equivalent to those obtained with the HDM for an infinitesimal thickness, where the separation between the branches of high and low frequencies becomes bigger when D ! 0. For internal trajectories, the agreement between the DRM and the HDM is very good, although the results differ at least in a factor 2, but might provide an initial idea of the energy losses due to plasmons excitation in experimental problems. References [1] M. Endo, Ph.D. Thesis Dissertation, Orleans University, France, 1975. [2] S. Iijima, Nature 354 (1991) 56. [3] S. Iijima, T. Ichihashi, Nature 363 (1993) 603.
[4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
S.J. Tans et al., Nature 386 (1997) 474. N. Hamada et al., Phys. Rev. Lett. 68 (1992) 1579. J.P. Salvetat et al., Phys. Rev. Lett. 82 (1999) 944. J.W. Mintmire et al., Phys. Rev. Lett. 68 (1992) 631. R. Saito et al., Phys. Rev. B 46 (1992) 3. R. Saito et al., Imperial College Press, 1998. A.C. Dillon et al., Nature 386 (1997) 377. P.C. Collins et al., Science 292 (2001) 706. T. Cohen-Karni et al., Nature-nano 1 (2006) 36. N.R. Arista, M.A. Fuentes, Phys. Rev. B 63 (2001) 165401. J.L. Gervasoni, N.R. Arista, Phys. Rev. B 68 (2003) 235302. S. Seguí, J.L. Gervasoni, N.R. Arista, Nucl. Instr. and Meth. B 233 (2005) 227. A. Aligia et al., Phys. Rev. B 70 (2004) 235331. T. Stockli et al., Phys. Rev. B 64 (2001) 115424. T. Stocki et al., Appl. Phys. Lett. 80 (2002) 29820. D.J. Mowbray et al., Phys. Rev. B 70 (2004) 195418. T. Pichler et al., Phys. Rev. Lett. 80 (1998) 4729. S. Seguí. Personal communication. Y.N. Wang, Z.L. Miskovic, Phys. Rev. A 66 (2002) 042904. Y.N. Wang, Z.L. Miskovic, Phys. Rev. A 69 (2004) 022901. D.J. Mowbray et al., Nucl. Instr. and Meth. Phys. Res. B 230 (2005) 142.