Curvature effects on collective excitations in dumbbell-shaped hollow nanotubes

ARTICLE IN PRESS Physica E 42 (2010) 1151–1154

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Curvature effects on collective excitations in dumbbell-shaped hollow nanotubes Hiroyuki Shima a,b,, Hideo Yoshioka c, Jun Onoe d a

Department of Applied Physics, Graduate School of Engineering, Hokkaido University, Sapporo 060-8628, Japan  Universitat Polite cnica de Catalunya, Barcelona 08034, Spain Department of Applied Mathematics 3, LaCaN, c Department of Physics, Nara Women’s University, Nara 630-8506, Japan d Research Laboratory for Nuclear Reactors and Department of Nuclear Engineering, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro, Tokyo 152-8550, Japan b

a r t i c l e in fo

abstract

Available online 28 October 2009

We investigate surface-curvature induced alteration in the Tomonaga–Luttinger liquid (TLL) states of a one-dimensional (1D) deformed hollow nanotube with a dumbbell-shape. Periodic variation of the surface curvature along the axial direction is found to enhance the TLL exponent significantly, which is attributed to an effective potential field that acts low-energy electrons moving on the curved surface. The present results accounts for the experimental observation of the TLL properties of 1D metallic peanut-shaped fullerene polymers whose enveloping surface is assumed to be a dumbbell-shaped hollow tube. & 2009 Elsevier B.V. All rights reserved.

Keywords: Tomonaga–Luttinger liquid Fullerene polymer Geometric effect

1. Introduction When the motion of a quantum particle is constrained to a geometrically curved surface, the surface curvature produces an effective potential field that affects spatial distribution of the wavefunction amplitude [1]. Such the geometric curvature effect on quantum states has been discussed in the early stage development of the quantum mechanics theory [2,3]. Still in the last years, the effect has focused renewed attention in the field of condensed matter physics, mainly due to technological progress that enables to fabricate low-dimensional nanostructures with complex geometry [4–10]. Several intriguing phenomena as to single-electron transport in curved geometry have been theoretically predicted [11–22], in addition to curvature-induced anomalies in classical spin systems [23–27] which imply untouched properties of quantum counterparts. Besides curvature ones, geometric torsion effects on quantum transport of relativistic [28] and non-relativistic [29] particles through twisted internal structures have been also discussed very recently. In the present study, we investigate the surface curvature effects on collective excitations of electrons confined in onedimensional (1D) dumbbell-shaped hollow nanotubes (see Fig. 1). This work is largely stimulated by the successful synthesis of peanut-shaped C60 polymers [30–33]. It was discovered that

 Corresponding author at: Department of Applied Physics, Graduate School of Engineering, Hokkaido University, Sapporo 060-8628, Japan. E-mail address: [email protected] (H. Shima).

1386-9477/$ - see front matter & 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2009.10.030

under electron-beam radiation of a C60 film, C60 molecules coalesce to form a peanut-shaped C60 polymer being metallic [34] and having a 1D hollow tubule structure whose radius are periodically modulated along the tube axis. Hence, the periodic variation in curvature is expected to provide sizeable effects on quantum transport, in which the system goes to Tomonaga– Luttinger liquid (TLL) states due to the 1D nature [35,36]. In fact, we shall see below that periodic surface curvature in the dumbbell-like tube enhances the TLL exponent a describing the singularity of spectral functions, which is qualitatively in agreement with the experimental results of 1D metallic peanutshaped C60 polymers obtained using in situ photoelectron spectroscopy [37].

2. Model and method Fig. 1 shows schematic illustration of a 1D dumbbell-shaped hollow nanotube. It consists of an infinite chain of equi-separated spherical surfaces with radius r0 , in which the chain is threaded by a thin hollow tube with radius r0  dr. The neck length s serves as the unit of length, and r0 =s ¼ 4 is fixed throughout calculations to mimic the actual geometry of peanut-shaped C60 polymers. Periodic modulation of the tube radius rðzÞ along the z direction is defined by 8 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi < r 2  z2 ; jzjo s ðregion IÞ; 0 ð1Þ rðzÞ ¼ : r  dr; s o z o s þ s ðregion IIÞ 0

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r (z) r r0

s

 0

z

Fig. 1. (Color online) Top: schematic illustration of a dumbbell-shaped hollow nanotube. It consists of a chain of large spherical surfaces threaded by a thin hollow tube. Bottom: cross section of the dumbbell-shaped hollow tube along with the tube axis z. The neck length s serves as the unit of length, and the radius of spheres r0 =s ¼ 4 is fixed throughout calculations. The width s of spherical regions is determined by dr as well as r0 (see text).

and rðzÞ ¼ rðz þ lÞ, in which pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s ¼ 2r0 dr  dr2 and l ¼ s þ2s:

ð2Þ

¨ We have established the Schrodinger equation HC ¼ EC for non-interacting spinless electrons confined to the dumbbell surface to deduce the TLL exponent a on the basis of the bozonization procedure [35,36]. Because of the axial symmetry, single-particle eigenfunctions of the system have the form of Cðz; yÞ ¼ einy cn ðzÞ. We used the confining potential approach [1] incorporated with the variable transformation from z to x defined by [38] Z z qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x ¼ xðzÞ ¼ ð3Þ 1 þ ðdr=dzÞ2 dz0 ;

Fig. 2. Two-period profiles of the curvature-induced effective potential Un ðxÞ, where (a) dr ¼ 1:0, L ¼ 6:78 and (b) dr ¼ 3:0, L ¼ 11:5 in units of s. The abscissa x, defined by Eq. (3), represents the line length along the curve on the surface with a fixed y. The integer n characterizing the angular momentum of eigenstates in the circumferential direction ranges from n ¼ 0 (solid), n ¼ 1 (dashed) to n ¼ 2 (dashed-dotted).

0

or equivalently ( r0 sin1 ðz=r0 Þ x¼ z þ r0 sin1 ðs=r0 Þ

for region I;

ð4Þ

for region II:

Then, we obtain the differential equation for the transformed wavefunction pffiffiffiffiffiffiffiffi cn0 ðxÞ  rðzÞcn ðzÞ ð5Þ such as [39,40] " # d2 s2 2 þ Un ðxÞ cn0 ðxÞ ¼ ecn0 ðxÞ; dx

e¼

2m s2 E

‘2

;

ð6Þ

where   2 1 s s2 Un ðxÞ ¼ n2   : 4 rðxÞ2 4r02

Fig. 3. (Color online) Energy-band structures of dumbbell-shaped hollow cylinders with (a) dr ¼ 1:0 and (b) dr ¼ 3:0 in units of s. Branches belonging to the angular momentum index n ¼ 0; 1; 2; 3 are represented by the symbols of circle, square, triangle, and diamond, respectively.

ð7Þ

Fig. 2 shows the spatial profile of Un ðxÞ for two periods, i.e., for 0 o x o2L with L ¼ xðlÞ. At the neck region, Un for n ¼ 0 takes minimum while those for n Z1 take maximum as understood from Eq. (7). It is worthy to note that the curvature dependence of Un ðxÞ is different from that of the curvature-induced potential

suggested by Ref. [1], in the latter of which the potential in the spherical region (i.e., region I) equals zero precisely. This difference comes from the transformation (5) that was used to eliminate the first-order derivative with respect to x involved in ¨ the original Schrodinger equation of the system; see Ref. [40] for details.

ARTICLE IN PRESS H. Shima et al. / Physica E 42 (2010) 1151–1154

Eq. (6) is numerically solved by the Fourier expansion method [40]. Fig. 3 shows low-energy band structures for dr=s ¼ 1:0 in (a) and dr=s ¼ 3:0 in (b), corresponding to those depicted in Fig. 2. Energy gaps at the Brillouin zone boundary, k ¼ p=L, become wider for a larger dr, as expected from the large amplitude of jUn ðxÞj with increasing dr.

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anomalies observed in TLL states; they include the decay in Friedel oscillation [43] and that of temperature (or voltage)-dependent conductance [44], where the power-law exponents depend on a. It is worthy to note that the similar increasing behavior of a was observed in sinusoidal hollow tubes [40], which implies that the presence of periodic curvature modulation rather than structural details is important for the increase in a to occur.

3. Results and discussions 4. Conclusion We now consider the TLL states of dumbell-shaped tubes in which the single-particle density of states nðoÞ near the Fermi energy EF obeys the form [35,36] nðoÞpj‘ o  EF ja :

ð8Þ

To make concise arguments, we assume EF to lie in the lowest energy band. We thus obtain [35,36]

a¼

K þ K 1  2 ; 2

ð9Þ

In conclusion, we demonstrated the curvature-induced enhancement of the TLL exponent a in dumbbell-shaped hollow nanotubes. The increase in a is attributed to the effective potential Un ðxÞ, thus can be regarded as a geometric curvature effect on quantum transport that is in the realm of existing materials of real peanut-shaped C60 polymers. We believe that experimental confirmation of the predictions will open a new field of science dealing with quantum electron systems on curved surfaces.

and K¼



2p‘ vF þ Vð2kF Þ 2p‘ vF þ 2Vð0Þ  Vð2kF Þ

Here, vF ¼ ‘ VðqÞ ¼ 

1

1=2 :

ð10Þ

dE=dkjk ¼ kF is the Fermi velocity, and

e2 log½ðq2 þ k2 Þr02  4pe

ð11Þ

with the dielectric constant e and the screening length k1 . According to the bosonization procedure [35,36], we set ks ¼ 1:0  103 so as to be smaller than all kF s values that we have chosen. We also set the interaction-energy scale e2 =ð4pesÞ ¼ 1:1  ‘ 2 =ð2m s2 Þ by simulating that of C60 - related materials [41,42]. Fig. 4 shows the dr- dependence of a for different kF values. Significant increases in a with dr for dr=a 43:0 are clearly observed, until reaching the plateau region at dr 4 3:7. These drdriven shifts in a originate from the enhancement of the potential amplitude jUn ðxÞj that results in a monotonic decrease in vF with dr at dr=a 43:0. The insets in Fig. 4 shows the kF dependence of a at dr=a ¼ 3:0, 3.5, 3.9, among which the last one presents an almost linear dependence on kF . The present results demonstrate that nonzero surface curvature yields diverse alterations in various kinds of power-law

Acknowledgments We acknowledge K. Yakubo, T. Ito and Y. Toda for stimulating discussions. H.S. thanks M. Arroyo and A. Fabregas for their supports and hospitalities during the completion of this work. He is also thankful for the financial supports from Kajima Foundation. This study was supported by a Grant-in-Aid for Scientific Research on Innovative Areas and the one for Young Scientists (B) from the MEXT, Japan. A part of the numerical simulations were carried out using the facilities of the Supercomputer Center, ISSP, University of Tokyo. References [1] [2] [3] [4]

[5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]

Fig. 4. dr- dependence of the TLL exponent a. The Fermi wavenumber kF is increased from kF s ¼ 0:2 to 1:6 from bottom. Inset: kF dependence of a at dr=s ¼ 3:0 (dashed), 3.5 (thin), and 3.9 (thick).

[32] [33] [34]

R.C.T. da Costa, Phys. Rev. A 23 (1981) 1982. B. De Witt, Phys. Rev. 85 (1952) 635. P.C. Schuster, R.L. Jaffe, Ann. Phys. 307 (2003) 132. V. Ya Prinz, V.A. Seleznev, A.K. Gutakovsky, A.V. Chehovskiy, V.V. Preobrazhenskii, M.A. Putyato, T.A. Gavrilova, Physica E (Amsterdam) 6 (2000) 828. D.N. McIlroy, A. Alkhateeb, D. Zhang, D.E. Aston, A.C. Marcy, M.G. Norton, J. Phys. Condens. Matter 16 (2004) R415. H. Terrons, M. Terrons, New J. Phys. 5 (2003) 126. I. Arias, M. Arroyo, Phys. Rev. Lett. 100 (2008) 085503. H. Shima, M. Sato, Nanotechnology 19 (2008) 495705. J. Zou, X. Huang, M. Arroyo, S.L. Zhang, J. Appl. Phys. 105 (2009) 033516. H. Shima, M. Sato, Phys. Status Solidi (a) 206 (2009) 2228. G. Cantele, D. Ninno, G. Iadonisi, Phys. Rev. B 61 (2000) 13730. H. Aoki, M. Koshino, D. Takeda, H. Morise, K. Kuroki, Phys. Rev. B 65 (2001) 035102. D.V. Bulaev, V.A. Geyler, V.A. Margulis, Phys. Rev. B 69 (2004) 195313. A.V. Chaplik, R.H. Blick, New J. Phys. 6 (2004) 33. A. Marchi, S. Reggiani, M. Rudan, A. Bertoni, Phys. Rev. B 72 (2005) 035403. J. Gravesen, M. Willatzen, Phys. Rev. A 72 (2005) 032108. H. Taira, H. Shima, Surf. Sci. 601 (2007) 5270. G. Ferrari, G. Cuoghi, Phys. Rev. Lett. 100 (2008) 230403. V. Atanasov, R. Dandoloff, A. Saxena, Phys. Rev. B 79 (2009) 033404. K.J. Friedland, A. Siddiki, R. Hey, H. Kostial, A. Riedel, D.K. Maude, Phys. Rev. B 79 (2009) 125320. S. Ono, H. Shima, Phys. Rev. B 79 (2009) 235407. Y.N. Joglekar, A. Saxena, Phys. Rev. B 80 (2009) 153405. H. Shima, Y. Sakaniwa, J. Phys. A 39 (2006) 4921. H. Shima, Y. Sakaniwa, J. Statist. Mech. (2006) P08017. S.K. Baek, H. Shima, B.J. Kim, Phys. Rev. E 79 (2009) 060106. S.K. Baek, P. Minnhagen, H. Shima, B.J. Kim, Phys. Rev. E 80 (2009) 011133. Y. Sakaniwa, H. Shima, Phys. Rev. E 80 (2009) 021103. B. Jensen, Phys. Rev. A 80 (2009) 022101. H. Taira, H. Shima, arXiv:0904.3149. J. Onoe, T. Nakayama, M. Aono, T. Hara, Appl. Phys. Lett. 82 (2003) 595. J. Onoe, T. Ito, S. Kimura, K. Ohno, Y. Noguchi, S. Ueda, Phys. Rev. B 75 (2007) 233410. J. Onoe, T. Ito, S. Kimura, J. Appl. Phys. 104 (2008) 103706. Y. Toda, S. Ryuzaki, J. Onoe, Appl. Phys. Lett. 92 (2008) 094102. T.A. Beu, J. Onoe, A. Hida, Phys. Rev. B 72 (2005) 155416.

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[35] J. Voit, Rep. Prog. Phys. 57 (1994) 977. [36] T. Giamarchi, Quantum Physics in One Dimension, Oxford University Press, Oxford, 2004. [37] T. Ito, J. Onoe, H. Shima, H. Yoshioka, S.I. Kimura, unpublished. [38] The new variable x corresponds to the line length along the curve on the surface with a fixed y. [39] N. Fujita, J. Phys. Soc. Jpn. 73 (2004) 3115.

[40] H. Shima, H. Yoshioka, J. Onoe, Phys. Rev. B 79 (2009) 201401(R). [41] A.F. Hebard, R.C. Haddon, R.M. Fleming, A.R. Kortan, Appl. Phys. Lett. 59 (1991) 2109. [42] A. Oshiyama, S. Saito, N. Hamada, Y. Miyamoto, J. Phys. Chem. Solid 53 (1992) 1457. [43] R. Egger, H. Grabert, Phys. Rev. Lett. 75 (1995) 3505. [44] C.L. Kane, M.P.A. Fisher, Phys. Rev. B 46 (1992) 15233.