Fullerenes, Nanotubes, and Carbon Nanostructures, 20: 1–16, 2012
Copyright © Taylor & Francis Group, LLC
ISSN: 1536-383X print / 1536-4046 online
DOI: 10.1080/1536383X.2011.586618

One-Dimensional Uneven Peanut-Shaped C60
Polymer: A Quantum Electronic System in
Riemannian Space

Downloaded by [Tokyo Kogyo University] at 19:02 26 October 2011

JUN ONOE1 , TAKAHIRO ITO2 , SHIN-ICHI KIMURA2 ,HIROYUKI SHIMA3 , YASUHIRO TODA3
AND HIDEO YOSHIOKA4
1

Research Laboratory for Nuclear Reactors, Tokyo Institute of Technology,
Tokyo, Japan
2
UVSOR Facility, Institute for Molecular Science and School of Physical Science,
The Graduate University of Advanced Studies (SOKENDAI), Okazaki, Japan
3
Division of Applied Physics, Faculty of Engineering, Hokkaido University,
Sapporo, Japan
4
Department of Physics, Nara Women’s University, Nara, Japan
When a C60 film is irradiated with a 3 kV electron-beam in an ultrahigh vacuum(base pressure: 10−7 Pa), C60 molecules are coalesced to form a one-dimensional(1D) uneven peanut-shaped C60 polymer. From a topological viewpoint, since the polymer has both positive and negative Gaussian curvatures, it can be regarded as a new
allotrope of nanocarbons different from graphene, fullerenes, and nanotubes. We will
present our recent results of the electronic, optical, and electron-transport properties of
the 1D uneven peanut-shaped C60 polymer and discuss its novel electronic properties
on the basis of quantum mechanics in Riemannian space.
Keywords One-dimensional uneven peanut-shaped C60 polymer, gaussian curvature,electronic properties, optical properties, electron-transport properties

Introduction
Since the first report on C60 photo-polymerization (1), there have been many reports on
C60 polymers obtained using various kinds of methods, as shown in Table 1 (2). We
have produced C60 polymers from photo and electron-beam (EB) irradiation of pristine
C60 films and investigated their structural and electric properties. As shown in Figure 1,
photo-polymerization provides two-dimensional (2D) semiconducting dumbbell-shaped
C60 polymers (left) with a resistivity of 103 !cm (3–9), whereas EB-polymerization does
metallic peanut-shaped C60 polymers with that of 1–10 !cm (10–18). These resistivity
This article was part of the Proceedings of the 9th Biennial International Workshop “Fullerenes
and Atomic Clusters” held in 2009 in St. Petersburg, Russia. Other articles in the proceedings werepreviously published in Fullerenes, Nanotubes, and Carbon Nanostructures 18(4–6) and 19(1–2).
Address correspondence to Jun Onoe, Research Laboratory for Nuclear Reactors, Tokyo
Institute of Technology, O-okayama, Meguro, Tokyo 152-8550, Japan. E-mail: jonoe@nr.titech.ac.jp

1

2

J. Onoe et al.
Table 1
Summary of fullerene polymers obtained using various methods

Downloaded by [Tokyo Kogyo University] at 19:02 26 October 2011

MethodReactant

Cross-linkage

Structure

Photo-irradiation

C60 film

dumbbellsingle-bondHigh-Temperatureand high- pressure

Solid C60

dumbbellCharge transfer

AC60 ∗
Na2 RbC60
Na4 C60
C60 powder
KCN
Solid C60
C60 film

dumbbellsingle-bondsingle-bonddumbbell

1D
2D hexagonal
1D
1D orthorhombic
2D tetragonal
2D hexagonal
3D tetragonal/
hexagonal
1D orthorhombic
1D orthorhombic
2D tetragonal
dimer/trimer

dumbbellpeanut-shaped
dissociation

dimer
1D
3D amorphous

Solid catalysis
Electron-beam∗

A denotes alkali metals.

Figure 1. Schematic illustration of 2D hexagonal dumbbell-shaped C60 polymers (left) and 1D
uneven peanut-shaped C60 polymer (right) (taken from Refs. 11 and 14) (color figure available
online).

values were obtained using four-probe measurements at room temperature under atmospheric conditions, which are much smaller than that of 108 –1014 !cm of solid
C60 (19,20).
As shown in Figure 1 (left), the 2D C60 polymer has a [2 + 2] four-membered crosslinkage between adjacent C60 molecules by dissociation of C = C double bonds of each
C60 . In this case, the whole system is not regarded as a π -electron conjugated system, thus
exhibiting semiconducting. On the other hand, as shown in Figure 1 (right), the peanutshaped polymer has a coalesced structure via the general Stone-Wales rearrangement

Quantum Electronic System in Riemannian Space

3

Table 2
Classification of nanocarbon allotropes based on Gaussian curvature
π -electron system

Gaussian curvature

N-membered ringgraphene
fullerene
nanotubes
Mackay crystal∗ (hypothetical)
1D peanut-shaped polymers0+0(body) + (cap)
—
+, —

6
5,6
5,6
6,8
5–8

Downloaded by [Tokyo Kogyo University] at 19:02 26 October 2011∗

see Figure 13 (taken from Ref. 40).

(21,22) between adjacent C60 molecules. The EB-polymer can be regarded as a π -electron
conjugated system, thus exhibiting metallic. In addition, when we focus on the geometrical aspect of the EB-polymer, this peanut-shaped structure has both positive and negative
Gaussian curvatures (23), which are different from well-known π -electron conjugatednanocarbon allotropes (see Table 2).
In the present paper, we will introduce our recent works on the electron-transport (24),optical (25), and electronic properties (26,27) of the peanut-shaped C60 polymer with positive and negative Gaussian curvatures. In particular, we will discuss the effects of Gaussian
curvatures on the electronic states of the polymer on the basis of the Schrodinger equationin Riemannian space (28,29). Finally, we will describe the perspective of the peanut-shaped
C60 polymer from the viewpoint of topological science.

Experiments
Electron-transport Properties
To investigate the electron-transport properties of the peanut-shaped C60 polymer, we performed four-probe measurements in vacuum (24). After 12 hours of EB irradiation (applied
voltage: 3 kV) of a C60 film (100 nm thick) deposited on a cesium iodide (CsI) substrateby sublimation of C60 powder (99.98% pure) at 673 K in a ultra-high vacuum (UHV)
chamber (base pressure: 2 × 10–7 Pa) equipped with infrared spectroscopy, the film wastaken out of the chamber and moved into the other vacuum chamber (base pressure: 4 ×
10–4 Pa). Prior to a C60 film deposition on the CsI, the powder was preheated at 473 K until
residual organic solvents were completely removed from the powder. Gold (Au) contacts
were deposited on the peanut-shaped C60 polymer film through a metal mask, and the film
resistance was recorded as a function of the substrate temperature in the range of 9–400 K.
Optical Properties
To investigate the optical properties of the peanut-shaped C60 polymer, we examinedthe dynamics of photo-excited carriers in the polymer, using two-color pump-probe femtosecond spectroscopy (a temporal resolution of 200 fs), in which a coaxial configuration
between the pump (1.07 eV) and probe (1.55 eV) laser beams provides a high signal-tonoise ratio (S/N) (25,30). C60 films (200 nm thick) were formed on a CsI substrate in the
UHV chamber described in the previous section by sublimation of the C60 powder at 673 K.

4

J. Onoe et al.

The peanut-shaped C60 polymer was prepared by 3 kV EB-irradiation, and its formation
was confirmed using in situ infrared spectroscopy (31). We also prepared a pristine sample
without EB irradiation as a reference. The films formed were mounted on the cold finger ofan optical cryostat, and the lifetime of photo-excited carriers in the films was recorded as afunction of the substrate temperature (20–300 K). The details of the measurement systemand conditions were described in Ref. (25).

Downloaded by [Tokyo Kogyo University] at 19:02 26 October 2011

Electronic Structure
To clarify the origins of the metallic I-V characteristics of the peanut-shaped C60 polymer,
we examined the valence electronic structure around the Fermi level (EF ), using in situ
high-resolution ultraviolet photoemission spectroscopy (PES). Prior to a C60 film deposition, 100 mg of C60 powder (99.98% pure) introduced into a Knudsen cell (K–cell) was
heated at 473 K for a few hours to remove residual organic solvents from the powder in a
preparation chamber (base pressure: 1 × 10–7 Pa). Thereafter, a C60 film (20–30 nm thick)
was formed on a copper (Cu) substrate by sublimation of the pretreated C60 powder in theK–cell at 673 K for 3 minutes in the preparation chamber. Subsequently, the pristine C60
film was transferred to an analysis chamber (base pressure: 1 × 10–8 Pa) and measured insitu by PES with an energy resolution of 12 meV and a monochromatic He Iα (21.218 eV)
and/or IIα (40.806 eV) emission line. Although we used He Iα in Ref. (26), we used a He
IIα emission line to examine the π -electron behavior near EF for the outermost layers of
the EB-irradiated film in Ref. (27). After the measurements, the film was returned to the
preparation chamber and EB-irradiated for 0.5, 1.5, 2.5, 5.0, and 12 hours. We confirmed
by in situ infrared spectroscopy that 12 hours of EB-irradiation was sufficient to allow the
C60 molecules to completely coalesce, thus forming a peanut-shaped C60 polymer all overthe film. After each EB-irradiation, the film was transferred to the analysis chamber for insitu PES measurements. The zero value in binding energy for the present measurements
was determined from EF (= 0 eV) of a Au film, which was deposited on the same Cu substrate as used for the pristine and EB-irradiated C60 films, by fitting its PES spectrum witha Fermi distribution function at the measurement temperature (350 K). In the present substrate temperature, EF of a semiconductor was located in the middle between the valence
and conduction bands by thermal broadening and was almost the same position as that ofa metal with thermal broadening at the temperature.

Results and Discussion
Electron-transport Properties
We examined the temperature dependence of the electrical resistance of the peanut-shaped
C60 polymer. Figure 2 shows the Arrehnius plot of the resistance of the polymer as afunction of the substrate temperature. It was found that the electron–transport mechanism
changed at around 90 K. For a substrate temperature above 90 K, the electron–transport
was based on a thermally excited mechanism, and the activation energy of the electron
conduction was obtained to be 60 meV, which is smaller than that of 100 meV for multiwalled carbon nanotubes (MWCNTs) (8). Because the peanut-shaped C60 polymer can beformed only by EB irradiation of a C60 film, which can be deposited on any substrate bydry (evaporation) or wet (spin coat) processes, this polymer can be superior to carbon nanotubes (CNTs) when applied to electronic devices. On the other hand, for a temperature
below 90 K, although the electron–transport mechanism may change to a variable range

Downloaded by [Tokyo Kogyo University] at 19:02 26 October 2011

Quantum Electronic System in Riemannian Space

5

Figure 2. Arrehnius plot of the resistance of the peanut-shaped C60 polymer in the range of 9–400 K(taken from Ref. 24) (color figure available online).

hopping mechanism, in situ four-probe measurements are further needed to clarify whether
or not the hopping mechanism is based on 1D metallic conduction, Mott–type, or Anderson
localization.

Optical Properties
Typical transient $T(t), the difference in transmission from that of an equilibrium condition as a function of the delay time between the pump and probe pulses, for thepeanut-shaped C60 polymer and pristine C60 films are shown in Figures 3(a) and 3(b),respectively. In the high temperature range exceeding 60 K, $T(t) was identical for both
samples and showed an instantaneous response (hereafter called the “fast component”).
This common feature remained largely unchanged over the whole temperature range. On
the other hand, in the low temperature range below 60 K, $T(t) for the polymer sample
showed an additional relaxation component with a slow decay (hereafter called the “slowcomponent”), which became dominant in the low temperature range. It is noted that the fastcomponent was still present even at the lowest temperature examined in the present work,
though its outline was largely obscured by the slow component, as shown in Figure 3(a).
As a result of considering the temperature-dependent (T-dependent) fast component (25),since the excitation energy of the incident pulses (pump: 1.07 eV, probe: 1.55 eV) was
well below the optically allowed inter-band transitions of C60 molecules (17), the present
observed instantaneous response can be attributed to two-photon absorption (TPA). In addition, the transmission change was found to be linearly dependent on the pump laser power,
thus indicating that the signal consisted of a TPA process involving one pump photon andDownloaded by [Tokyo Kogyo University] at 19:02 26 October 2011

6

J. Onoe et al.

Figure 3. Typical transmission change $T(t) vs. decay time for (a) 1D peanut-shaped C60 polymer
and (b) pristine C60 film over a wide temperature range of 20–300 K (taken from Ref. 25) (color
figure available online).

one probe photon. This assignment agrees well with the transient profile, which is largely
consistent with the correlation between the incident pulses.
We next consider the T-dependent slow component appearing at a lower temperature for the peanut-shaped C60 polymer. Using the two-component exponential decayfunction,$T(t) = Af exp(−t/τf ) + As (T) exp[−t/τs (T)]where A and τ , respectively, denote the amplitudes and relaxation times as a function oftemperature, and the notations f and s refer to the fast and slow components, respectively.
We obtained a good fit with the data of Figure 3(a) for the polymer sample. Because thefast component could be regarded as T-independent, Af and τ f were fixed at constant values throughout the fitting procedure. The results of τ s (T) and As (T) obtained using theformula are shown in Figures 4(a) and 4(b), respectively. A signal with a decay timeof a few picoseconds started to appear below 60 K and became dominant with further
decreasing the temperature. A T-dependent transient signal with a similar decay time has
been observed for various quasi-1D compounds (30,32), in which the appearance of relaxation bottleneck reflects an energy gap associated with a Peierls transition. Therefore, theslow component appearance indicates an energy gap formation below 60 K for the peanutshaped C60 polymer in a similar manner to that for quasi-1D compounds, such as K0.3 MO3 .
Consequently, the results of Figures 3 and 4 suggest that the peanut-shaped C60 polymer is
a 1D or quasi-1D metallic carbon polymer.
Figures 5(a) and 5(b) show a schematic diagram of the transition processes accounting
for the fast and slow transient components, respectively. For an excitation energy smaller
than the inter-band energy gap, only the TPA process contributes to the $T(t), resulting
in an instantaneous response during the correlation between the incident two pulses. The
TPA transition is also applicable to the pristine sample. On the other hand, in the presenceof a narrow energy gap, even when the gap includes large fluctuations, the photo-excited
carriers from higher excited states to the gap states and vice versa contribute to the $T(t).

Downloaded by [Tokyo Kogyo University] at 19:02 26 October 2011

Quantum Electronic System in Riemannian Space

7

Figure 4. Temperature dependence of (a) decay time τ s and (b) amplitude As for the slow component
obtained from an exponential fitting to $T(t). The dashed line indicates the background level (takenfrom Ref. 25) (color figure available online).

Figure 5. (a) Schematic diagram of a TPA process consisting of one pump photon and one probe
photon. (b) Inter-band carriers relaxation processes in the presence of an energy gap at a lower
temperature (taken from Ref. 25).

The T-dependent $T(t) reflects the T-dependence of the energy gap and order parameter
fluctuations.
In typical quasi-1D compounds exhibiting the Peierls transition, the decay time
diverges near the transition temperature at which efficient phonon transitions (both emissions and re-absorptions) greatly make the relaxation time longer at the beginning of thegap formation. However, as shown in Figure 4(a), there was no clear divergence at around
60 K, and instead the decay time and amplitude increased monotonically with decreasing temperature down to 20 K. We consider that the different behavior of the decay timeis presumably due to a topological effect driven from the positive and negative Gaussian
curvatures at present, but it is further necessary to study the present subject.
Electronic Properties
Figures 6 shows the PES spectra of the EB-irradiated C60 film for several irradiation
times (350 K): 0 hour (red), 0.5 hour (orange), 1.5 hours (green), 2.5 hours (sky blue),and 5.0 hours (blue), along with that of the pristine C60 film measured at 300 K. For thepristine C60 film (red line), some intensive narrow bands such as HOMO and HOMO–1

Downloaded by [Tokyo Kogyo University] at 19:02 26 October 2011

8

J. Onoe et al.

Figure 6. EB-time evolution of valence photoemission spectra of a C60 film in a wide binding energy
region. All PES spectra were obtained at 350 K except that of pristine C60 film at 300 K (taken from
Ref. 27) (color figure available online).

were observed because of the weak interaction between adjacent C60 molecules. On theother hand, as EB-irradiation time increased, each band became broadened in response to
the disappearance of molecular character caused by EB-induced polymerization between
adjacent C60 molecules, and the valence electronic states spread monotonically toward EF
with increasing EB-irradiation time. After 5 hours of EB-irradiation, the electronic states
reached EF , as illustrated by the blue line.
To understand this behavior near EF in details, we next measured the EB-irradiation
time–dependence of PES spectra of the film in the binding energy of 3 eV below EF .
Figure 7 shows the evolution of PES spectra of the C60 film in the binding energy region
as a function of EB–irradiation time, where insets shows the spectra in the vicinity of
EF : 0 hour (red), 0.5 hour (orange), 2.5 hours (sky blue), and 5.0 hours (blue), whichwere symmetrized to remove the effect of thermal broadening from those own spectra.
Because the S/N of the 1.5 hour-irradiated C60 film PES spectrum (the green line shown in
Figure 7) was poor in comparison with that of the others, it was omitted in this figure. As
shown in Figure 7, the electronic states in the binding energy range of 0–2 eV gradually
increased in response to the broadening of the HOMO band located at around 2.4 eV
as the EB-irradiation time increased. This indicates that EB-irradiation time reduced theenergy gap, which is related to the electron-transport properties of a C60 film continuously.
This finding implies that the electronic properties of a C60 film can be controlled from aninsulator to a metal via a semiconductor continuously only within an EB-irradiated area,
which is interesting when the EB-irradiated C60 polymer is applied to the fabrication ofelectronic devices.
Why did the electronic properties of the C60 film change continuously from insulating to metallic via semiconducting as EB irradiation time increased? One reason is
attributed to the change in polymerization degree upon EB irradiation. We performed thefirst-principle calculations of the energy gap for the dimer (C60 -C60 ), trimer (C60 -C60 -C60 ),and one-dimensional system (-C60 -) with a give cross-linked structure between adjacent
C60 molecules (18) and found them to be 0.33, 0.11, and 0.02 eV, respectively, along with

Downloaded by [Tokyo Kogyo University] at 19:02 26 October 2011

Quantum Electronic System in Riemannian Space

9

Figure 7. EB-time evolution of valence photoemission spectra of a C60 film in the binding energy of
3 eV below the Fermi level. Inset shows the symmetrized spectra in the vicinity of the Fermi level inorder to remove the effect of thermal broadening (taken from Ref. 27) (color figure available online).

Figure 8. In situ high-resolution PES spectra of pristine C60 film (blue line) and 1D peanut-shaped
C60 polymer (red line) obtained at 300 K and 350 K, respectively. Inset shows the PES spectrum of
the 1D peanut-shaped C60 polymer near EF (taken from Ref. 26) (color figure available online).

that of 1.67 eV for C60 , which is in an excellent agreement with experimental results of
1.6–1.8 eV (33). Another reason is attributed to the general Stone-Wales rearrangement
between adjacent C60 molecules, which proceeded as EB irradiation time increased.
We next discuss the PES results obtained after 12-hour EB irradiation. Figure 8 shows
PES spectra of pristine C60 (blue line) and peanut-shaped C60 polymer (red line) films
obtained at a substrate temperature of 300 K and 350 K, respectively, where the inset showsDownloaded by [Tokyo Kogyo University] at 19:02 26 October 201110

J. Onoe et al.

the PES spectrum of the peanut-shaped C60 polymer near EF . For the peanut-shaped C60
polymer, each band was broadened in response to the disappearance of molecular character
caused by EB-induced polymerization between adjacent C60 molecules, and the valence
electronic states reached EF . In order to clarify the behavior of the electronic structureof the peanut-shaped C60 polymer in the vicinity of EF , we measured more precise PES
spectra of the pristine C60 and peanut-shaped polymer films in the binding energy of 3.0 eV
below EF , as shown in the inset of Figure 8. It is found that the photoemission spectrum of
the peanut-shaped polymer was much greater than that of the pristine C60 film near EF andclearly came across EF edge. These findings suggest that the peanut-shaped C60 polymer
has a metallic electronic structure. However, the results shown in the inset of Figure 8 were
measured at 350 K; thus, the effect of a thermal broadening on the PES spectra should betaken into account. This implies that it is possible that the peanut-shaped polymer is not
only metal or semimetal but also a narrow-gap semiconductor.
We next examined the T-dependent PES spectra in the vicinity of EF for the peanutshaped C60 polymer because the thermal broadening effect reduces with decreasing
temperature. For example, Figure 9 plots the Fermi distribution function near EF in therange of 50–350 K (350 K: red, 200 K: orange, 150 K: green, 100 K: sky blue, and 50 K:blue) (34). At 350 K, there is a large thermal broadening in the distribution function ina similar manner to the spectrum shown in the inset of Figure 8. As temperature lowers from 350 K to 50 K, thermal broadening becomes smaller and the Fermi step at EF
appears clearly at 50 K, as shown in Figure 9. We examined the T-dependent PES spectra
near EF for the peanut-shaped C60 polymer. Figure 10 shows the PES spectra measured
at 50 K (blue, top), 100 K (sky blue, second top), 150 K (green, middle), 200 K (orange,
second bottom), and 350 K (red, bottom) (34). It is found that the spectrum came across
EF even at 50 K, thus indicating that the peanut-shaped polymer is metal. Interestingly, thespectral function does not clearly show a Fermi step even at 50 K, unlike the prediction
shown in Figure 9. Two- and three-dimensional metal such as TaSe2 and Rhodium, respectively, always exhibit a Fermi step, but quasi-1D metal such as K0.3 MO3 and (TaSe4 )I
does not show such the step (35). Accordingly, the results of Figure 10 suggest that the

Figure 9. Plot of the Fermi distribution function as a function of temperature (red: 350 K, orange:
200 K, green: 150 K, sky blue: 100 K, and blue: 50 K) (taken from Ref. 34) (color figure available
online).

Downloaded by [Tokyo Kogyo University] at 19:02 26 October 2011

Quantum Electronic System in Riemannian Space

11

Figure 10. Temperature dependence of PES spectra in the vicinity of EF : 350 K (red), 200 K(orange), 150 K (green), 100 K (skyblue), and 50 K (blue) (taken from Ref. 34) (color figure available
online).

peanut-shaped C60 polymer is a 1D or quasi-1D metal, which is consistent with the resultsof the photo-excited carriers dynamics described in the previous section.
If the peanut-shaped polymer is 1D metal, the electrons should exhibit a TomonagaLuttinger liquids (TLL) behavior in which the density of states (DOS) shows a powerlaw dependence of the binding energy and temperature (the value of exponent α is lessthan unity) (36). For example, for metallic single-walled carbon nanotubes (SWCNTs), a
TLL behavior is observed (α = 0.5) (37), which is the direct evidence of 1D metal. Ina similar manner to SWCNT, we examined the T-dependent PES spectra near EF in therange of 30–350 K using a monochromatic He IIα emission line. We also observed a TLL
behavior for both binding energy and temperature and obtained α to be ca. 0.6 (29), thus
demonstrating that the peanut-shaped C60 polymer is 1D metal as well as SWCNTs. Here
we have a next question: what is the difference in the exponent (α) value between SWCNT
and the peanut-shaped polymer? Is the difference physically meaningful?
Riemannian Geometrical Effects on TLL Exponent
To understand the difference in the TLL exponent value between the SWCNT and peanutshaped C60 polymer, we focused on the difference in their topology based on Gaussian
curvature, as shown in Table 2. At the first step, we regarded the electron motion on the 1D

12

J. Onoe et al.

Downloaded by [Tokyo Kogyo University] at 19:02 26 October 2011

Figure 11. Schematic illustration of a quantum hollow cylinder with periodic radius modulation(taken from Ref. 28) (color figure available online).

peanut-shaped C60 polymer as free electrons moving on a quantum hollow cylinder withperiodic radius modulation (see Figure 11), resulting in a 1D curved surface with positive
and negative Gaussian curvatures aligned alternatively and periodically. On such the curved
surface, the Schrodinger equation dealing with electron motion should be written not in
Euclidean but rather in Riemannian space (28). As a consequence, the single-particle DOS,n(ω), near EF exhibits a power-law singularity as follows:n(ω) ∝ |!ω − EF |α , α =K + K −1−12Here, K (> 0) is a parameter to express the strength of electron-electron interactions andcan be written as
K = limq→0!

2π !υF + g4 (q) − g2 (q)2π !υF + g4 (q) + g2 (q)Here, vF denotes the Fermi velocity, and g4 (q) = V(q, m) and g2 (q) = V(q, m) – V(2kF , m)
are q-dependent coupling constants. V(q, m) is the Fourier transform of the screened
interaction. The details of the present theoretical treatments were described in Ref. (28).
Figure 12 shows the δr dependence of both K and α for different kF values. The insets
in Figure 12 show the kF dependence of K and α at δr/a =2.0. Figure 12 demonstrates
that K decreased and α increased significantly with increasing δr for δr/a exceeding 2.5.
Such δr-driven shifts in K and α are attributed to the effects of Gaussian curvatures on
the nature of TLL states. Consequently, it is interesting to note that the difference in TLL
exponent α value between SWCNT and 1D peanut-shaped C60 polymer can be explainedby Riemannian geometrical effects.

Summary and Perspectives
We introduced our recent results of the electron-transport, optical, and electronic propertiesof the 1D uneven peanut-shaped C60 polymer formed by EB irradiation of C60 films. This
polymer was found to be a 1D metal and can be regarded as a new allotrope of nanocarbon
materials based on Gaussian curvature. Indeed, the 1D metallic polymer exhibits a different behavior of optical and electronic properties from well-known 1D metal materials:
there is no divergence in the decay time upon an energy gap formation associated with
the Peierls transition, and the TLL exponent value is larger than that of 1D metal SWCNT

Downloaded by [Tokyo Kogyo University] at 19:02 26 October 2011

Quantum Electronic System in Riemannian Space

13

Figure 12. δr dependence of K and α defined by the equations described in section 3.4, respectively.
Insets show nonmonotonic behaviors of K and α as a function of kF at δr/a=2.0 (taken from Ref.
28) (color figure available online).

Figure 13. Schematic illustration of the unit cell of Mackay crystal consisting of 6- and 8-memberedrings colored by blue and red, respectively (taken from Ref. 40) (color figure available online).

14

J. Onoe et al.

Downloaded by [Tokyo Kogyo University] at 19:02 26 October 2011

with zero Gaussian curvature. In particular, the increase in TLL exponent is caused by
the Gaussian curvature that drives effective potentials acting on electrons motion along
Riemannian surface.
There have been some reports on carbon materials with a negative Gaussian curvatureexperimentally (38–40). However, they were just observed using a transmission electron
microscope, and their physical properties have not yet been examined. To the best of
our knowledge, the 1D metallic uneven peanut-shaped C60 polymer is the first material
with well-known physical properties that are affected by Riemannian geometry. Curvatureeffects on electronic, electric, and magnetic properties of condensates have been predicted
theoretically (41–45), for example, so-called Mackay crystal shown in Figure 13 (41). We
believe that the 1D uneven peanut-shaped C60 polymer is a good material to verify thetheoretical predictions and open a new science in Riemannian space.

Acknowledgments
One of the authors (J.O.) is greatly thankful to PRESTO/CREST-JST, Murata Foundation,
Sumitomo Foundation, and SCAT-Telcom for supporting the present studies financially. In
addition, he also thanks Dr. S. Ryuzaki and Mr. A. Takashima for PES measurements.

References
1. Rao, A. M., Zhou, P., Wang, K.-A., Hager, G. T., Holden, J. M., Wang, Y., Lee, W.-T. et al.
(1993) Photoinduced polymerization of solid C60 films. Science, 259: 955–957.
2. Onoe, J., Nakayama, T., Nakao, A., Hashi, Y., Esfarjani, K., Ohno, K., Kawazoe, Y., Aono,
M., and Takeuchi, K. (2002) The nano-structure of C60 photopolymers. In Clusters andNanomaterials, Springer-Verlag: Berlin, pp. 135–169, and references therein.
3. Onoe, J. and Takeuchi, K. (1996) In situ high-resolution infrared spectroscopy of a photopolymerized C60 film. Phys. Rev. B, 54: 6167–6171.
4. Onoe, J., Nakao, A., and Takeuchi, K. (1997) XPS study of a photopolymerized C60 film. Phys.
Rev. B, 55: 10051–10056.
5. Onoe, J. and Takeuchi, K. (1997) How many [2+2] four-membered rings are formed on a C60
molecule when photopolymerization is saturated? Phys. Rev. Lett., 79: 2987–2989.
6. Esfarjani, K., Hashi, Y., Onoe, J., Takeuchi, K., and Kawazoe, Y. (1998) Vibrational modes andIR analysis of neutral photopolymerized C60 dimers. Phys. Rev. B, 57: 223–229.
7. Nakayama, T., Onoe, J., Nakatsuji, K., Nakamura,K., and Aono, M. (1999) Photo√ J., Takeuchi,√
3x
3R30◦ -Ag : An STM study. Surf. Rev.
induced products in a C60 monolayer on Si(111)
Lett., 6: 1073–1078.
8. Onoe, J., Nakayama, T., Nakao, A., Hashi, Y., Esfarjani, K., Kawazoe, Y., Aono, M.,
and Takeuchi, K. (2000) In situ FTIR, XPS, and STM studies of the nano-structure of aphotopolymerized C60 film. Mol. Cryst. Liq. Cryst., 340: 689–694.
9. Onoe, J., Nakayama, T., Aono, M., and Hara, T. (2004) Electrical properties of a twodimensionally hexagonal C60 photopolymer. J. Appl. Phys., 96: 443–445.
10. Hara, T., Onoe, J., and Takeuchi, K. (2000) In situ Fourier-transform infrared study of electronirradiation-induced reaction in a C60 film. Jpn. J. Appl. Phys., 39: 1872–1876.
11. Hara, T., Onoe, J., and Takeuchi, K. (2002) Field emission from an electron-beam irradiated C60
film. J. Appl. Phys., 92: 7302–7305.
12. Onoe, J., Nakayama, T., Aono, M., and Hara, T. (2003) Structural and electrical properties ofan electron-beam irradiated C60 film. Appl. Phys. Lett., 82: 595–597.
13. Hara, T. and Onoe, J. (2003) Vibrational analysis of peanut-shaped C120 fullerenes. Eur. Phys.
J. D, 24: 389–392.

Downloaded by [Tokyo Kogyo University] at 19:02 26 October 2011

Quantum Electronic System in Riemannian Space

15

14. Onoe, J., Nakayama, T., Aono, M., and Hara, T. (2004) The electron transport properties ofphoto- and electron-beam-irradiated C60 films. J. Phys. Chem. Solids, 65: 343–348.
15. Onoe, J., Nakao, A., and Hida, A. (2004) Valence photoelectron spectra of an electron-beamirradiated C60 film. Appl. Phys. Lett., 85: 2741–2743.
16. Beu, T. A., Onoe, J., and Hida, A. (2005) First-principle calculations of the electronic structureof one-dimensional C60 polymers. Phys. Rev. B, 72: 155416.
17. Ueda, S., Ohno, K., Noguchi, Y., Ishii, S., and Onoe, J. (2006) Dimension dependence of theelectronic structure of fullerene polymers. J. Phys. Chem. B, 110: 22374–22381.
18. Beu, T. A. and Onoe, J. (2006) First-principles calculations of the vibrational spectra of onedimensional C60 polymers. Phys. Rev. B, 74: 195426.
19. Wen, C., Li, J., Kitazawa, K., Aida, T., Honma, I., Komiyama, H., and Yamada, K. (1992)
Electrical conductivity of a pure C60 single crystal. Appl. Phys. Lett., 61: 2162–2163.
20. Mort, J., Ziolo, R., Machonkin, M., Huffman, D. R., and Ferguson, M. I. (1991) Electricalconductivity studies of undoped solid films of C60/70 . Chem. Phys. Lett., 186: 284–286.
21. Stone, A. J. and Wales, D. J. (1986) Theoretical studies of icosahedral C60 and some related
species. Chem. Phys. Lett., 128: 501–503.
22. Ueno, H., Osawa, S., Osawa, E., and Takeuchi, K. (1998) Stone-Wales rearrangement pathways from the hinge-opened [2+2] C60 dimer to IPR C120 fullerenes. Vibrational analysis ofintermediates. Fullerene Sci. Technol., 6: 319–338.
23. “Gaussian curvature” is defined to be the product of the maximum and minimum curvatures at
a given point on a curved surface.
24. Onoe, J., Ochiai, Y., Ito, T., Kimura, S., Ueda, S., Noguchi, Y., and Ohno, K. (2007) Electronicand electron–transport properties of peanut-shaped C60 polymers. J. Phys. Conf. Ser., 61:
899–903.
25. Toda, Y., Ryuzaki, S., and Onoe, J. (2008) Femtosecond carrier dynamics in an electron-beamirradiated C60 film. Appl. Phys. Lett., 92: 094102.
26. Onoe, J., Itoh, T., Kimura, S., Ohno, K., Noguchi, Y., and Ueda, S. (2007) In situ high-resolution
photoelectron spectroscopic and density-functional study of a peanut-shaped C60 polymer. Phys.
Rev. B, 75: 233410.
27. Onoe, J., Ito, T., and Kimura, S. (2008) Time dependence of the electronic structure of anelectron-beam-irradiated C60 film. J. Appl. Phys., 104: 103706.
28. Shima, H., Yoshioka, H., and Onoe, J. (2009) Geometry-driven shift in the Tomonaga-Luttinger
exponent of deformed cylinders. Phys. Rev. B, 79: 201401(R).
29. Ito, T., Onoe, J., Shima, H., Yoshioka, H., and Kimura, S. (to be submitted).
30. Shimatake, K., Toda, Y., and Tanda, S. (2007) Selective optical probing of the charge-densitywave phases in NbSe3 . Phys. Rev. B, 75: 115120.
31. Onoe, J. and Takeuchi, K. (1995) In situ high-resolution FT-IR study of the orientational phasetransition in C60 film. J. Phys. Chem., 99: 16786–16791.
32. Demsar, J., Bijakovic, K., and Mihailovic, D. (1999) Single particle and collective excitations in
the one-dimensional charge density wave solid K0.3 MoO3 probed in real time by femtosecond
spectroscopy. Phys. Rev. Lett., 83: 800–803.
33. Kremer, R. K., Rabenau, T., Maser, W. K., Kaiser, M., Simon, A., Haluska, M., and Kuzmany,
H. (1993) High-temperature conductivity study on single-crystal C60 . Appl. Phys. A, 56:
211–214.
34. Onoe, J. (2009) Synthesis of new topological nanocarbons and their functional properties. In
Topology Designing, NTS Publisher: Tokyo, pp. 312–324.
35. Dardel, B., Malterre, D., Grioni, M., Weibel, P., Baer, Y., and Levy, F. (1991) Unusualphotoemission spectral function of quasi-one-dimensional metals. Phys. Rev. Lett., 67:
3144–3147.
36. Voit, J. (1994) One-dimensional Fermi liquids. Rep. Prog. Phys., 57: 977–1116.
37. For example Ishii, H., Kataura, H., Shiozawa, H., Yoshioka, H., Otsubo, H., Takayama, Y.,
Miyahara, T. et al. (2003) Direct observation of Tomonaga-Luttinger liquid state in carbon
nanotubes at low temperatures. Nature, 426: 540.

Downloaded by [Tokyo Kogyo University] at 19:02 26 October 201116

J. Onoe et al.

38. Luzzi, D. E. and Smith, B. W. (2000) Carbon cage structures in single wall carbon nanotubes: a
new class of materials. Carbon, 38: 1751–1756.
39. Terrones, M., Banhart, F., Grobert, N., Charlier, J.-C., Terrones, H., and Ajayan, P. M. (2002)
Molecular junctions by joining single-walled carbon nanotubes. Phys. Rev. Lett., 89: 075505.
40. Nasibulin, A. G., Pikhitsa, P. V., Jiang, H., Brown, D. P., Krasheninnikov, A. V., Anisimov, A.
S., Queipo, P. A. et al. (2007) A novel hybrid carbon material. Nature Nanotech., 2: 156–161.
41. Mackay, A. L. and Terrones, H. (1991) Diamond from graphite. Nature, 352: 762.
42. Townsend, S. J., Lenosky, T. J., Muller, D. A., Nichols, C. S., and Elser, V. (1992) Negatively
curved graphite sheet model of amorphous carbon. Phys. Rev. Lett., 69: 921–924.
43. Park, N., Yoon, M., Berber, S., Ihm, J., Osawa, E., and Tomanek, D. (2003) Magnetism in
all-carbon nanostructures with negative Gaussian curvature. Phys. Rev. Lett., 91: 237204.
44. Terrones, H. and Terrones, M. (2003) Curved nanostructures materials. New J. Phys., 5:
126.1–126.37.
45. Taira, H. and Shima, H. (2007) Electronic states in cylindrical surfaces with local deformation.
J. Phys. Conf. Ser., 61: 1142–1146.