Structures, fundamental properties, and potential applications of low-dimensional C60 polymers and other nanocarbons: a review

ABSTRACT Since carbon (C) atom has a variety of chemical bonds via hybridization between s and p atomic orbitals, it is well known that there are robust carbon materials. In particular, discovery of C60 has been an epoch making to cultivate nanocarbon fields. Since then, nanocarbon materials such as nanotube and graphene have been reported. It is interesting to note that C60 is soluble and volatile unlike nanotube and graphene. This indicates that C60 film is easy to be produced on any kinds of substrates, which is advantage for device fabrication. In particular, electron-/photo-induced C60 polymerization finally results in formation of one-dimensional (1D) metallic peanut-shaped and 2D dumbbell-shaped semiconducting C60 polymers, respectively. This enables us to control the physicochemical properties of C60 films using electron-/photo-lithography techniques. In this review, we focused on the structures, fundamental properties, and potential applications of the low-dimensional C60 polymers and other nanocarbons such as C60 peapods, wavy-structured graphene, and penta-nanotubes with topological defects. We hope this review will provide new insights for producing new novel nanocarbon materials and inspire broad readers to cultivate new further research in carbon materials.


Introduction
Since the discovery of low-dimensional carbon materials such as zero-dimensional C 60 [1], one-dimensional (1D) carbon nanotubes (CNTs) [2], and 2D graphene [3] from the end of the 20th century to the beginning of the 21st century, they are well known to have cultivated new science and industrial applications so far [4][5][6].
Since C 60 has unique features such as solubility (against organic solvents such as toluene) and volatility (Chap.5 of Ref. [4]), it is easy to produce C 60 films on any substrate using spin-coat or thermal evaporation method, which is advantage for device fabrication.In addition, the electric conductivity of C 60 films can be controlled from insulating to semiconducting/metallic using photo/electron-induced polymerization (Chap.7 of Ref. [4]) with a variety of cross-linked structures based on the generalized Stone-Wales (GSW) transformation [7] in association with topological defects.Figure 1 shows schematic representation of the change in geometrical structure from dumbbell-shaped C 60 dimer to C 120 nanotube via the GSW transformation [7] in association with topological defects.Although the other polymerization methods such as high-pressure/high-temperature (HPHT) treatment [10], alkali-metal doping [11], and mechanochemical reaction [12] are also well known so far, these methods are however hard to be employed practically in device fabrication processes.On the other hand, since photolithography and electron-beam (EB) lithography are well known to be a standard technique for many semiconductor devices, the modification of the physicochemical properties by photo/electron-induced polymerization is useful to improve the device performance such as solar cells [13].
In this review, we will introduce the structures, fundamental properties, and potential applications of 1D peanut-shaped C 60 polymer and related materials formed via electron-induced polymerization in Section 2, and those of 2D dumbbell-shaped C 60 polymer and related materials formed via photo-induced polymerization in Section 3. Thereafter, we will introduce the other low-dimensional nanocarbon materials such as C 60 peapods, wavy-structured graphenes, and penta-nanotubes in Section 4. Finally, we will describe the outlook for future research in nanocarbon materials in Section 5.Although nanodiamonds are regarded as quantum dots, there have been many review papers so far [27][28][29].Thus, they are omitted in this review.[8].The structures of P01-P24 derived from the GSW transformation [7] were geometrically optimized using Gaussian03 package [9].Takashima et al. [8]; licensed under a Creative Commons Attribution (CC BY) license.

One-dimensional peanut-shaped C 60 polymers and related nanocarbons formed via electron-induced polymerization
functional theory (DFT) [8].Comparison between the experimental and theoretical IR spectra suggests that the cross-linked structure of the 1D peanut-shaped C 60 polymer is roughly close to that of P08 via intermediate cross-linkages of P01, P04, and P06 (see Figure 1) [30].Since the IR peaks originating from the 1D polymer are intense and narrow (the full width at half maximum is almost same as that of pristine C 60 film), the cross-linkage is considered to be one given structure.
Masuda et al. further examined the 1D polymer obtained from electron-beam irradiated C 60 single crystal (SC) films using high-resolution transmittance electron microscope (HRTEM).Judging from the cryo-HRTEM images and their corresponding fast Fourier-transformed (FFT) patterns [26] in comparison with the results of electron diffraction pattern [25], they demonstrated the structural change from pristine C 60 SC film to 1D C 60 polymer film by EB irradiation, as shown in Figure 3. Here, the unit cell of (a) and (b) is shown by a red solid line.C 60 molecules in the three-fold symmetrical [111] FCC layer (Figure 3(a)) are polymerized to form the 1D C 60 polymer [0001] HCP with an intermolecular distance of 0.93 nm along one of the three nearest neighboring directions of [−12 -10] HCP in association with the symmetry broken to change the stacking from FCC to HCP-m (Figure 4(b)).Here, FCC and HCP are the abbreviation of 'face-centered-cubic' and 'hexagonalclosed-pack', respectively.

Fundamental properties
The 1D peanut-shaped C 60 polymer with a concaveconvex periodic curved structure exhibited physical properties arising from 1D metal such as Peierls transition [31], charge-density-wave phonon anomaly [32], and 1D van Hove singularity of phonon dispersion [33].Especially, it is interesting to note that the 1D polymer film first demonstrates  [20].Onoe et al. [20]; licensed under a Creative Commons Attribution (CC BY) license.geometrical curvature effects on the electronic states [24,34], which has been a big puzzle in quantum mechanics since the theoretical predictions in 1950s [35][36][37].
The behavior of the free electrons on the curved surface is characterized by the Hamilton operator of the following equation (quantum mechanics of submanifold), Here, m* is the effective mass of the electron, (q 1 , q 2 ) is the curved coordinate system, g ij is the inverse matrix component of matrix [g ij ], g is the determinant of matrix [g ij ], and g ij is the metric tensor, respectively.h and k are the mean curvature and Gaussian curvature, respectively, and represent the degree of curvature at each point on the curved surface.Since h and k are the functions of g ij , the Hamilton operator is uniquely determined by g ij describing a shape once the shape of the surface is determined.It is worth noting that the second term appears as a new scalar potential [Jensen-Koppe-da Costa (JKC) potential] besides the first term corresponding to the kinetic energy of the electron.This is the effective electric field potential caused by the geometric curvature of the surface, and plays a role in driving surface-curvature induced changes in physical properties.
The 1D peanut-shaped polymer exhibits Tomonaga-Luttinger liquid (TLL) states [38,39] which is the direct evidence for 1D metal [24].Accordingly, the density-of-states (DOS: ρ) of 1D metal obey not Fermi-Dirac distribution function but a power-law dependence of binding energy or temperature when measured using photoelectron spectroscopy.Namely, Here, E, E F , and T denote binding energy, Fermi energy, and absolute temperature, respectively.When the exponent α is less than unity (α < 1), it can be regarded as 1D metal with TLL states.Actually, 1D metallic single wall (SW) CNTs show a power-law dependence with an exponent (α) of ca.0.5 [40].
As shown in Equation 1, to demonstrate whether the JKC potential affects the electronic states or not, Shima et al. first theoretically examined the change in the α using deformed cylinder models (Figure 1 of Ref. [34]), whose radius varies periodically, and found that the variation in the surface curvature inherent to the system gives rise to a significant increase in the powerlaw exponent of the single-particle DOS (Figure 5 of Ref. [34]).
Onoe et al. examined the exponent α of 1D peanutshaped C 60 polymer film using in situ high-resolution photoemission spectroscopy (PES) [24].As shown in Figure 4(a), the 1D peanut-shaped C 60 polymer shows a power-law dependence at 30 K, which indicates TLL states in case of 1D metal.By fitting the PES spectra with a power-law function of Equation 2 in the energy range of 18-70 meV, they obtained an exponent α of 0.66, as shown in Figure 4(c).Since the TLL exponent α value depends on the choice of an energy range set for fitting, they examined various energy range sets within 18-100 meV (a fitting accuracy of ±0.02) and obtained α to be 0.65 ± 0.08.In a similar manner, Figure 4(d) plots the temperature dependence of the ratio of the photoemission intensity at E F to the intensity at 0.5 eV in binding energy on a log -log scale, and demonstrates a power-law dependence on temperature in the range of 30-350 K, in which the exponent α was obtained to be 0.59 ± 0.04.Judging from the results of Figure 4, the TLL exponent α for the 1D peanut-shaped C 60 polymer can be concluded to be ca.0.6, which is significantly larger than that of ca.0.5 (0.43-0.54) for metallic SWCNTs [40,42,43].In the theoretical work [34], when the radial modulation degree (δr) of the uneven structure increased from 0 nm (a straight tube) to 0.16 nm (an uneven peanutshaped tube), the TLL exponent α increased from 0.5 to 0.6.As shown in Figure 4(a), the δr can be estimated to be ca.0.14 nm, which is in good agreement with the predicted value.Details of discussion have been described in Ref. [22].Thus, Onoe et al. first observed the geometrical curvature effects on electronic states that have been a big puzzle since 1950s.This is a novel property different from that of fullerenes, CNTs, and graphene.
It is important to consider the heat-resistance of the 1D peanut-shaped C 60 polymer for practical use.Nakaya et al. confirmed that the 1D polymer exhibits a heat-resistance at least up to 723 K annealing under UHV conditions [44], because the thermal heating system thus used can rise temperatures up to 723 K.This is comparable to that of polyimide film commercially available as the highest heat-resistance organic material.
In addition to the 1D peanuts-shaped C 60 polymer described above, electronic properties of the other peanut-shaped C 60 polymers have been investigated theoretically [45][46][47].Noda et al. performed first-principles calculations of several 1D peanut-shaped C 60 polymer models to examine their energetically stable structure and electronic properties [41].Figure 5 shows schematic illustration of 54 different 1D peanut-shaped C 60 polymer models optimized geometrically (T3 model was proposed by G. Wang et al. [45], and the other 53 models were derived from the T3 model via the GSW transformation).They consist of not only 5-/6-but also 7-and/or 8-membered rings.Unlike 1D CNTs consisting of only 6-membered ring, the 1D peanut-shaped C 60 polymer models have both positive and negative Gaussian curvatures respectively caused by the 5-and 7/8-membered rings beside 6-membered one.It is found from the results of Figure 5 that the FP5N model is most stable energetically among the 54 models (see Table 1 of Ref. [41]).Interestingly, the energetic stability of the 1D peanutshaped C 60 polymers depends on their geometric components: the more octagon and pentagon-octagon pairs (heptagon and hexagon-heptagon pairs) in their structures, the more stable (unstable) the 1D polymers.
Figure 6 shows the band structures of typical 1D peanut-shaped C 60 polymer models (T3, FP4K, FP5N, and FP6L) chosen among the 54 different models, and indicates that the metallic or semiconducting band structure of the 1D peanutshaped C 60 polymers depends on their carbonnetwork polygon patterns.Especially, the band structure of the FP5N model with the lowest total energy shows a metallic property (no band gap near E F ).

Ultrahigh density reversible memory atomically controlled by STM
Shrinkage in the intermolecular distances (center-tocenter distance) by ca.0.1 nm when two C 60 molecules form a dumbbell-shaped dimer [4] is required to be detected in real space using atomic force microscope (AFM) [48] and STM [49].This fact leads to readily discriminate C 60 monomers and its dimers, and gives rise to utilize the dimers as a carrier of digital information representing '1', whereas the monomers as that representing '0'.Thanks to the 1 nm size of a single C 60 molecule, when a C 60 dimer is used as a digital bit, it is expected that the density of digital bits reaches to more than tens of terabits per square inches ideally.This meets with the information-oriented society requiring innovative data storage technologies [50,51].
Utilizing molecular thin films as a recording media has been extensively proposed and investigated so far [52][53][54][55][56][57][58][59][60][61].When compared to other molecules, it is noted that C 60 molecule has following three advantages.Firstly, C 60 molecule and its thin film are relatively stable and easy to be handled in both dry and wet processes.Secondary, well-ordered high-purity C 60 thin films are easily produced on any substrate at RT. Thirdly, C 60 polymerization/depolymerization in its thin film can be controlled reversibly and locally at designated positions with a nanometer precision [53,55] using STM as described below.
To control the number of C 60 molecules chemically bound with each other at nanoscale, local excitation is a promising way.Electronic excitation or field ionization induced by atomically sharp STM tip [62][63][64] has been found to control the polymerization/depolymerization area at a single C 60 scale [53,55,60,65].For examples, Nakayama's group examined STM tip-induced dimerization/de-dimerization in a C 60 monolayer on a highly oriented pyrolytic graphite (HOPG) [65].Figure 7(a) shows a series of STM images acquired at a sample bias voltage (V s ) of +1.0 V and a tunneling current (I t ) of 20 pA in the same area at RT: pristine C 60 monolayer (left), dimerization (center) induced by negative (−2.7 V) Vs with a pulse width of 1 s, and dedimerization (right) induced by positive (+3.0 V) Vs with a pulse width of 1 s.Since C 60 molecule is freely rotated on an inert substrate such as HOPG, STM image of the molecule shows a spherical.On the other hand, since C 60 dimer with [2 + 2] four-membered cross-linkage is stopped rotating, STM image of the dimer shows a stripe with an internal structure [49].
Figure 7(b) shows the mesh imposed to the STM image after the dimerization (left) and schematically illustrates the corresponding dimerization (right).A slight shift in the molecular position of C 60 dimer can be measured from the STM images [49,65].
They further examined STM images of the intermolecular bonds between C 60 molecules in a multilayer film [53], and found that the probability of polymerization/depolymerization was higher than that in a monolayer film, because all the C 60 molecules in a monolayer inevitably interact with the underlying substrate surface more or less.In case of a C 60 trilayer film, the intermolecular distance in the dimer was estimated to be 0.9 nm [66], which is in good agreement with that of a dimer via the [2 + 2] cycloaddition reaction between C 60 molecules [12,49,[67][68][69].Furthermore, it is noted that C 60 trimers were also observed besides the dimers.Figure 8 shows that the unbound (a) and bound (b) states of C 60 molecules can be controlled locally and selectively just below the STM tip at RT only by changing the polarity of the V s .The high selectivity and reversible control between the monomer and polymer performed at RT is a specific feature of STM-induced C 60 polymerization/depolymerization when compared to alternative methods such as high-temperature/ high-pressure [10], photo-irradiation [70], and alkalimetal doping [11].The reason behind such the high selectivity is due to the electric-field-induced shifts in the molecular orbital energy levels associated with charge transfer to/from substrate, namely, ionization of C 60 molecules caused underneath the STM tip (Figure 8(c,d)).More details of the physical mechanisms of the selective induced polymerization/depolymerization have been theoretically investigated using first-principles calculations [68].
When we consider potential applications of polymerization/depolymerization control to memory devices and further to data storage technology, the density of digital bits, the speed of bit operations, and the nonvolatility of the bits play key roles in these applications.Since there have hitherto been some reports on improvement of the three key factors toward realization of polymerization/depolymerization-based data storage technologies, we will overview the status of those developments as below.
Data-storage centers are relying on the performance of magnetic hard disk drive (HDD), and it is highly demanded to increase the capacity of the data center by a higher density storage technology [53].Tremendous work has been devoted to increase the capability of magnetic recording technology.However, since there is a certain physical limitation coming from magnetic instability when decreasing the size of a bit, a lot of studies have been carried out for future data-storage devices beyond magnetic devices so far.Kalff et al. demonstrated an example of ultrahigh density data-storage using positional operation of single chlorine atoms on Cu(100) by STM [71], and achieved a bit density of 3.2 Pbits/inch 2 (P: peta-; × 10 15 ).However, the operation of bits and keeping the bit status require ultrahigh vacuum and low-temperature conditions (bit operation was demonstrated at 1.5 K).Deoxyribonucleic acid (DNA) molecules have also been investigated by Church et al. and Erlich et al., because they have a potential to provide a huge-capacity for store data [72,73].Although the information density of DNA-based storage device is expected to be 10 5 times larger than that of HDD [72], essential digital operations such as DNA sequencing (encoding) and reading (decoding) is still impractical in time and costs [73].
On the other hand, an ultrahigh density data storage based on local C 60 polymerization/ depolymerization has been shown to achieve 190 Tbits/inch 2 (T: Tera-; × 10 12 ) which is higher by two orders of magnitude than that of the state-ofthe-art HDD devices.Figure 9 shows bit operations at a single-molecule-level using a C 60 trilayer film at RT.The STM-induced formation of C 60 dimers/trimers is clearly observed as a depression (dark) area (Figure 9(b)).This is because C 60 molecules at the outermost layer react with other C 60 molecules in the underlying layers as illustrated in Figure 9(a).These facts indicate that not only writing/erasing digital bits but also readout of the recorded bits can be performed with the same STM tip (Figure 9(c)), thus being advantageous for realizing ultra-high density molecular memory devices.In addition, the nonvolatility of the recorded bits was confirmed to be more than 1 week [53].Although the density of digital bits using STM-induced C 60 polymerization/depolymerization is smaller than that using atomic-scale bit operation or DNA-based data storage technology, bit operations do not require low temperatures for single chlorine atoms on Cu(100) [71] or complicated and expensive procedures for DNA encoding and decoding [73].
When the STM-induced C 60 polymerization/depolymerization technique is practically used for highdensity data storage, there are several issues to be solved.From a viewpoint of storage speed, the speed of STM-induced polymerization/depolymerization must be at late comparable to that (10 6 -10 8 bits/sec) of the present HDD.Currently, the storage speed was reached to 3.6 × 10 2 bits/s at fast [58], which is actually limited by a pulse width (a few ms) of the applied voltage to induce C 60 polymerization/depolymerization.Further study is necessary to achieve data writing and erasing with a speed of ca. 10 6 bits/s for practical use by improving the quantum efficiency (η) of both polymerization (η p = 2 × 10 −7 reaction/hole) and depolymerization (η d = 3 × 10 −8 reaction/electron).

CO 2 immobilization
Immobilization and reuse of environmentally hazardous compounds such as CO 2 , NO x , or SO x will play key roles in solving our environmental and energy issues [74,75].One possible way is to use porous materials with nanospace [76].Given that these nanoporous materials exhibit high electrical conductivity or high heat-resistance, electrochemical or heat treatments can be utilized to enhance chemical reactions in their nanospace.
As described in Sections 2.1 and 2.2, the 1D peanutshaped C 60 polymer film has a HCP structure shown in Figure 3, which results in formation of sub-nm spaces with a size of ca.0.3 nm and 0.6 nm periodically arranged (Figure 1 of Ref. [76]).In addition, the 1D polymer exhibits physical properties arising from 1D metal and a high heat-resistance at least up to 723 K.These findings suggest that the robust sub-nm space can act as a specific reaction field in a similar manner to those of conventional porous materials [74][75][76][77][78][79][80][81].
After the 1D peanut-shaped C 60 polymer film was exposed to atmospheric air for 30 min at RT, Nakaya et al. examined the change in in situ infrared (IR) and mass spectra of the 1D C 60 polymer under UHV conditions in combination with first-principles calculations based on DFT when compared to those of the pristine polymer (Figure 3 and 4 of Ref. [82]), and found that CO 2 is immobilized as a carbonate ion (CO 3 2-) in the sub-nm space of the 1D polymer film via [CO 2 + H 2 O] reaction [51], despite the fact that the reaction [CO 2 + H 2 O] is hard to proceed at RT in the gas phase (the activation energy, E a = ca. 2 eV [83]).Although a pristine C 60 film also has a similar sub-nm space inside, it was confirmed that no significant differences in their IR spectral features before and after atmospheric-air exposure are observed (Figure S2 in the supporting information of Ref. [82]).Thus, the 1D metallic peanut-shaped C 60 polymer as a framework of the sub-nm space also plays a crucial role in promoting the reaction at RT.
To reveal the mechanisms behind the reaction occurred at RT in the sub-nm space, Kitagawa's group theoretically examined the most energetically preferable location of CO 2 molecule in the sub-nm space, using a sub-nm space model (Figure 10(a)).The electronic structure of the model obtained at the PM6 level of theory exhibited a weak charge polarization especially around the bridged concave portions (Figure S3a of Ref. [82]). Figure 10  optimized geometrical structure of the CO 2 -H 2 O transition state (TS) in the sub-nm space, whereas Figure 10(b) shows that of H 2 CO 3 formed from the reaction.Only under a lower convergence condition, one TS was found along the reaction coordinate (Figure S4 in the supporting information of Ref. [82]), whereas they found product H 2 CO 3 directly via no TSs under a higher convergence condition (Figure 10(b)).This suggests that the E a of [CO 2 + H 2 O] reaction is very small or negligible in the sub-nm space of the 1D polymer film, which resembles a proton transfer process [84].Figure 10(c) shows the dependence of valence molecular orbital (MO) energies on the ∠O-C-O bond angle.The lowest unoccupied MO (LUMO) energy was lowered monotonically to be a negative value at bond angles below 170°.This implies that the LUMO has an electron affinity.Furthermore, the insets show that a carbon pσ-type orbital of the LUMO expands more toward the obtuse-angle side of CO 2 at smaller ∠O-C-O angles.This suggests that the bending motion promotes [CO 2 + H 2 O] reaction from the obtuse-angle side, giving rise to a decrease in the E a .Thus, factors such those mentioned above work together to promote the reaction at RT via no or negligible TSs.Alternative mechanisms were also discussed, but they can be excluded (Supporting information in Ref. [82]).
There have been many conventional methods to activate CO 2 radioactive-chemically, thermochemically, biochemically, photochemically, electrochemically, and catalytically [85].The common idea among them is to add electrons to LUMO of CO 2 , because the LUMO features an antibonding orbital.Thus, the reduction of CO 2 makes O=C=O bonding weakened to become reactive.On the other hand, in the nanospace of 1D peanut-shaped C 60 polymer film, CO 2 is activated at RT via enhancement in its angular vibration by CO 2 molecule pinning due to local Coulomb interactions, which reduces the LUMO energy to be negative (electron affinity emerges).
The CO 2 uptake performance of the 1D C 60 polymer film is estimated to be 1.60 mmol/g, which is comparable to that of zeolites such as Li-LSX (1.34 mmol/g) and Na-LSX (0.87 mmol/g) obtained under the similar exposure conditions (in 1 atm air at RT) [86].

1D N-doped peanut-shaped C 60 polymer
To mitigate the greenhouse effects, considerable efforts have been made to convert CO 2 to valuable industrial chemicals.The widely used catalysts for CO 2 reduction are metals [87], metal oxides [88], metal chalcogenides [89] and metal -organic complexes [90].However, these metal-containing catalysts are costly and their energy efficiency is limited.Peanut-shaped carbon nanotubes (PSNTs) are of particular interest because of their exotic geometry with a large specific surface area and with positive and negative Gaussian curvatures respectively with negative and positive polarizations.Thus, PSNTs are expected to have good performance for CO 2 capture and conversion [91,92].
As shown in Figure 5, there are many types of 1D PSNTs studied theoretically, and FP5N was found to be energetically stable with a high symmetry and to exhibit a metallic feature [41], which share the structural features of C 60 cages and 1D uneven peanutshaped structures.Figure 11 shows the geometric structure of front (a) and side (b, d) views for FP5N, along with the electronic band structure and DOS (c).There exist 5-, 6-and 8-membered rings where the C atoms have much more complicated bonding environments that offer more flexibility to tune the reactivity when compared to conventional 1D CNTs.This enables us to improve the catalytic performance.Figure 11(c) shows that FP5N is metallic as the partially occupied bands cross the E F .When one nitrogen (N) atom is substitutionally doped into the C-framework of FP5N, there are four chemically nonequivalent doping sites (α, β, γ, and δ shown in Figure 11(b)).
The total energy calculations for the four N-doped FP5N configurations were carried out to determine the most preferred doping site.The configuration with N at γ site was found to be most stable, which is lower in energy by 0.04, 0.07 and 0.17 eV than the configurations with N at α, β and δ sites, respectively.Then, Wang et al. calculated the band structure of the most stable structure of N-doped FP5N.
Figure 12(a) reveals that the metallic feature still remains after N doping at γ site.When more N atoms are substitutionally doped into FP5N, the possible N doping sites were categorized into three different types: pyridinic N (in the hexagonal ring), pyrrolic N (in the pentagonal ring), and octatomic N (in the octatomic ring), as shown in Figure 12(d  Even though the energy change (0.65 eV) for the graphitic N is larger than that of the pyrrolic N (0.52 eV), pyridinic N (0.52 eV) and octatomic N (0.60 eV), it still lowers the overpotential by 0.92 V as compared to that in the pristine FP5N, which produces CO at a higher overpotential of 1.57 V.This significantly improves CO 2 conversion performance by N doping.Furthermore, since the binding energy of •COOH to the graphitic N (2.67 eV) is higher than that of the pristine FP5N (1.58 eV), •COOH binds more strongly on the graphitic N.
During the CO 2 conversion, the first electron transferred to the chemically stable CO 2 usually possesses a high overpotential.On the contrary, in the case of FP5N, the free energy changes of [CO 2 → •COOH] reaction for pyrrolic N, octatomic N and pyridinic N are estimated to be only 0.04, −0.01 and 0.06 eV, respectively.These free energy changes are rather small in contrast to the pristine FP5N (1.57eV), the graphitic N (0.48 eV) and many more promising electrocatalysts including transition metal Cu (0.41 eV).The binding energies of •COOH to pyrrolic N, octatomic N and pyridinic N are estimated to be 3.12, 3.17 •CHO] reactions, respectively.The corresponding overpotentials of CO 2 conversion are estimated to be 0.52, 0.52 and 0.60 V, respectively, which makes the whole reaction being spontaneous and exergonic.For octatomic N, the CO 2 conversion pathways are identified to be as shown in Figure 13(b) (front view).
Since the large free energy change of [•CO → CO (g)] reaction makes •CO further hydrogenated, the final product is CH 3 OH for pyrrolic N, octatomic N and pyridinic N.This is rare for all kinds of N-doped carbon nanomaterials as catalysts for CO 2 conversion.When compared to metal catalysts, the N-doped PSNTs also have some merits.For instance, the overpotential of Mo-Bi bimetallic chalcogenide for reducing CO 2 into CH 3 OH is 0.7 V [94], and among the metal electrocatalysts studied previously, copper is the only metal that is capable of reducing CO 2 to significant amounts of hydrocarbons and oxygenates with high overpotentials (0.9-1.1 V) [95,96], which is higher than that of N-doped PSNT.Thus, this finding demonstrates that the N-doped PSNT is a promising candidate to replace metal catalysts for effective CO 2 reduction.
In addition, it is also important to note that PSNTs possess much lower lattice thermal conductivity as compared to conventional carbon nanotubes (CNTs) besides the high performance for CO 2 immobilization and conversion.By using non-equilibrium molecular dynamics simulations and lattice dynamics together with DFT, Sun et al. found that the thermal conductivity of the PSNT is reduced by more than 90% as compared to that of CNTs and remains almost the same when different strains applied [97,98].This exhibits very different behaviors from those of CNTs (the thermal conductivity decreases monotonically with increasing strain).The insensitive response of thermal conductivity against strain is due to the insensitivity of its phonon DOS and group velocity against strain.Furthermore, when 5-8 membered defect is introduced to PSNT, the lattice thermal conductivity of PSNT with the defect is less by one-tenth than that of pristine CNT with a similar radius.This arises from the low phonon group velocity, short relaxation time, large lattice vibrational mismatch and strong anharmonicity [97].These findings provide new insight for 1D PSNT going beyond conventional CNTs.

Structures
Since the first report on C 60 polymerization by Ar-ion laser or UV-vis lamp irradiation of a pristine C 60 film at RT [70] by Rao et al., Onoe et al. investigated the structure of photopolymerized C 60 films using in situ FT-IR spectroscopy [16], in situ x-ray photoelectron spectroscopy (XPS) [17,18], and in situ STM [20,49] under UHV conditions.The results of FT-IR, XPS, and STM found that C 60 molecules are polymerized to form 1D/2D C 60 polymer with a cross-linkage of [2 + 2] cycloadditional four-membered ring.
Figure 14 shows (a) the dependence of the IR intensity ratio (I 2D /I dimer ) on photo-irradiation time for 100 nm-(red) and 200 nm-thick (black) C 60 films on cesium iodide (CsI) substrates, (b) schematic model for the spatial distribution of the dimer (blue) and 2D polymer (red) in each film, and (c) STM image of 100-h photo-irradiated C 60 thin film formed on the Si(111)/∏3×∏3-Ag substrate.Figure 14(a) shows that the IR intensity ratio of the 200 nm-thick C 60 film (blue) increased remarkably until 100-h irradiation and became saturated after 150-h irradiation, whereas that of the 100 nm-thick C 60 film (red) increased remarkably until 200-h irradiation and is extrapolated to be saturated after 500-h irradiation.It is found that the saturated (I 2D /I dimer ) value of ca.1.15 for the 100 nm-thick film was larger than that of ca.0.8 for the 200 nm-thick film.This difference can be well explained by assuming that the 2D dumbbell-shaped C 60 polymer is formed within a few surface layers, as shown in Figure 14(b).This assumption is strongly supported by the results obtained using XPS (photoelectrons were measured from a few nm in depth of the film) [17,18].In addition, it was observed from the result in Figure 14(c) that the 2D [2 + 2] dumbbellshaped C 60 polymer is formed on the outermost C 60 layer.Furthermore, even if the film thickness increases, the formation region of the 2D polymer from the outermost layer should remain unchanged as long as the same photo-irradiation time.Thus, the saturated (I 2D /I dimer ) value of the 100 nm-thick film was greater than that of the 200 nm-thick one.
For practical use of the 2D dumbbell-shaped C 60 polymer, it is important to consider its heat resistance.Unlike the 1D peanut-shaped C 60 polymer film, the [2 + 2] four-membered ring was drastically dissociated at temperatures exceeding 373 K with an activation energy of 1.25 eV [69].Accordingly, the 2D C 60 polymer is limited to be used for devices that not only work below 373 K but also are fabricated in all processes below the temperature.
More recently, semiconducting monolayer 2D C 60 networks (band gap: ca.1.6 eV) via cross-linking of C-C single and [2 + 2] cycloaddition bonds have been reported to be synthesized by Hou et al. and indicated no thermal decomposition after heating at 600 K for 10 min [99].However, since the decomposition of the 2D polymer proceeds on the basis of the kinetics with some activation energy as heating time becomes long, the heating condition thus examined is too short to confirm the heat resistance of the 2D monolayer C 60 networks.Accordingly, it is further necessary to investigate the heating-time dependence of the 2D network decomposition.Section 3.3.2will discuss the kinetics of thermal decomposition of a photopolymerized C 60 film with [2 + 2] cycloaddition bonds.

Fundamental properties
Onoe and Nakayama et al. examined the electrical properties of the 2D C 60 photopolymer using fourprobe measurement [19].Although Okada and Saito theoretically predicted the 2D polymer to be a semiconductor with a band gap smaller by onethird [100] than that (1.85 ± 0.04 eV) for pristine C 60 solids [101], no attempts to examine the electrical properties of the 2D C 60 photopolymer have been made until then.
Figure 15 shows the I-V characteristics and sheet resistance of photopolymerized C 60 film (after 400 h UV-vis irradiation) obtained in atmospheric air at RT. Here, the red circles represent the I-V characteristics measured using probes 1 and 4 (outer probes), whereas the blue squares represent the sheet resistance derived from voltages measured by probes 2 and 3 (inner probes), as shown in the inset.The sheet resistance and I-V characteristics of a pristine C 60 film could not be obtained, because its sheet resistance was beyond the range of the measurement system at that time.This was presumably due to oxygen in air which is well known to act as an electron capture (O 2 is well known to become O 2 -as a superoxide radical).Judging from the inset of the optical micrograph taken during the corresponding four-probe measurements, the 2D dumbbell-shaped C 60 polymer film seems to be uniformly smooth and to have no grain boundaries within the measurement area.The typical inter-probe distance was set in the range of 8-40 μm, ensuring an appropriate film thickness for the four-probe measurements.When the inter-probe distance (L = 8-40 μm) is much larger than the film thickness (t = 70 nm), the sheet resistance (R s ) of the film can be evaluated in terms of R s = 4.532 × (V/I), and the resistivity (ρ) can be calculated from ρ = R s × t.Here, V is the difference in voltage between the two inner probes, and I is the current flow between the two outer probes.
As shown in Figure 15, the I-V curve demonstrates that the 2D C 60 polymer is a non-doped semiconductor (except during the plateau of the I -V curve).Since the R s depends on the current flow between the two outer probes, the resistance thus obtained does not correspond to the carrier density of the 2D C 60 polymer film.The carriers in the film are apparently supplied by the biased metal probe during the measurements.Namely, the 2D C 60 polymer has no carriers (electrons or holes) contributing to the conductivity at RT.This makes it difficult to estimate the resistivity of the polymer using the four-probe measurement system at that time.In addition, although the temperature dependence of R s should be measured for the 2D C 60 polymer in order to determine the energy gap of the film from Arrhenius plot, the experimental system resulted in temperature variations only from RT to ca. 373 K, which were insufficient to estimate the energy gap.However, considering that R s and I-V characteristics of the pristine C 60 film could not be obtained using the measurement system, these results nonetheless demonstrate that the 2D photopolymer is a semiconductor, which is consistent with the theoretical prediction [100].According to first-principles calculations of C 120 dumbbell-shaped dimer (Figure 4 of Ref. [19]), it is clear that nodes where no electrons exist are formed at the [2 + 2] four-membered ring between adjacent C 60 molecules for the HOMO.This indicates that electrons cannot move from one C 60 to another without excitationinduced hopping.In other words, there is some energy gap for the electron hopping between adjacent C 60 molecules via the ring.Consequently, the 2D dumbbell-shaped C 60 polymer has a semiconducting property.On the other hand, electrons can move over the whole dumbbell dimer for the LUMO, indicating that the 2D C 60 polymer exhibits conducting properties when electrons are excited from the HOMO to the LUMO.
For a further study, we will examine the temperature dependence of R s for the 2D C 60 polymer film using in situ UHV four-probe measurement apparatus equipped with a helium cryostat [102,103].

Organic solar cells
Since C 60 has a large electron affinity of 2.68 eV [104], C 60 is often used an electron acceptor for organic solar cells (OSCs) that expects to be one of the wearable power generators.After photogenerated excitons are separated to electron/hole carriers at a donor/acceptor interface, the electrons diffuse to a cathode through the C 60 film.However, since the resistivity of pristine C 60 film is very high (2 × 10 6 Ωcm [43]) even under UHV conditions, it is one of the main factors to lower the energy conversion efficiency of OSCs.As described in the previous section, photopolymerized C 60 film exhibits a semiconducting property with the LUMO where electrons can move from one to another via the [2 + 2] bond between adjacent C 60 molecules (Figure 4 of Ref. [19]).
From this standpoint, Kato et al. [13] investigated the modification of the structural and optical properties of C 60 film used as an acceptor in terms of photopolymerization between adjacent C 60 molecules in order to improve the external quantum efficiency (EQE) of Zinc phthalocyanine (ZnPc)/C 60 OSC. Figure 16   schematically illustrates the OPV cell structures), and (b) UV-vis-NIR spectra in a wavelength region of 400-900 nm.As shown in Figure 16(a), the EQE increased in a wavelength region both of 400-480 nm and of 520-580 nm for photopolymerized C 60 film (named as p-C 60 film) used instead of the pristine one, which well corresponds to the increase in the absorbance of p-C 60 film in both the wavelength regions as shown in Figure 16(b).In addition, since the structure of ZnPc film used as a donor remained unchanged after 37-h photo-irradiation (Figure 3 of Ref. [13]), its optical characteristic is considered to be unchanged.In fact, they confirmed no changes in UV-vis-NIR spectra of 40 nm-thick ZnPc film even after 74-h UV-vis photoirradiation (photon energy: 2-4 eV, fluence: 0.4 W cm −2 ), as shown in Figure S1 of Ref. [11].Consequently, the increase in EQE in those wavelength regions is mainly due to an increase in the number of photogenerated intra-and inter-molecular excitons in p-C 60 film [105,106].Details of discussion on the other structural and physical properties of C 60 film influenced by photo-irradiation have been described in Ref. [11].

Organic thermoelectric materials
C 60 films exhibit a giant Seebeck coefficient (S) of more than 150 mV K −1 (at 300 K) [107] and a lower thermal conductivity (k) of 0.4 W m −1 K −1 (at 300 K) [108] when compared to those (S = 0.27 mV K −1 , k = 1.6 W m −1 K −1 at 300 K) of Bi 2 Te 3 commercially available [109], thus being expected to be one of powerful candidates for next-generation wearable power generator used to wearable devices (sensors, etc.).However, the electrical conductivity (σ) of C 60 films was reported to be an order of 10 −5 Ω −1 cm −1 [44], which is quite smaller than that (307 Ω −1 cm −1 ) of Bi 2 Te 3 [109].According to the figure of merits (Z = S 2 σk −1 ) used as a performance index, it is necessary to increase the σ of C 60 films drastically while suppressing the decrease in the S for practical use, because there is the trade-off relationship between S and σ (the Mott's formula) [110].For instance, a typical way to increase the σ of C 60 films is to use dopants such as alkali metals (potassium: K) [44,111] and organic di-metal complexes [112] into the films.However, while the σ was increased with a fraction of the complexes in the film, the S was comparably decreased.
One possible way to solve this issue is to utilize C 60 photopolymerization, because a part of polymerization in C 60 films contribute to increase the σ while suppressing the decrease in the S. Actually, the polymerization reduces their intermolecular distance by ca.10% via the [2 + 2] four-membered ring, thus increasing the σ [19,20].This idea is quite similar to that of the carrier energy filtering effects [113,114].This inversely consider that a perfect C 60 polymer network via the [2 + 2] four-membered ring is randomly broken to suppress a phonon-mediated thermal conductivity.When C 60 molecules are polymerized, the absorption band of C 60 films correspondingly becomes broadened toward a longer wavelength [13].Accordingly, the modification of the electrical and optical properties by photopolymerization is useful to improve the device performance such as OSCs, as shown in Figure 16.
Thus, the control of the proportions of individual C 60 photopolymers is expected to further modify the physicochemical properties of C 60 films for the device performance.To the best of our knowledge, there have been no reports to discuss the proportion distribution for C 60 monomer, dimer, trimer and larger oligomers kinetically so far, though Metelov et al. reported on depolymerization of C 60 dimer [115] and two-dimensional (2D) rhombohedral C 60 polymer [116] synthesized by the HPHT method.
They further examined the decomposition kinetics, and it is interesting to note from Figure 17(b) that x 2 (the dimer) decreases at temperatures exceeding 100°C, whereas x 3 (the oligomers) remains constant up to 140°C.Correspondingly, the E a of decomposition for (C 60 ) 2 dimer and (C 60 ) n (n ≧ 3) oligomers was obtained to be 1.26 eV and 1.78 eV from the Arrhenius plots of k 2 (c) and k 3 (d), respectively.Details of discussion on the reason behind the E a of the trimers larger than that of the dimer have been described in Ref. [117].Accordingly, one given annealing temperature in the range of 100-140°C makes it possible to proceed the decomposition of (C 60 ) 2 dimer with remaining the proportion of (C 60 ) n (n ≧ 3) oligomers unchanged.Figure 17(e) shows the results of x 1 , x 2 , and x 3 at an annealing temperature of 110°C.The proportion of the (C 60 ) 2 dimer (x 2 ) becomes smaller than that of the (C 60 ) 3 trimers (x 3 ) after more than 300 h.Thereafter, further photoirradiation will induce the polymerization of [C 60 + (C 60 ) n-1 (n > 3) → (C 60 ) n ] to increase the x n value.Therefore, the iterations of [photopolymerization/ 110°C-thermal decomposition] cycle can control the proportion of (C 60 ) n oligomers to modify the physicochemical properties of photopolymerized C 60 films for application to C 60 -based thermoelectric materials, as shown in Figure 18.[119].However, the C 60 peapods now can be synthesized conveniently with a well reproducibility by using sublimation of C 60 (dry process) or 'nano-extraction' (wet process) technique [120] in the presence of CNTs with open-ended tips.The arrangement of the encapsulated C 60 molecules can be controlled utilizing the diameter of CNTs used as a container.In particular, when using SWCNTs with a diameter of 1.4 nm, C 60 molecules with a diameter of 0.7 nm can efficiently achieve van der Waals contact within the internal space of SWCNTs.This results in formation of an ideal 1D C 60 array (Figure 19(b)).

Other low-dimensional nanocarbons
The distinctive structure of C 60 peapods has attracted considerable attention from the perspectives of physics and engineering.Especially, their electronic structures, optical, electrical, and magnetic properties have been extensively investigated so far [121].The impracticality of extracting the 1D C 60 molecules from the peapods has been focused to examine the properties of C 60 peapods treated as a C 60 /CNT composite nanocarbon.
Notably, the interactions between CNTs and C 60 are utilized to control the properties of the CNTs.For instance, the strong electron affinity of C 60 [105] causes electron transfer from SWCNTs to C 60 molecules, resulting in a partial positive charge distribution along the axis of SWCNTs.This makes SWCNTs featured as a p-type material.For example, Roth's group demonstrated a single-electron transistor using C 60 peapods [122].In addition, this property has also been reported to enhance a field emission efficiency [123].Furthermore, the dependence of C 60 filling ratio on the thermal conductivity of C 60 peapods has been investigated by equilibrium molecular dynamics (MD) simulations and obtained that the maximum thermal conductivity of C 60 peapod at filling ratio of 50% is greater by ca. 30 times than that of pristine CNT [124].
Attempts to produce 1D C 60 polymers via polymerization between the adjacent encapsulated C 60 molecules in the peapods have been extensively made since the material's discovery [119].Fusion induced by EB irradiation was confirmed in the first report [119], and  it was shown that fusion reactions between C 60 molecules progress upon 100 kV EB irradiation, eventually converting the internal 1D C 60 array in SWCNT to form double-walled carbon nanotubes (DWCNTs).DWCNTs can also be formed from C 60 peapods under heat treatment at 1200°C [130].This method is useful to synthesize DWCNTs with precisely controlled diameters at a bulk scale.Another way to polymerize C 60 in peapods is chemical doping, which emerges a metallic property of the peapods [131].
When C 60 peapods are considered as a starting material for the synthesis of 1D C 60 polymers, CNTs can be used as a test tube in order to track the polymerization reaction of C 60 using HRTEM [132].In particular, the development of spherical aberration correctors and high-speed imaging detectors has improved the resolution and frame rates of TEM significantly.This leads to investigate the structure and kinetics of reactions between C 60 molecules precisely at atomic scale.Nakamura and Harano et al. recorded TEM video with a millisecond time-resolution to capture EB-induced fusion reaction from two C 60 molecules to one coalesced C 120 and identified several metastable C 120 intermediate species based on their shapes and sizes [133].These reactions occur stochastically, and the lifetime of individual intermediate species varies correspondingly (Figure 20(a)).From the statistical analysis of numerous recorded images of cascade reactions, the average lifetime of various intermediate species was determined.High-energy short-lived intermediate species such as P02 were successfully observed as shown in Figure 20(b).Moreover, by examining both the frequency of individual reaction events in various temperature ranges and the temperature dependence of the reaction rates, it is possible to estimate the E a from the excited states [134].This revealed that the fusion reaction causes via the excited states of C 60 and C 120 intermediate species that depend on EB kinetic energy [135].

Wavy-structured graphenes
Noda et al. theoretically examined wavy-structured graphenes as new metallic carbon allotropes using first-principles calculations based on DFT [136].Although an ideal graphene is exactly a 2D flat layer atomically, the wavy-structured graphene has convex and concave curved structures induced by the topological defects consisting of 5-and 7-/8-membered rings, respectively.Figure 21(a) shows schematic illustration of the wavy-structured graphene with the defects, which consists of sixty C atoms per unit cell.Such the wavy-structured graphene can be formed from graphene by introducing topological defects via the GSW transformation [137,138].Thus, we examined the effects of the topological defect consisting of pentagon-octagon-pentagon membered rings (named Mickey-Mouse-shaped defect) on the wavy-structured graphene.
Unlike the 2D flat graphene which exhibits strongly dispersive bands intersecting at the Dirac point near E F , the band structure of the wavy-structured graphene with the Mickey-Mouse-shaped defects shows simultaneous occurrence of both flat and dispersive bands, which are parts of an anisotropic Dirac cone tilted from the isotropic one of the pristine graphene (see Figure 21(b)).Owing to the presence of the flat bands, the wavy-structured graphene has an extremely high DOS near E F , which is quite similar to that of high-performance metal catalysts such as platinum and palladium [139][140][141].As far as we examined, similar characteristics of the flat band and high DOS near E F have been found for the other wavy-structured graphene models (see Supplementary Information of Ref. [136]).

Penta-nanotubes
It is well known that CNTs can be obtained from graphene, namely (n, m) nanotube can be defined using a chiral vector, C nm = na 1 + ma 2 .Here, a 1 and a 2 are the unit vectors, and n and m are the integers to indicate the number of unit vectors in graphene.Due to the gapless semi-metallic feature of graphene, the electronic properties of CNTs exhibit a high chiralitydependence.Namely, a CNT is metallic only when its C nm satisfies (n-m) = 3l, where l is an integer.The difficulty in fabricating and separating CNTs with certain conductance (metallic or semiconducting) greatly hinders its application in nanoelectronics.
Going beyond the conventional graphene, pentagraphene exclusively consisting of only 5-membered rings was proposed by Wang et al. [142], which exhibits many interesting properties due to its unique geometric structure (negative Poisson's ratio, intrinsic piezoelectricity, giant out-of-plane, and second harmonic generation susceptibility) [143,144].In similar, pentananotubes can be constructed by rolling up the pentagraphene sheet along the (n, m) chiral vectors with n=m ranging from 2 to 8, as shown in Figure 22(a) [142].The optimized geometry of a (3, 3) penta-tube is presented in Figure 22(b).Figure 22(c,d) shows the dynamic and thermal stabilities of this penta-nanotube obtained using phonon spectra calculations and Ab initio MD (AIMD) simulations, respectively.All the (n, n) pentananotubes are dynamically robust as well as thermally stable up to 1000 K.
More interestingly, the penta-nanotubes inherit the semiconducting feature of penta-graphene.As shown in Figure 23, they are all semiconducting with medium size energy band gaps from 2.008 to 2.731 eV [145].When the diameter of the inner layer increases from 5.69 Å in (4, 4) to 12.10 Å in (8,8), the band gap decreases rapidly with increasing the diameter.When the inner diameter is larger than 20.22 Å in (13,13), the bad gap becomes insensitive to the inner diameter due to the small curvature effects for large penta-nanotubes.This is very different from the conventional CNTs with a chiralitydependent conductance.Namely, they can be metallic or semiconducting that depends on their chirality.This unique feature of penta-nanotube demonstrates the advantages for application to nano electronic devices.In addition, the band gap can be effectively tuned by strain (Figure 23(b) for (9,9) nanotube), where the bad gap reduces linearly under pressive strain, while increasing nonlinearly with stretch.
Mechanically, the elastic modulus of pentananotubes was reported to be up to 1000 GPa [146][147][148][149], which is comparable to that of conventional CNTs [150].In addition, several other exceptional features such as the auxeticity at ambient conditions, the ultimate tensile stress of 85-110 GPa, and the critical strain up to 25% have confirmed the mechanical superiority of penta-nanotubes [146,151] over conventional CNTs.
In contrast to the feature of the bandgap, the fracture patterns of penta-nanotubes strongly depend on the chirality of the nanotubes.For example, Sousa et al. [148] showed that the armchair-type pentananotubes are suddenly fractured at lower tensile strain values due to the differences in bonding alignments when compared to zigzag-type penta-nanotubes.Interestingly, penta-nanotubes can have direct α-phase and inverted β-phase, which depend on the inner dimers being parallel or perpendicular to the main axis of the tube [152].In addition, strain engineering can be used to improve the structural performance of penta-nanotubes and to enhance their electronic and mechanical tunability under applied conditions.For instance, Wang et al. found that the compressive strain can drive a transition from semiconductor to metal in β-phase (4, 4), (5,5), and (6,6) tubes [152].Sousa et al. [146] showed that (11,11) βphase penta-nanotube experiences a structural transition to (11,11) α-phase under tensile stretching at RT.In addition, elasticity -plasticity transition can also be induced by tensile strain engineering [149].
Using large-scale MD simulations, Chen et al. [149] found that penta-tube has comparable mechanical properties with that of the conventional CNTs.However, unlike the brittle CNTs, penta-tube exhibits a large extensibility (with failure strain exceeding 60%) and behaves plastically during tensile deformation.The plasticity is inherently caused by the phase transition from pentagonal to polygonal (including trigonal, tetragonal, hexagonal, heptagonal, and octagonal) carbon rings.The plastic feature of penta-tubes depends intrinsically on the tubediameter and strain-rate.
Recently, Shah et al. [153] reported that porous carbon penta-nanotubes can be used as a sensor for halogen gas (F 2 , Cl 2 , Br 2 , I 2 ), where the host carbon atoms were selectively removed to create the nanopores on the tube surface, and the I -V curves suggested the performance of gas detection.Furthermore, interaction energy graphs showed the efficient separation of various halogen molecules by functionalizing the pores with F 2 , Cl 2 and H atoms.
Besides the penta-nanotubes formed from pentagraphene, the other penta-tubes have also been studied.For example, based on the experimental synthesis of penta-PdSe 2 sheet, Kuklin et al. [154] found that the electronic properties of (n, 0) penta-PdSe 2 tubes, like that of the penta-PdSe 2 monolayer, are semiconducting with similar band gaps, whereas (n, n) tubes exhibit indirect -direct band gap transitions following the increase in the tube diameter.Motivated by the recent synthesis of penta-NiN 2 sheet, Ati et al. [155] explored the H 2 adsorption on (n, 0) penta-NiN 2    The studies on penta-nanotubes have significantly expanded the family of 1D carbon nanomaterials with new features and provided more options for device applications.

Outlook
This review has focused on the structures, fundamental properties, and potential applications of low-dimensional C 60 polymers and related nanocarbons formed via photo-and electron-induced polymerization and those of other low-dimensional nanocarbons such as C 60 peapods, wavy-structured graphenes, and penta-nanotubes, as shown in the summary of Figure 24.
As described in Section 2.2, the 1D peanutshaped C 60 polymer is the only one material that can examine the properties in curved quantum systems experimentally.This is of great significance to cultivate a new interdisciplinary field between geometry and materials science by pioneering the unexplored doctrine of curved quantum systems (quantum mechanics of submanifolds).More recently, discrete geometry analysis has been applied to several 1D peanut-shaped C 60 polymers  for exploring the correlation between mathematical and physical quantities [156].Figure 25 shows the mapping of low-dimensional nanocarbons with respect to Gaussian curvature (k) and dimension (D), along with negatively curved graphitic 3D carbon crystals proposed by Mackay, Terrones, and the other groups [157][158][159].Interestingly, Park et al. theoretically predicted to exhibit magnetism due to trivalent carbon radicals in the tetrapod core with a concave surface (Figure 1 of Ref. [160]).As shown in Figure 25, a flexibility of bond direction by sp 2 -sp 3 combinations further discovers new novel carbon materials different from fullerenes, nanotubes, and graphene.
In our materials world that has hitherto been composed of two axes: physics and chemistry, novel materials have been discovered by 'knowledge', 'experience', and 'intuition' (for example, fullerenes, nanotubes, and graphene) so far.This is so-called 'Serendipity'.If modern mathematics is incorporated into the materials world as a new axis, novel materials discovered without 'Serendipity' (Beyond Serendipity) will no longer be a dream (Figure 3.21 of Ref. [15]).

Figure 4 .
Figure 4. (a) Schematic illustration of the 1D metal C 60 polymer with an uneven peanuts-shaped structure similar to the crosslinked structure of the P08 C 120 stable isomer shown in Figure 1.The area colored in sky blue represents a geometrical curved surface in which π-electrons move one-dimensionally.The δr denotes the degree of uneven deformation.(b) Temperature dependence of the photoelectron spectra of the 1D peanut-shaped C 60 polymer in the vicinity of E F .(c,d) The power-law dependence of the photoelectron spectral function shown in (b) on the binding energy (c) and temperature (d) [24].Onoe et al. [24]; licensed under a Creative Commons Attribution (CC BY) license.

Figure 7 .
Figure 7. (a) High-resolution STM images showing control of polymerization (left to central) and depolymerization (central to right) between C 60 molecules in C 60 monolayer film using STM.(b) Local modification of molecular arrangement in C 60 monolayer films after polymerization [65,65].Superimposed triangular mesh in left STM image shows two-dimensional periodicity of pristine C 60 molecules.Right picture shows schematic side view of a C 60 dimer in the monolayer film.

Figure 8 .
Figure 8. Effective probabilities of (a) Polymerization and (b) Depolymerization between C 60 molecules induced by an STM tip in a C 60 trilayer film [53].The measurements were made as a function of sample bias V s from about −4.0 to 4.5 V at different tunneling currents I t = 20, 80, and 200 pA.Insets in (a) and (b) Schematically show that local C 60 molecules are negatively and positively ionized due to an STM tip when negative and positive V s 's are applied, respectively.(c,d) Schematic potential energy diagrams of the whole experimental system consisting of the substrate, C 60 trilayer film, vacuum gap, and STM tip when negative and positive V s 's were applied, respectively.
(b) shows the

Figure 9 .
Figure 9. Ultrahigh-density data storage using C 60 trilayer film as a recording media [53,58].(a) Schematic side views of binary digital data ('0' and '1') stored in C 60 film.(b) STM images observed before (left) and after (right) data writing.(c) STM images showing data erasing (left to central) and rewriting (central to right).All of data storage operation were carried out at RT.

Figure 10 .
Figure 10.(a) Model structure of 1D peanuts-shaped C 60 polymer film for quantum chemical calculations.(B) Optimized molecular structure of (a) CO 2 , (b) transition state (TS) obtained at a lower convergence, and (c) H 2 CO 3 in the sub-nm space.Symbols p and q correspond to those in the front views of Figure 5(a).The contacting carbon atoms and adsorbed molecules are shown as ballmodels, and are connected by dashed lines together with their distances.(C) Change in the frontier orbital energies of CO 2 with respect to ∠O-C-O bond angle.At each angle, only the C-O bond distance was optimized.As the angle decreases, only the LUMO energy decreases monotonically and becomes negative below 170°, implying that the LUMO has an electron affinity [82].Nakaya et al. [82]; licensed under a Creative Commons Attribution (CC BY) license.
).Each of these sites is unique in the N-doped FP5N structure.The preferred electrocatalyst candidates were screened by calculating the free energy changes for both •COOH and •OCHO radicals at all the catalytic sites in the first proton-coupled electron transfer (PCET) step[93].The free energy change of •COOH adsorption was found to be much smaller than that of •OCHO.This suggests that the first intermediate PCET step for CO 2 electrocatalytic reduction (CO 2 ER) is •COOH.The geometrical structure of •COOH adsorbed on top of the C atom near the graphitic N is shown in Figure 12(b) (front view) and (c) (side view).

and 3 .
10 eV, respectively, which are all higher than that of graphitic N.This is because pyrrolic N, pyridinic N and octatomic N are electron-rich, which bind CO 2 while electrons of graphitic N are located in the π* antibonding orbital, making them less accessible for CO 2 binding.The strong binding of CO 2 to pyrrolic N, pyridinic N and octatomic N sites accounts for the small free energy change for the first step: [CO 2 → •COOH].The free energy changes of [•CO → CO (g)] reaction at pyrrolic N, octatomic N and pyridinic N are all larger than that of [•CO → •CHO] reaction.This indicates that •CO is likely to produce •CHO rather than CO gas and further hydrogenated.The potential-limiting steps for pyrrolic N, octatomic N, and pyridinic N are found to be [•H 2 COH → CH 3 OH (g)], [•CHO → •HCOH], and [•CO →

Figure 12 .
Figure 12.(a) Electronic band structure and DOS of N-doped FP5N structure.(b) Front and (c) Side view of •COOH adsorbed on N-doped FP5N.(d) Schematic diagram of different N doping catalytic sites on FP5N [92].Reprinted with permission from Zhou et al. [92].Copyright 2019 Elsevier.

Figure 14 .
Figure 14.(a) The dependence of the intensity ratio (I 2D /I d ) on photo-irradiation time for 100 nm-(red) and 200 nm-thick (black) C 60 films on CsI, (b) Schematic model for the distribution of the dimer (blue) and 2D polymer (red) in each film, and (c) STM image of 100-h photo-irradiated C 60 multi-layers formed on Si(111)/√3×√3-ag substrate [20].Onoe et al. [20]; licensed under a Creative Commons Attribution (CC BY) license.

Figure 15 .
Figure 15.Sheet resistance (blue square) and current -voltage curve (red circle) of the photopolymerized C 60 film using four-probe measurements in air at RT. Inset shows the optical micrographs obtained during the measurements.A typical distance between adjacent probes was set in the range of 8-30 μm [19].Onoe et al. [19]; licensed under a Creative Commons Attribution (CC BY) license.

4. 1 .
C 60 peapods C 60 peapods can be regarded as composite materials in which C 60 molecules are encapsulated within the internal cavities of CNTs (Figure 19(a)), which was first reported in TEM images (Figure 19(b)) of C 60 molecules aligned in a single-walled CNT (SWCNT) that was accidentally observed by Smith et al.

Figure 18 .
Figure 18.Schematic illustration for controlling the proportion of (C 60 ) n oligomers via the iterations of [photopolymerization/ 110°C-thermal decomposition] cycle.

Figure 20 .
Figure 20.Millisecond-scale TEM video imaging of multistep conversion of C 60 peapods under electron-beam irradiation [133]: (a) Cinematographic frames (3.125 ms/frame) of a C 60 dimer undergoing generalized Stone-Wales transformation.The time begins with the start of recording, and the numbers on the right bottom are the intermolecular distances in nm.The scale bar is 1 nm.(b) Experimental, simulated images and proposed models of coalesced dimers for observed intermediates in TEM.Reprinted with permission from Shimizu et al. [133].Copyright 2022 American Chemical Society.

Figure 21 .
Figure 21.(a) Geometrical structure and (b) Band structure along Γ-X-M-Y-Γ directions and DOS of the wavy-structured graphene with Mickey-Mouse-shaped defects that are periodically aligned parallel to the y-direction.The inset in (b) shows the Brillouin zone of the wavy-structured graphene, where fractional coordinates at Γ, X, M, and Y points are given by (k x , k y ) = (0.00, 0.00), (0.50, 0.00), (0.50, 0.50), and (0.00, 0.50), respectively.A tilted Dirac cone is located between Γ and X points at the Fermi level.Horizontal dashed red lines in the band structure and DOS indicate the Fermi level [136].Reprinted with permission from Noda et al. [136].Copyright 2014 the Royal Society of Chemistry.

Figure 22 .
Figure 22.(a) Illustration of chiral vectors of penta-tube.Dashed lines with arrows denote the lattice basis vector.(b) Side view of the optimized structure of (3, 3) penta-tube, (c) The corresponding phonon spectrum, and (d) evolution of potential energy of (3, 3) penta-tube during AIMD simulations at 1000 K, where the insets show snapshots of atomic configuration at the end of the simulation [142].
tubes, and found that when the sheet is curved into nanotubes, the Ni-N distance is enlarged due to the stress, which weakens the orbital hybridizations between Ni and N, and reduces the charge transfer from Ni to N. The Ni ion with a less charge on the tube shows a weak polarizing ability for H 2 molecule.

Figure 24 .
Figure 24.Schematic illustration of C 60 -based low-dimensional nanocarbons in this review: structures and potential applications based on their fundamental properties.

Figure 25 .
Figure 25.Schematic mapping of low dimensional nanocarbons with respect to Gaussian curvature (k) and dimension (D), along with 3D Mackay crystal.