Keywords

1 Introduction

Carbon is one of the essential elements in the world; in terms of abundance, it holds the sixth position of typical elements in the universe, fourth in our solar system, and about seventeenth in the Earth’s crust [1]. The approximated relative abundance for carbon ranges 180–270 parts per million [2]. It is also noteworthy that the presence of carbon in human beings as an element is only subsequent to oxygen [3] and therefore acquires around 18% of human body weight. One of the remarkable characteristics of carbon is that it can occur in a broad area of metastable phases modeled near ambient environments along with their extensive kinetic stability. Despite the fact that carbon in its elemental form is relatively scarce on the earth’s crust [1, 2, 4], it plays a significant role in the ecosystem of the earth. With the ongoing research toward the development of various unique forms of carbon, the current century can be rightly called “The era of carbon allotropes” [5]. Carbon nanoforms or nanostructures comprise various low-dimensional allotropes such as buckminsterfullerene or C60, carbon nanotubes, graphene, poly-aromatic molecules, and carbon quantum dots. The uses of these nanostructures have been explored in different areas like nanoscience, materials science, engineering, and technology [6,7,8,9,10,11,12]. Recently, nanotechnology has gathered much attention because of its direct application in developing novel materials comprising significant properties like better directionality, high surface area with flexibility, etc. [13,14,15,16,17,18]. These properties uncover various applications of carbon nanomaterials design in almost all research domains [9, 19,20,21,22,23]. Ergo, in recent past decades, carbon science has become a trending topic along with its nanoscience discipline.

Carbon is traditionally understood to occur in only two naturally occurring allotropic configurations known as graphite and diamond. Nevertheless, the crystal structure and properties of graphite and diamond are significantly different [24,25,26,27,28]. Chemically, the tendency of carbon atoms to create covalent bonds with other carbon atoms leads to the formation of novel allotropes in the carbon family [29] such as buckminsterfullerene [30, 31], carbon nanotubes [32, 33], and graphene [11]. Although the existence of carbon and its applications has been known to us for centuries, the modern timeline for the development of carbon science is represented in Fig. 1.

Fig. 1
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Timeline of carbon nanostructures

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A new chapter in the exploration of the carbon family began with the discovery of buckminsterfullerene’s (“buckyballs”) [30] in the mid-1980s accompanied by the discovery of fullerene nanotubules (“buckytubes”) [33]. The breakthrough discovery of these nanostructures triggered increased research efforts in the exploration of carbon materials. Table 1 presents some predictions and discoveries of carbon nanostructures.

Table 1 Timeline of predictions, discoveries, and observations of carbon nanostructures
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The theoretical and computational approach has made significant contributions in the field of carbon nanostructures (graphene, fullerenes, and carbon nanotubes) by offering a framework with predictive structures along with their chemical and physical properties. Computational framework in nanoscience has consistently complemented the experiments for the development of carbon nanostructures with the prediction of their properties. The theoretical approach also provides an understanding of the reaction and separation mechanisms of carbon nanostructures. Experimental methodologies like X-ray diffraction and nuclear magnetic resonance are used for probing and solving the crystal structure of any material. A computational approach can be used alternatively. Several methodologies were developed to deal with the problem of structure prediction. One prominent and effective model comprises investigating material’s crystal structure, energy, and thereby choosing the material with the lowest energy as the “best guess” solution. In this context, various methods have been established. Random crystal structure prediction is an easy way that produces random atomic compositions with optimization to stabilize those compositions (inside the limits of bond lengths) [43]. While random crystal structure prediction is simplistic, unbiased, and easy to parallelize, it necessitates sampling various configurations to achieve better results. Another widespread approach to improve efficiency is evolutionary algorithms [44], which initially starts with a random structure and then enriches guesses with the lowest-energy results with each iteration [45]. In order to improve the results of structural prediction, different algorithms, force statistics, and data mining [46,47,48,49,50] are used to study criteria for crystallization such as in the Inorganic Crystal Structure Database [51]. However, the drawback of data mining methodology is that it is identified by the compounds analogous to previously observed ones, hence, lacking in novel and distinct structural phases.  The recent approach for efficient crystal structure prediction involves partial experimental information to apply limitations on symmetry [52]. Every method has its own significance for different applications.

To accomplish electronic structure calculations of carbon nanostructures like fullerenes and model CNTs, many-body empirical potentials, empirical tight-binding molecular dynamics, and local density functional (LDF) means were utilized at beginning of the past decade [53, 54]. The Huckel approximation was used to investigate electronic structure for large Ih point group fullerenes [55]. The geometry optimizations of these large fullerenes were also carried by methodologies like molecular mechanics (MM3), semi-empirical methods [56], AM1 [57], PM3 [58], and Semi-Ab Initio Model 1 (SAM1) [59]. The computational strength has also been extensively evolving due to the availability of more powerful computing resources. Consequently, theoreticians are delighted in examining and developing carbon nanostructures past molecular mechanics and semi-empirical methods. An analysis of theoretical and computational approaches utilized to explore different nanostructures of the carbon family is provided in this chapter.

2 Zero Dimensional (0D) Carbon Nanostructures

2.1 Fullerenes

Fullerenes form a hollow cage-like arrangement of carbon atoms comprising solely of hexagon and pentagon rings. Buckminsterfullerene (C60) was the first in the series of developments of such carbon nanostructures [30]. Kroto, Curl, and Smalley were awarded the Nobel Prize in Chemistry in 1996 for this discovery. However, before the experimental realization of these fullerenes, they were first hypothesized by many researchers. In 1966, graphite molecules with hollow-shell were described in the scientific column “Daedalus” [34]. Subsequently, various theoretical hypotheses were made on the capability of 60 carbon atoms with truncated icosahedron [35, 36, 60, 61]. The occurrence of C60 was primarily predicted by Osawa in 1970 [35]. These results were later confirmed by mass synthesis of C60 by Krätschmer in 1990 using the carbon arc method accompanied by infrared (IR) spectroscopy for structure verification [62]. The aforementioned findings since then sparked widespread novel research for C60 along with other fullerene derivatives.

The study for fullerene with the early graphite laser vaporization was initiated and observed by Rohlfing et al. [63]. The carbon clusters formed in the experiments were noticeably bimaximal comprising of even and odd forms of Cn (where n <  = 25), while only even forms in Cn (where n >  = 40) relying upon their experimental situations. According to ab initio and various spectroscopic investigations, carbon clusters varying from n = 2 to 9 tend to present linear chain structures with single and triplet electronic ground states in odd and even clusters, respectively [63]. Contrary to that, some ab initio studies suggest that even number clusters in the range n = 2–8 show cyclic equilibrium structures with lower electronic states [64, 65]. Furthermore, Cn clusters ranging from n = 10 to 25 present monocyclic ground state configurations. The above-said conversion from linear chains to monocyclic rings is attributed to the fact that additional bonding associated with ring closure ultimately surpasses the strain energy acquired with the twisting of the polyyne chain to create a ring. Through semi-empirical molecular orbital theory calculations, the transformation point with 10 carbon atoms is predicted [66, 67]. However, according to the intensities in the high-mass region, these carbon nanostructures were indecisive and needed plausible explanations [68]. The photoionization time-of-flight mass spectrum (PI-TOF-MS) of these carbon nanostructures ranging from 1 to 100 atoms is presented in Fig. 2.

Fig. 2
figure 2

Reproduced with permission from Rohlfing et al. J. Chem. Phys 81, 3322 (1984). Copyright 1984 AIP Publishing

PI-TOF-MS spectrum (involving the amalgamation of two different spectra) for carbon clusters attained through doubled Nd:YAG vaporizing laser energy (40 mJ) and unfocused ArF ionizing laser energy (1.6 mJ and 193 run). The vertical deflection plate voltage of 300 V is utilized for Cn+, 1 < n < 30, leading to the optimization of C20+ collection while 600 V was utilized for C2n+, 20 < n < 50, for the optimization of C100+.

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Evidently, these elucidations must justify the detail that ion signals of even Cn were observed in the high-mass region. Subsequently, this instantly eliminates a variety of probable configurations for Cnclusters, for instance, fractions of diamond lattice/graphite sheet. These structures tend to present both even and odd peaks of mass by means of linear chains and monocyclic rings. Moreover, it does not exclude the possibilities of other configurations like “carbyne” [63]. The second probable reason consistent with this analysis would be that the second sets of high-mass region carbon clusters are all fullerenes (Fig. 2). This is the well-known fullerene hypothesis and has gathered much attention for the reason that closed cages bypass the dangling bonds of edges that are anticipated to destabilize fractions of diamond and graphite lattices [30] and additionally, due to the fact that trivalent cages fulfill the valence necessities of carbon atoms compared to linear chains and monocyclic rings. These qualitative theoretical aspects of the fullerene hypothesis along with electronic structure calculations provided support to the experimentation of C60 in 1985.

One of the important investigations performed was the comparison of carbon cages with chains, rings, and toroids along with fractions of infinite diamond and graphite lattices using semi-empirical models [69]. The analysis suggested that cage structures with atoms greater than 25 would be the most stable carbon clusters. Furthermore, the existence of solely pentagonal and hexagonal rings along with the unavailability of adjacent pentagonal rings were conditions for the stability of cage structures [69]. The affinity of fullerenes comprising limited adjacent pentagonal rings was also addressed by Kroto in 1987 using empirical arguments derived from chemical and geodesic rules [70]. The structures studied by Kroto in 1987 are presented in Fig. 3. It is noteworthy that both the aforementioned studies suggested that C60 was the smallest fullerene without adjoining pentagonal rings consists of D5h isomer of C70.

Fig. 3
figure 3

Reproduced with permission from Kroto, Nature 329, 529 (1987). Copyright 1987 Springer Nature

Structures of fullerenes by Kroto et al. a C70, most stable fullerene created by splitting two halves of C60 through 10 extra carbon rings, b C50, comprising isolated singlet and doublet pentagonal structures, c structure of C32 with threefold axis, and d C28 which is a tetrahedral fullerene.

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Kroto's inference of C60 with Ih symmetry as foundational fullerene was supported by Krätschmer et al. [62] in 1990 through four-band IR absorption spectrum and latterly in the same year by Taylor et al. [71] through 13C nuclear magnetic resonance (NMR) spectroscopy. Thereafter, several other configurations of fullerenes were synthesized including C76 [72], C78 [73, 74], and C84 [74, 75].

Various configurations of fullerenes are shown in Fig. 4. Each fullerene molecule shows the features of a carbon cage, as each atom is bonded to the other three carbon atoms in the same manner as in graphite [73]. The extensive series of techniques to synthesize fullerenes observed that C60 is the most plenteous among fullerenes accompanied by C70 [76]. C60 with Ih symmetry comprises two C–C bonds with (i) one at the link of two hexagonal rings denoted and (ii) one at the link of pentagonal and hexagonal rings. Contrarily, C70 with D5h symmetry consists of eight C–C bonds. It is noteworthy that two pentagonal rings sharing similar C–C bonds are energetically unfavorable. Mathematically, 1812 methods are known to build isomers of 60 carbon atoms, while C60 holds its uniqueness and special place with stability due to the fact that all of its pentagonal rings are secluded by its hexagonal rings. The state is known as the “isolated pentagon rule” (IPR) [77]. C60 being the smallest member of fullerene family obeying the IPR, C62, C64, C66, and C68 fullerenes does not follow the IPR.

Fig. 4
figure 4

Reproduced with permission from Yan et al. Nanoscale 8, 4799 (2016). Copyright 2016 Author(s), licensed under the Creative Commons Attribution 3.0 Unported License

Structures of fullerenes along with their symmetries. Reprinted with permission from Ref. [73].

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Figure 5 presents that the number of IPR isomers is directly proportional to the size of fullerenes. The studies of IPR with possible isomers of fullerenes assisted the experimentalists to identify and characterize them [78,79,80]. For instance, in C78 (consisting five isomers), isomers with C2v and D3 symmetry were identified using 13CNMR spectra [74]. Theoretical investigations of C82 lead to experimental characterization of its three isomers having C2 symmetry also using 13C NMR spectra [74, 81]. Additionally, several computational investigations were performed since the discovery of fullerenes to thoroughly study their isomers and subsequently to predict the lowest-energy configurations of giant fullerenes [82,83,84,85,86,87,88,89]. Becke, 3-parameter, Lee–Yang–Parr(B3LYP)hybrid functional along with various basis sets were used to examine C86 along with its 19 isomers following IPR [87]. Their studies suggested that isomer 17 (C2 symmetry of C86) is the most stable among them followed by isomer 16 (Cs symmetry of C86). Similarly, several theoretical calculations played a crucial role in predicting accurate lowest-energy structures of the fullerene family [83, 87].

Fig. 5
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(the details of isomers were taken from Ref. [77])

Size of fullerenes with respect to the number of isolated pentagon rule (IPR) isomers

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Several theoretical and computational studies in the last decades have been dedicated to exploring C60 along with its chemical and physical properties. Theoretical investigations by Fowler and Steer [90] suggested that Cn (n = 60 + 6 k, k = an integer except one) should comprise closed-shell electronic structures. Schmalz et al. showed that the aromaticity of C60 is less than that of benzene [69] through resonance circuit theory and Huckel molecular orbital (HMO) theory. The stability occurring through bond delocalization was explained by Amic and Trinajstic [91]. The electronic and vibrational properties of C60 were evaluated through the two-dimensional HMO method [92]. Semi-empirical calculations involving overlapping of non-planar π-orbital were also given by the free-electron model in the Coulson–Golubiewski, self-consistent Huckel approximation for the curvature system [93]. The large-scale restricted Hartree–Fock calculations were carried out presenting electron affinity of 0.8 eV and ionization potential to be 7.92 eV with ΔHf = 415–490 kcal/mol [94,95,96]. On the basis of ab initio self-consistent field (SCF) theory, the heat of formation was also evaluated by Schulman and Disch [97]. To measure structural parameters, electronic spectra, and oscillator strength, the Pariser–Parr–Pople method and the CNDO/S method (with CI) were used by many researchers [98,99,100,101]. The ground and excited states of C60 presenting π-bonding character were determined by the tight-binding model using electron–phonon coupling [102]. The primarily vibrational properties of C60 were investigated by Newton and Stanton using MNDO theory [103]. It was observed that C60 contains four IR active modes because of its high symmetry (“t1u” symmetry) and 10 Raman active modes involving eight “hg” and two “ag” symmetries. The 174 vibrational modes of C60 contribute to 42 elementary modes with different symmetries. Proceeding to understand magnetic properties of C60, by means of HMO and London theories, the ring current magnetic susceptibility was evaluated with less than 1 ppm shielding because of the termination of the contribution of both diamagnetic and paramagnetic spins [104, 105]. The theory also presented the absence of usual aromatic behavior [104, 105]. Some investigations proposed that the diamagnetic part has been underestimated [106]. Fowler et al. (using coupled Hartree–Fock calculations) in their study proposed that the aforementioned shielding has to be approximately similar as for analogous aromatic structures [106]. Later on, Haddon and Elser addressed the shielding of fullerenes [104, 105, 107] and reinterpreted the study done by Fowler et al. [106], concluding that their study is inconsistent with the results of small delocalized susceptibility. The chemical shift observed in NMR analysis of C60 done by Taylor et al. indicated the presence of aromatic systems; these were confirmed by Fowler and group subsequently [71].

Several theories and computational studies have also been dedicated to exploring doping, defects, functionalization, etc. in fullerenes for their possible applications in antiviral activity, DNA cleavage, photodynamic electron transfer, lightweight batteries, lubricants, nanoscale electrical switches, cancer therapies, and astrophysics [109, 110].

2.2 Carbon Quantum Dots

Carbon quantum dots or carbon dots are relatively newer members among the carbon nanostructure family. These are quasi-spherical nanoparticles involving sp2/sp3 amorphous or nanocrystalline forms having size generally <10 nm carrying oxygen/nitrogen groups [111, 112]. Surprisingly, carbon dots were discovered unintentionally in 2004 in an experimental study of carbon nanotubes through electrophoretic fractionation of arc-discharge soot [42]. Carbon dots have gained much attention due to the fact that they possess strong fluorescence with better solubility, biocompatibility, and non-toxicity [113]. However, these fluorescent carbon nanostructures gained significant attention due to improved fluorescence emissions through the surface passivation synthesis approach [114]. The carbon quantum dots along with their STEM and absorption spectra are shown in Fig. 6.

Fig. 6
figure 6

Reproduced with permission from Sun et al., J. Am. Chem. Soc. 128, 7756 (2006). Copyright 2006 American Chemical Society

a Carbon dots attached with PEG1500N in aqueous solution. b STEM images of carbon dots. c The absorption (ABS) and luminescence emission spectra of carbon dots in an aqueous medium; the graph is plotted with 20 nm increment from longer excitation wavelengths 400 nm on the left and the intensities of emission spectral are normalized to quantum yields (inset is the normalized spectral peaks).

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Experimental and theoretical investigations have been used to understand the chemical and physical properties of carbon quantum dots for their applications in various fields like sensing, bio-imaging, nano-medicine, catalysis, optoelectronics, and energy conversion/storage. However, there are considerably rare theoretical studies on carbon quantum dots, and many of them are based on the graphene nanoflakes model [115,116,117,118,119,120].

Analogous to other quantum dots, the emission of carbon quantum dots is associated to their respective sizes. Carbon quantum dot size <1.2 nm showed UV light emission [121], visible light emissions were reported for quantum dots with size from 1.5 to 3 nm while near-infrared emissions were observed for quantum dots with sizes ~3.8 nm [122]. These observations have also been supported using theoretical calculations. The observation of indirect dependence of the HOMO–LUMO gaps on the size of the carbon quantum dots lead to the conclusion that strong emission of carbon quantum dots is a result of its quantum size rather than carbon–oxygen surface [123]. The photoluminescence mechanism, electronic structures, and frontier molecular orbitals of carbon quantum dots have also been studied using time-dependent density functional theory (TD-DFT) as implemented in Gaussian 09 with B3LYP hybrid functional and the 6-31G(d) basis set [124]. The carbon quantum dots were categorized in two forms: class I representing graphitized carbon core and class II representing disordered carbon core. These classes are depicted in Fig. 7 along with their photoluminescence mechanism.

Fig. 7
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Reproduced with permission from Zhu et al., J. Mater. Chem. C 1, 580 (2013). Copyright 2013 Royal Society of Chemistry

Photoluminescence (PL) mechanism of class I and class II carbon quantum dots. The number of hexagonal rings is indicated after fused aromatic rings (FARs) and the number of repeating units of cyclo-1,4-naphthylene (CN) is indicated by a number.

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The study showed that the HOMO–LUMO gap decreases with an increase in the size of class I carbon quantum dots while an opposite trend on the size-dependency of the HOMO–LUMO gap is observed for class II carbon quantum dots. Several studies related to the electronic structure of carbon quantum dots have been explained using molecular orbital (MO) theory [121, 123, 125,126,127]. In the majority of these reports, carbon quantum dots show n → π* and π → π* transitions because of their well-available transition energies. The π-states of carbon quantum dots are attributed to the sp2 hybridized carbon in their core, while the n-states are attributed to the functional groups attached. It is found that the energy gap (Eg) among π-states reduces consistently with the increase in the number of aromatic rings of carbon quantum dots similar to organic molecules [121, 123]. The electronic properties of amorphous carbon nanodots were explored using semi-empirical molecular–orbital theory using the EMPIRE13 code [128]. Unexpectedly, electronic structures were found to rely weakly on parameters like elemental composition and atomic hybridization. Contrarily, the geometry of sp2 arrangement describes the band gap of carbon quantum dots. The existence of localized electronic surface states resulting in amphoteric reactivity and near-UV/visible range optical band gaps was predicted [128]. The molecular orbitals, molecular electrostatic potential (MEP), local electron affinity (EAL), and ionization energy (IEL) maps along with excitation energies are depicted in Fig. 8. There have been fewer theoretical studies to understand their optical and electronic mechanisms and in-depth theoretical studies are further expected.

Fig. 8
figure 8

Reproduced with permission from Margraf et al., J. Phys. Chem. B 119, 24, 7258 (2015). Copyright 2015 American Chemical Society

A 2 nm carbon dot with a molecular orbitals; left side presents band-like and right side presents surface states with iso-density surfaces of 0.01 e−Å−3. b Electron iso-density surface maps, MEP (left part) from −50 (blue) to 50 kcal mol−1 (red), EAL (middle part) from −150 (blue) to 5 kcal mol−1 (red) and IEL (right part) from 270 (blue) to 500 kcal mol−1 (red). c Excitation energies calculated with different methods for different sized carbon dots.

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3 One Dimensional (1D) Carbon Nanostructures

The first-ever proof for the existence of one-dimensional carbon allotrope was reported in 1993 [40]. Single-walled carbon nanotubes (SWCNT) discovered by Iijima and Bethune lead all scientists for a hunt to utilize this a new form of carbon in many applications for technological advancement like field emission displays, energy storage and energy conversion devices, sensors, hydrogen storage, and semiconductor devices [129,130,131,132,133,134].

CNT is one of the exceptional inventions which has enriched the field of nanotechnology. It has been consistently studied since the past 20 years due to its potential application in varied areas. The fullerenes discovered by Kroto et al. [30] were the building blocks of the CNTs. CNTs have a variety of physical properties such as stiffness, elasticity, deformation, and tensile strength along with electronic properties showing superconducting, metallic, semiconducting, or insulating behavior.

The discovery of CNTs was reported as a “worm-like” structure long before this tubular form of carbon could be imagined, in 1952 by Radushkevich and Lukyanovich [135]. Dimensionally, SWCNTs are around 1 nm in diameter while their length is in order of a few micrometres. Nevertheless, the size and the shape of nanotubes can vary. The ratio of the diameter and length of the nanotubes, also known as aspect ratio, is typically around 1000 due to which it is generally considered nearly as a one-dimensional structure [136].

The different types of CNTs depend on the number of carbon layers present in them. Monolayered tubes are called single-walled carbon nanotube (SWCNT), while tubes having more than one layer are known as multi-walled carbon nanotubes (MWCNTs). The SWCNTs are generally understood to form by rolling a graphene sheet. Density functional theory calculations have shown the possibility of forming CNTs from bilayer graphene nanoribbons under different pressure conditions depending on the edges of nanoribbons involved [137]. The CNTs are classified into three different types: armchair, zigzag (see Fig. 9), and chiral carbon nanotubes (see Fig. 10). These are formed by rolling graphene sheets along a different axis. The axis of rolling is the chiral vector which is represented by n and m pair (n, m) of indices corresponding to the unit vectors along different directions in the graphene honeycomb crystal lattice sheet. When m = 1, 2,… and n = 0, the nanotube is “zigzag” and if m = n, the nanotube is then termed as “armchair” while the remaining configuration iscalled chiral [136, 138, 139]. Due to the rolling of the sheet into a tube, the symmetry of the plane breaks and forms a new symmetry in a distinct direction of the hexagonal lattice and the axial direction. This develops a peculiar electronic behavior of the nanotube, which is metallic or semiconducting. In the case of the semiconducting tube, its bandgap is sensitive toward its diameter; the small diameter tube has a large band gap while the wide diameter consists of a lower band gap [140]. The diameter of the nanotube thus makes it a conductor with conductivity higher than copper as well as a semiconductor comparable to the potential of silicon. In the structure of a nanotube, every carbon atom is bonded covalently with three nearby carbon atoms with its sp2 molecular orbital, creating one (the fourth) valence electron free in every hexagonal unit, which is delocalized over all atoms providing the nanotube its electrical nature. Some CNTs which show metallic nature have the resistivity in the range of 0.34 × 10–4 to 1.0 × 10–4 Ω/cm [141]. The semiconducting CNTs generally show p-type semiconducting behavior [142]. The SWCNTs can also be described as quantum wires due to their ballistic electron transport, while the electronic transport in MWCNTs is quasi-ballistic [143]. Apart from the well-known electronic properties of CNTs, they show equally good mechanical properties as well. The sp2 carbon–carbon bonds present in the CNTs result in exceptional mechanical properties which were not observed in previously explored material systems. From some previous studies, we get an idea about the stiffness of CNTs, basically in their axial direction [144]. Among all carbon materials, CNTs show extremely high value for Young’s modulus (~1TPa) which is even five times higher than steel, and provides a measure of the stiffness of the material [145, 146]. All the studies regarding the mechanical properties of CNTs were first predicted theoretically [53, 147,148,149]. The transformation from the hexagonal ring of carbon to pentagon–heptagon in CNTs was proposed by Yakobson [150] and Ru [151] when uniaxial tension is applied. DFT calculations suggest that SWCNTs form novel quasi-two-dimensional sheets when subjected to high pressure [152]. In a theoretical study done by Guanghua et al. [153] on the CNTs’ mechanical properties, their nature of dependence on diameter is revealed. They found Young’s modulus in the range of 0.6–0.7 TPa for nanotubes with diameter >1 nm. The closest agreement with the experimental value of Young’s modulus of MWCNTs (1–1.2 TPa) was theoretically calculated by Hernandez et al. [154]. In this study, they also predicted that mechanical properties depend on the diameter of the tube; when the diameter increases, the properties are also enhanced to a certain value and ultimately reach the values corresponding to that of graphene. Calculated values of Young’s modulus for individual SWNTs were found in the range from 320 to 1470 GPa [144, 155] while the breaking strength ranged from 13 to 52 GPa [156]. The vibrational properties of CNTs are studied by the normal mode analysis as this technique is standard to understand the dynamics of nanotubes. This technique investigates the harmonic potential analytically for normal mode analysis. The linear combination of Cartesian co-ordinates provides the co-ordinates for normal mode. This method provides a natural description of molecular vibration as it includes the motion of all atoms simultaneously during the vibration.

Fig. 9
figure 9

Reproduced with permission from Maeda et al., Physica B 263–264, 479 (1999). Copyright 1999 Elsevier

a Unit cell for two different carbon nanotubes (armchair and zigzag) depicting the primitive azimuthal angle θ (=2π/N). b Phonon dispersion curve for armchair and zigzag SWCNT.

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Fig. 10
figure 10

Reproduced with permission from Saito et al. Appl. Phys. Lett. 60, 2204 (1992). Copyright 1992 AIP Publishing

a Icosahedral C140 fullerene-based hemispherical cap covered end chiral fiber with chiral vector Ch = (10, 5). b Different probable vectors for the construction of chiral fibers. The two different combinations of circled dots and dots denote the metallic and semiconducting behavior for corresponding chiral fiber constructed.

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Apart from the small size, CNTs show quantum effects leading to the low-temperature specific heat and thermal conductivity; CNTs are also of great importance for their thermal properties [149, 157, 158]. The thermal conductivity can be modulated and increased by incorporating different materials with pristine CNTs. The thermal conductivity measured at room temperature for MWCNTs was found to be 3,000 W/K [159], while in a similar study the MWCNTs were found to have thermal conductivities ~200 W/mK higher as compared to the SWCNTs [160]. The main factor which influences the thermal properties is the number of active phonon modes along with a free path of phonon and boundary surface scattering [160,161,162]. Properties of CNTs are observed to depend on the atomic arrangement, length and diameter of tubes, structural defects, and impurities [163,164,165].

4 Two-Dimensional (2D) Carbon Nanostructures: Graphene

Graphene is a single atom layer of carbon atoms arranged in a hexagonal honeycomb pattern. It is one of the most studied two-dimensional (2D) materials to date. Figure 11 illustrates a graphene sheet as a 2D building block for different carbon materials in all dimensions such as 0D buckyballs by wrapping up the graphene sheet, 1D nanotube by rolling it, and in 3D graphite by stacking it. Thus, it is known as the mother of all graphitic forms of carbon material. The research has exponentially developed after 2004 when Geim and Novoselov isolated graphene for the first time using the “Scotch Tape” method and characterized it. In the current scenario of the material world, ongoing research is overwhelmed after focusing on characterization, mass production of ultra-thin carbon films including graphene for various applications [166,167,168,169,170,171,172].

Fig. 11
figure 11

Graphene sheet is a 2D building block for different carbon materials in all dimensions like 0D buckyballs which is formed by wrapping of graphene sheet, 1D nanotube can be made by rolling it and 3D graphite is formed by stacking it, therefore it is known as the mother of all graphitic forms of carbon material

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A unit cell of graphene with the Bravais lattice along with the band structure is shown in Fig. 12. The unique linear dispersion of the band structure near the K point is illustrated by a pseudo-spin direction which is indicated by the arrows. From the past one and half decades, promising applications in the field of corrosion prevention [173], super capacitors [174, 175], long-lasting batteries [176], display panels [177], efficient solar cells [178], desalination [179], and water purification [180,181,182] have emerged.

Fig. 12
figure 12

Reproduced with permission from Fuhrer et al. MRS Bulletin 35, 289 (2010). Copyright 2010 Cambridge University Press

a Unit cell of graphene with the triangular Bravais lattice having lattice vectors a1 and a2; unit cell comprises two atoms in the honeycomb lattice. b Band structure of graphene calculated by the tight-binding method displaying the pi bands, with only nearest neighbor hopping. The inset E, kx, and ky are the energy and the wave vector components in x- and y- directions, respectively. c The unique linear dispersion of the band structure near K point with the pseudo-spin vector direction indicated by the arrows.

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The electronic properties of single-layer graphite were investigated by Wallace even before its isolation [183] and introduced the term “graphene” back in 1947. The electronic band structure was investigated theoretically by the tight-binding (TB) approach. The TB approach is more suitable for handling larger systems than the plane waves method, due to its low computational costs. The method was at first described as an interpolation scheme by Slater and Koster [184]. It has been developed comprehensively and now it is a well-established technique to explain the electronic structure of solids [185].

The tight-binding (TB) calculations were performed using the Hamiltonian

$$H= \sum_{i{l}_{1}\sigma }{\epsilon }_{{l}_{1}}{a}_{i{l}_{1}\sigma }^{\dag}{a}_{i{l}_{1}\sigma }+ \sum_{ij}\sum_{{l}_{1}, {l}_{2}, \sigma }\left({t}_{ij}^{{l}_{1}{l}_{2}}{a}_{i{l}_{1}\sigma }^{\dag}{a}_{j{l}_{2}\sigma }+H.c.\right)$$
(1)

here, the spin σ of the electron is capable of jumping from the orbital l1 with its onsite energies \(\left({\epsilon }_{{l}_{1}}\right)\) existing in the ith unit cell to orbital l2 in the jth unit cell. The hopping interaction strength labeled as \({t}_{ij}^{{l}_{1}{l}_{2}}\) relies on the nature of the orbitals participating as well as on the lattice geometry [184]. Following that, a least-squared error fitting is executed through the alteration of the \(\epsilon\)’s and t’s, leading to the calculation of band dispersions at various high-symmetry points. Graphite layer shows semiconducting behavior with zero activation energy at zero temperature, but at higher temperatures due to excitation, the highest bands are filled and show metallic nature. Large anisotropic diamagnetic susceptibility which is greatest across the layers is observed. The study done by Boehm in 1962 provided the concept of single-layer graphite sheet through the reduction of graphite oxide (GO) in dilute sodium hydroxide and also by deflagration of heated GO [186]. To describe the atom intercalation in graphite, effective-mass-approximation differential equations were used at that time for self-consistent screening. At room temperature, graphene displays a strong ambipolar electric field effect between the valence and the conduction bands. This results in ballistic electron transfer at a speed which is slower than light speed and 10–100 times greater than that in silicon chips. Graphene is the thinnest material and is 200 times stronger than steel and harder than diamond but at the same time, it is flexible and transparent [10, 187, 188].

Investigation of the physical properties of graphene reveals that it has a tremendously high optical transparency of up to 97.7%, which makes it a potential material for transparent electrodes for its use in solar cell applications [189]. It also consists of high thermal conductivity of 5000 Wm−1 K−1 [190], and exceptional mechanical properties like high Young’s modulus of 1 TPa [191], and most importantly large specific surface area of 2630 m2 g−1 [192]. Still, there is a need to find a method for the utilization of graphene in many applications and also to guarantee cost-effective production by avoiding some major obstacles. It is a great need to develop a method with the help of which ideally flat graphene membrane without any defects can be achieved. The need to fill the large gap between the theoretical prediction and actual fabrication of graphene is essential. Irreversible agglomerates and the restacking are a key challenge in the synthesis of graphene which need to be addressed.

5 Summary and Outlook

Nanomaterials provide exotic properties, exclusive of the framework of their periodic solid counterparts. Additionally, novel phenomena emerge at the nanoscale level that is not observed in microcrystalline materials. Among all, carbon nanostructures like three dimension (3D—Graphite, diamond), two dimension (2D—graphene), one dimension (1D—carbon nanotubes), and zero dimension (0D—fullerenes and carbon quantum dots) have gained significant attention due to their unique properties. The discovery of C60 and carbon quantum dots (0D), CNTs (1D), and graphene (2D) has led to the increased research activity in novel multidisciplinary areas, from synthesis to their theoretical and computational investigations for potential applications. In the present chapter, the theoretical and computational development of carbon nanostructures, specifically on fullerenes, carbon quantum dots, carbon nanotubes, and graphene have been introduced and discussed. The underlying mechanism of size dependency of these carbon cage structures (fullerenes and carbon nanotubes) is essential for modifying their properties according to the potential nanotechnology applications. Computational and theoretical studies have found significant role in predicting and designing their properties accordingly. By the means of powerful supercomputers, performing static and dynamic calculations at high-level ab initio and DFT methodologies is achievable for these carbon nanostructures. Still, the application of futuristic quantum chemical approaches to investigate the structures and properties of large carbon nanostructures (fullerenes, carbon quantum dots, graphene, and CNTs) is a daunting task. The theory of isolated pentagon rule (IRP) in fullerenes chemistry has been discussed. The knowledge on the computational and theoretical aspects of accidentally discovered carbon quantum dots were explored which is still in its developing stage. The theoretical prediction of carbon nanotubes (armchair, zigzag, and chiral) and graphene before their experimental realization is provided. Obtaining insight of the electronic structures along with their chemical and physical properties is still needed for constructing new materials based on carbon-based nanostructures for certain applications. The synergy among theoreticians and experimentalists will expand the applications of carbon nanostructures promptly.