Optical and magnetic properties of free-standing silicene, germanene and T-graphene system

Methodology

In recent years, DFT has emerged as an important theoretical tool for predicting various physical properties related to exotic materials. This method based on supercell approach, however, takes into account the relaxation of atoms. Traditional calculation such as local density approximation (LDA) and generalized gradient approximation (GGA) underestimate the bandgaps of semiconductors and insulators. This can be, however, overcome by going beyond DFT and taking appropriately the many-body effect as is done through self-energy computation involving Green’s function (G) and the screened coulomb (W) interaction or in short through GW approach.

Hybrid functionals, on the other hand, have also been proven to be another powerful technique in DFT which can produce consistent band structure comparable to experimental situation and a reliable description of charge localization often used in low-dimensional system. In particular, the screened hybrid functional due to Heyd, Scuseria and Ernzerhof (HSE) [48] has been observed a reliable one in predicting formation energy and other defect levels in semiconductors. Another hybrid functional due to Perdue, Burke and Ernzerhof (PBE) [49] has also been instrumental in addressing the electronic properties based on atomic structures.

In DFT, optical properties of any system can be calculated with the help of frequency-dependent dielectric function which is complex in nature : ϵ(ω)=ϵ1(ω)+iϵ2(ω), where ϵ1(ω) and ϵ2(ω) are, respectively, the real and imaginary part of the dielectric function as a function of energy of the incident electromagnetic (EM) wave. These two quantities are not independent of each other. Rather, they are connected by a relation which is known as the Kramers–Kronig (KK) relation [50, 51]. In any numerical simulation, the imaginary part of the dielectric function is calculated with the help of a time-dependent perturbation theory in the simple dipole approximation. In the long-wavelength limit (q→0), the imaginary part of the dielectric function is given by the following expression :

In the above expression (1), ω is the frequency of the EM radiation in energy unit. Ω represents the volume of the supercell and ϵ0 is the free space permittivity. CB and VB represent the conduction band and the valence band (VB), respectively. u→ and r→ denote the polarization vector and position vector of EM field, respectively. The matrix element of this dot product of these two vectors is computed between the single-electron energy eigen states. Since the magnetic field effect is weaker by a factor of v/c, the transition matrix elements between the eigen states of CB and VB have been calculated only due to the electric field. Phonon contribution, local field and excitonic effects are not taken into consideration. By definition, the imaginary part of the dielectric function is positive for any polarization and frequency [6, 52, 53, 54, 55].

As mentioned before, ϵ1(ω) can be obtained from ϵ2(ω) by KK transformation which is given by,

Various sum rules involving ϵ1(ω) and ϵ2(ω) can be verified to check the consistency in the numerical computation [20].

The complex refractive index (N~) of any material is related to the complex dielectric function (ϵ(ω)) by the relation N~=ϵ(ω). From this relation, one can get the real as well as the imaginary part of the refractive index in the form N~=n(ω)+ik(ω),where,

From n(ω) and k(ω), one can compute the reflectivity at normal incidence of EM wave as,

The reflectivity modulation (RM(ω)) [56] can be obtained from the reflectivity data by the following expression,

Although R(ω) is positive for all frequencies, however, RM(ω) can take either of sign.

The absorption coefficient can be evaluated by using the imaginary part of the refractive index as,

where c represents the speed of light in vacuum and ω is in the energy unit. The direct measure of the collective excitation is the quantity called EELS. It is given by the relation L(ω)=Im(−1(ω))L(ω)=Im(−1ϵ(ω)) or in terms of ϵ1 and ϵ2,

Typical energy of the plasmons of a system can be estimated by looking at the peak positions of any loss function.

The optical conductivity σ(ω) is related to the dielectric function via the following relation :

All the optical property calculations are performed in the long-wavelength limit q→0 of EM wave.

Optical properties

Because of its buckled structure, it is convenient to study the modification of electronic as well as optical properties of silicene under an external electric field. Applying external electric field is both theoretically and experimentally convenient. So, it is natural to explore the effect of external electric field on silicene sheet. But external electric field can significantly change the electronic as well as the optical properties of silicene. Kamal et al. [57] have first studied the effect of this external electric field on the electronic and the optical properties of pure silicene and hydrogenated silicene by employing DFT. It has been observed by them that the optical properties strongly depends on the direction of the incident EM field, i.e. whether it is in the plane of the silicene sheet (parallel polarization) or perpendicular to it (perpendicular polarization). The authors have found that the low-energy regime (below ∼5 eV) and the high-energy regime (above ∼5 eV) are mainly dominated, respectively, due to parallel and perpendicular polarization. Optical anisotropy observed in this system may be due to inherent buckling present in this 2D material system. A careful analysis of the data obtained from absorption spectra calculation has clearly indicated that there is no absorption of light below a cut-off frequency. The physical reason of this phenomena is due to the symmetry breaking between two sublattices due to the application of external electric field. The more the strength of the external electric field, the more the cut-off value has been seen. When a silicene-based device is irradiated by THz EM field which is circularly polarized, the topologically insulated spin-up and spin-down bands are respectively red- and blue-shifted [58]. While in the case of band insulator, both bands are shown to be continually blue-shifted with increasing staggered potential.

First-principles DFT calculations on hybrid S/G nanocomposite have revealed that nanocomposite exhibits stronger optical absorption in the frequency range from 0 to 15 PHz compared to silicene on graphene monolayer [59]. The interest in this composite structure originates from the fact that silicene interacts overall weakly over graphene through van der Waals interaction in such a way that their intrinsic electronic structure is restored. Besides, the interlayer interaction can interestingly induce p-type and n-type doping of silicene and graphene, respectively.

Changes in material properties with the application of strain have always been interesting in the field of material science. Mohan et al. [60] have studied the strain-dependent electronic and optical properties of silicene. They consider both asymmetric and symmetric strain (up to 20%) of equal magnitude. Within 4 eV, they have found some characteristic peaks corresponding to inter-band transitions in electronic band structure. But interestingly, it has been observed that the characteristic peaks vanish with increasing magnitude of strain. Those transitions, which occur above 2 eV, shifted to lower frequency with increasing magnitude of tensile and asymmetric strain. But with compressive strain, the transitions are blue-shifted. If we plot the real part of the dielectric function as a function of energy, then the points where it cuts the x-axis (energy) from both positive and the negative sides denote the collective excitations of electron. These excitations produce oscillations of electron’s density in the system, which are known as plasma oscillations. The peaks in the EELS correspond to these oscillations. In pure silicene, two distinct EELS peaks appear. One is within 7–8 eV which is due to π+σ plasmons. The other one is within 2 eV, which is due to π plasmons. But with increasing magnitude of compressive strain, the positions of the π plasmons are not affected. But the disappearance of π plasmon peaks are noticed for tensile and asymmetric strain. Optical conductivity is also very important quantity for identifying the electronic transitions within the systems. Matthes et al. [61, 62] have studied the optical conductivity of silicene. Their calculation is based on independent quasiparticle approximation. It has been revealed from the study that the low-energy peak around 2 eV is due to π–π∗ transition. The high-energy peak near 5 eV is due to σ–σ∗ transitions. It is due to the fact that σ bonds are much stronger than the π bonds so naturally their excitation peak should occur at higher energy.

Motivated by these intriguing features, recently Das et al. [63] have explored the optical properties of disordered silicene nanosheet. Where the pristine nanosheet has been made disordered by doping with Al and P atoms (∼16% for Al/P doping and ∼31% for codoping). The concentrations of Al and P atoms have been varied from 3.12% to 15.62%. While for Al-P codoped systems, it has been varied from 6.25% to 31.25%. There are several observations in the optical properties of these doped systems. One is below 3 eV, which can be associated with π–π∗ transitions. Another one is above 3 eV, which can be associated with σ–σ∗ transitions. It has been found that for perpendicular polarization, two new small yet significant EELS peaks have been emerged. Absorption coefficient study reveals that the maximum value of the absorption coefficient of the doped system is higher than that of pristine system. For parallel polarization, the maximum value of the absorption coefficient is blue-shifted irrespective of the doping concentrations. But, for perpendicular polarizations, a red-shift is observed for Al and P doped systems. In Figure 1, we have depicted the real and the imaginary part of the refractive index for Al and P doped systems for parallel polarization. In the case of parallel polarization, the low-energy (below 10 eV) sector of the energy spectra is mainly affected. Here, again, it is observed that the signature of defects is appearing mainly in the low-energy sector of the spectrum. All of these transitions can be associated with π band transitions.

Nanostructure having closed edges are known as nanodisks. Nanodisks, which have hexagonal symmetry, are mainly derived from many benzene rings [64]. It can be experimentally synthesized by using an experimental technique called soft-landing mass spectrometry [65]. Regarding the optical properties of nanodisks, Sony and Shukla [66] first have studied the optical absorption spectra of various graphene nanodisks by using Pariser–Parr–Pople (PPP) model.

Figure 2:

The schematic structures of four differently shaped nanodisks: (a) zigzag trigonal (ZT), (b) armchair trigonal (AT), (c) diamond shaped (DS) and (c) bowtie shaped (BS).

The magnetic and optical properties of differently shaped nanodisks have been first studied by Chowdhury et al. [67] by employing DFT. Four differently shaped nanodisks : zigzag trigonal (ZT), armchair triangular (AT), diamond shaped (DS) and bowtie shaped (BS) have been considered for the study. In Figure 2, we have schematically illustrated the structures of four differently shaped nanodisks. Edge atoms play a very important role in these kinds of nanodisks. Now, the percentage of the atoms at the edges of ZT, AT, DS and BS are, respectively, given by 69.2, 66.6, 62.6 and 60%. From the real part of the dielectric function data, an oscillatory behaviour up to 9 eV has been found for ZT and AT nanodisks. Beyond this energy range, the optical response becomes negligible. The static values of the real part of the dielectric function of all the nanodisks are found to be less than that of bulk Si and silicene. Different values of the dielectric function for different nanodisks can be traced due to anisotropy and different shape geometry. The comparison of optical properties of four different nanodisks shows that DS nanodisk has the best optical response, whereas AT bears the most poor optical response. This kind of study is very important for device fabrication in nano-optoelectronic technology and material characterization techniques.

Figure 3:

(Top panel) Real (ϵ1) and (bottom panel) imaginary (ϵ2) parts of the dielectric function as a function of energy for mono- and DV-induced silicene nanosheet for perpendicular polarization.

Magnetic properties of single-vacancy (SV), DV, Al and P doped silicene have already been studied by Majumdar et al. [68]. In Figure 3, we have illustrated the real and imaginary parts of the dielectric function for SV and DV-induced silicene nanosheet as a function of energy for perpendicular polarization. From the figure, it can be observed that the signature of vacancy is mainly appeared within 6–10 eV. Also, the effect of both SV and DV is quite similar in the optical spectrum. Due to the presence of vacancy within the sample, the optical response is poor as can be seen from the magnitude of the dielectric functions. The maximum value of the imaginary part of the dielectric function is only 0.25. In the case of the real part, it varies between 0.85 and 1.15. Wei et al. [69] in 2013 and in 2016 Zakerian and Berahman [70] have studied the optical absorption of SV-induced silicene sheet by employing both DFT and Bethe Salpeter equation (BSE). As BSE takes into account the electron–hole interaction which DFT does not, so BSE employed data is expected to be more accurate than DFT. They have obtained two peaks in pristine silicene, one is located around 1.2 eV and the other one is around 4 eV [69]. SV-induced silicene has also shown two peaks like the pristine one, but here interestingly the two peaks are red-shifted. This happens due to the presence of dangling bond in the defected silicene. This characteristic feature can help the experimentalist to distinguish pristine silicene and defected silicene apart from Raman study.

DFT can only predict the ground-state properties of a many-body system as it is based on frozen atom approximation. But, if one can incorporate appropriate lattice dynamics and also choose thermally equilibrated configurations, then various optical properties can be computed at finite temperature which can be matched with the experiment. Now, to generate various thermally equilibrated configurations, molecular dynamics (MD) is a commonly adopted computational tool. From those configurations, one can compute the ensemble average of dielectric functions [71]. Yang and Liu [72] adopted this technique to calculate the dielectric function of monolayer silicene sheet. Apart from 0 K, they have considered two more temperatures 300 and 600 K. At each configuration, they have calculated the imaginary part of the dielectric function and then they have plotted the ensemble average of the imaginary part of the dielectric function. It has been revealed from their study that for parallel polarization, absorption peak around 1 eV energy range is enhanced consistently with increase of temperature. It has been explained through zero-energy gap and intraband transition which generally dominates the optical absorption occurring at low energy. With the increase of temperature, the amplitude of lattice vibrations is enhanced. As a result, more and more free carriers participate in the intraband transition which are thermally excited. But in the case of perpendicular polarization, due to structural disorder, the bands tend to spread out which reduce the absorption peaks. It will be interesting if these kinds of study can be done with silicene with defects.

Ye et al. [73] have studied the optical properties of periodically removed hexagonal silicon chains from silicene sheet, which is known as silicene nanomesh. While plotting the imaginary part of the dielectric function, they have observed a striking difference with that of bulk Si for both polarized and unpolarized light. Bulk Si has a threshold value of 1.1 eV below which there is no direct optical transitions occurring between the valence band maximum (VBM) and the conduction band minimum (CBM) [74]. It has been noticed that the maximum value of the imaginary part of the dielectric function lie within the visible to infrared (IR)\ part of the spectrum. The peak positions also do not noticeably change for the unpolarized and for parallely polarized light; however, the magnitude is doubled for parallely polarized light. But a strong anisotropy signal is noticed for both kinds of polarizations. The authors further commented that silicene nanomesh may pave the ways towards the fabrication of solar cell.

It is to be noted that quite a lot of theoretical works have already been done on the optical properties of silicene. But experimental works related to the exploration of optical properties are still inadequate. Sugiyama et al. [75, 76] have synthesized and also studied the optical properties of phenyl-modified (oxygen free) organosilicon nanosheet. The consequence reported of this structure is its uniform dispersion in organic solvents. The material has been synthesized by the reaction of layered polysilane [Si6H6] with phenyl magnesium bromide [PhMgBr]. The IR spectrum reveals the vibrations of phenyl groups at 1,150 and 1,410 cm−1 that correspond to the Si–Ph bond. The signature of C–C bond has been observed at 1,700–2,000 cm−1. Also an asymmetric vibration of Si–H bond has been noticed at 2,100 cm−1. Photoluminescence (PL) spectrum study has also been done using 350 nm excitation. The emission spectra shows a peak at 415 nm, which is shorter than its bulk counterpart. The X-ray absorption spectrum of the silicon sheets has a peak at 268 nm, which corresponds to L→L critical point in the band structure [77]. Optical properties of alkyl-modified crystalline silicon nanosheets have been explored by Nakano et al. [78]. They have adopted different experimental techniques to characterize the system. This new structure is able to disperse uniformly in organic solvents. Fourier-transform infrared (FTIR) spectroscopy analysis produced different peaks due to bonding of different functional groups with silicon. The absence of characteristic peak at 1,600 cm−1 due to C–C double bond indicates that the organic molecules are covalently attached to the silicon surface. Borensztein and co-workers [79] have used an experimental technique called “surface differential reflectance spectroscopy” (SDRS) to explore the optical properties of single-layer Si on Ag (110). While analysing the SDRS signal, it has been observed that a sudden change in slope occurred at 3.83 eV. From theoretical DFT calculation, three significant peaks can be observed at 2.0, 3.8 and 4.6 eV in pristine silicene [80]. But, while silicene is deposited on Ag (110) surface, these peaks positions are supposed to change due to interlayer interface. But in the Ag surface-deposited case, no change of slope has been observed at 4.6 eV, and only a tiny effect has been seen at 2.0 eV. Through this work, the authors hope that this study will help the theoreticians in future to determine the exact atomic structure of Si layer on Ag surface.

Figure 4:

Comparison of absorbance spectra of graphene, silicene and germanene. Graphene: 0.02293, silicene: 0.02290, germanene: 0.02292. The universal behaviour is noticed for all of them, and this is independent of group IV elements, buckling and Fermi velocity (Reprinted with permission from Ref. [81]). Copyright (2012) by American Institute Physics.

Before ending this section, we would like to mention one more remarkable feature of the optical properties of these 2D materials. After the immediate discovery of graphene, it has been realized that the transmittance can be defined in terms of fine structure constant (α=e2ℏc) [82]. It was then natural to find this universal feature in other 2D materials like silicene, germanene, etc. In Figure 4, we compare the absorbance of graphene, silicene and germanene and confirms the universal character which is independent of the corresponding Fermi velocity and buckling [81]. It can be clearly observed that as ω→0 the absorbance A(0)=πα. It is also known [81] that as we go towards higher frequency, there is a considerable deviation of absorption spectra in the case of silicene and germanene compared to graphene. This universal feature of absorbance can be understood [62, 80, 81, 83] from the imaginary part of the dielectric function and crossing of bands linearly at the Dirac points. Here, it is noteworthy to point out that this feature is also independent of the choice of gauge, i.e. whether it is transverse or longitudinal.

Magnetic properties of doped FS silicene monolayer

Pristine FS monolayer silicene is non-magnetic like that of graphene [84]. But, the presence of defects like vacancy or adatoms can make silicene magnetic. This kind of metal free magnetism is now the subject of an intense research. This type of magnetism comes due to the formation of localized states caused by defects or molecular adsorption. Inducing magnetism into non-magnetic nanostructure is technologically very important for making quantum information and spintronic devices. Below, we will review some of the recent work based on magnetic properties of monolayer FS silicene sheet.

The phenomenon of ferromagnetism observed at reasonably high temperatures in some compounds which do not contain any atoms with open d or f shells is known as d0 ferromagnetism [85, 86] e.g. in HfO2 [87], ZnO [88, 89, 90] with vacancies and so on. In most cases, ferromagnetism does not appear in the bulk when the system is pure. Thus, lattice defect is necessary for the occurrence of d0 ferromagnetism. They may be point defects – atomic vacancies or interstitials – induced by irradiation or thermal treatments (annealing) or by lattice mismatch, grain boundaries and dislocations can also induce this d0 ferromagnetism. In other words, the defect states can give rise to an appreciable amount of magnetic moment in connection with the molecular orbitals, which are localized very close to the defect site [86]. The utility of d0 ferromagnetism lies in the fact that large magnetization like that of Fe, Co and Ni can be achieved in a nanomaterial having an active defect. Thus, one can visualize magnetism in non-magnetic semiconducting matrices with some appropriate impurities/voids of non-magnetic atoms. Hence, d0 magnetism can take an important key role in designing novel materials in spintronics at room temperature. The nanoparticle surface area in 2D systems seems to be an integral part in triggering this d0 ferromagnetism in contrast to the bulk one. Here, it is to be noted that, in some alkaline-earth metal nitrides XN (X=Ca, Sr, Ba), half-metallic d0 ferromagnetism has been predicted without any defects [91, 92]. Recently, we have studied the d0 ferromagnetic properties of a new kind of material which is the hybrid of graphene and silicene [93, 94, 95]. From our study, it has been revealed that, due to a single silicon atom vacancy, the system possesses a magnetic moment of 4 μB. However, for a single carbon atom vacancy, the ground state has been found to be non-magnetic. From the charge density analysis, we have noticed that, because of the silicon atom vacancy, the three undercoordinated carbon atoms are unable to form any covalent bond, which makes the system magnetic. However, for carbon void system, the undercoordinated silicon atoms form covalent bond, making the system non-magnetic [96].

Figure 5:

The total magnetic moments of the spin-polarized systems as a function of adsorption concentration (Reprinted with permission from Ref. [97]). Copyright (2016) by Elsevier.

Hydrogenation and halogenation is one of the viable route to induce magnetism in silicene. Zheng et al. [98] have employed the first-principle calculation to investigate the magnetic properties of silicene sheet adsorbed with H and Br atoms. It has been found that when the silicene sheet is fully saturated with H and Br atoms, the ground state is found to be non-magnetic. But when the sheet is half-saturated from one side, it shows ferromagnetic property. It is due to localized and unpaired electrons of the unsaturated Si atoms. Total energy calculation reveals that half-hydrogenated silicene exhibits ferromagnetic order [99], while half-brominated one exhibits antiferromagnetic ordering. Paszkowska and Krawiec [100] have explored the stability of magnetism in hydrogenated silicene under the influence of strain, charge doping and external electric field. It has been found that the magnetism is present in strained hydrogenated silicene unless a structural phase transition occurs. As long as the hydrogenated silicene maintains its LB structure, strain does not influence the magnetic property of the system. However, when an external electric field is applied, then it has been observed that the magnetic ground state is still maintained. But interestingly, the magnetism disappeared for both electron and hole doping. Zhang et al. [101] have investigated the magnetic properties of different hydrogenated conformer of silicene sheets. It has been noticed that half-hydrogenated chair-like conformer shows magnetic ground state. This is because of the pz electrons of the unhydrogenated Si atoms which are unpaired and strongly localized to induce magnetism in the system. On the other hand, fully saturated sample is non-magnetic because of the absence of these pz electrons due to bonding with H atoms. Zhang and Yan [102] also reported similar kind of results for fully and half-hydrogenated silicene sheet giving similar physical explanation. Adsorption of adatoms on silicene sheet is also found to be effective in tailoring magnetism in silicene sheet. Ju et al. [97, 103] have explored the magnetic properties of silicene nanosheet with changing adsorption coverage of H, F and C atoms. In Figure 5, the total magnetic moment of C and H adsorbed systems have been depicted with different coverage density. It has been reported that adsorption of F atoms makes the sample non-magnetic; however, H and C atoms induce magnetism within the sample for different adatom coverage. It has been observed that the total magnetic moment of the system is same (1 μB) for different H coverage density. This value, however, remains constant and not affected by H concentration. In the case of C atoms adsorbed systems, among five different configurations, only for three configurations with low coverage density, magnetic ground states have been reported with the value of 2 μB.

Inducing magnetism in a non-magnetic material by transition metal (TM) atoms [104] is a conventional process due to their half filled d and f orbitals. They show diverse structural, electronic and magnetic properties. If TMs are adsorbed, depending on the type of adatom and atomic radius, the system can exhibit metal, half-metal and semiconducting behaviour. Motivated by the diverse properties of TM atoms, it would always be interesting to study the effect of TM atoms on silicene surface which is more reactive than that of graphene. Le et al. [105] have studied the adsorption of 3d-TM atoms on silicene using DFT+U formalism. They have considered four types of TM atoms : Cr, Mn, Fe and Co. According to the data obtained by them, the total magnetic moments of the considered (2 × 2) supercell consisting of four Fe atoms is 2.21 μB/cell. Further, it has been reported that each Fe atom produces a ferromagnetic moment of 3.07 μB, whereas each Si atom produces an antiferromagnetic moment of 0.43 μB. Similarly, when Co atom is used in place of Fe, non-magnetic ground state is obtained. When the other two adatoms have been used, much higher ferromagnetic moment of magnitude 4.01 and 5.18 μB/cell, respectively, for Cr and Mn have been obtained. Strain-dependent magnetic properties of silicene doped with Cr and Fe have been explored by Zheng et al. [106]. They have mainly considered isotropic uniaxial tensile strain to study strain-tunable magnetism. The unstrained system possesses around 2 μB when it is doped with Cr atom. When a small isotropic tensile strain (∼3.5 %) is applied, the magnetic moment suddenly jumps to around 4 μB. Similar phenomena is observed in the case of Fe atom doped system. Where in the unstrained case, its magnetic moment is around ∼ 0.4μB. But when a small tensile strain (∼ 2 %) is applied, it reaches a very high magnetic state of magnetic moment of 4 μB. A hysteresis is also observed for both kinds of doped system specially for Fe doped systems. Sahin and Peeters [107] have also studied the adsorption of 3d-TM atoms along with alkali and alkaline-earth metal atoms. It has been demonstrated by them that alkali and alkaline-earth metal atoms are ineffective in inducing any magnetic moment in the adsorbed systems. However, significant amounts of magnetic moment have been noticed for 3d-TM atoms. Their study has revealed that the amounts of magnetic moment per unit cell for Ti, V, Cr, Mn, Fe and Co are, respectively, 2, 2.7, 4, 3, 2 and 1 μB/cell. Ozcelik and Ciraci [108] have also explored the adsorption of Si, H and Ti atoms on silicene by employing DFT. From the obtained data, they have noticed the magnetic ground state of silicene. In Table 2, we have provided the magnetic moment of different doped and adsorbed atoms induced monolayer silicene systems.

Figure 6

Magnetic moment (in μB) as a function of defect concentrations for (left panel) Al doped and (right panel) P doped silicene sheet.

Vacancy-induced magnetism has also been one of the interesting topic in condensed matter and material physics. Here also the absence of d orbital is noteworthy. SV, DV and triple-vacancy (TV)-induced buckled silicene have been reported to be non-magnetic [109]. However, for SV-induced graphene, ferromagnetic ground state has been found to be energetically favourable [109]. In SV-induced graphene, there are three atoms which are twofold coordinated, and each of them has a dangling bond. MD simulation reveals that, upon bond reconstruction, three of them reconstruct in an asymmetric way. Among three, two of them form a C–C bond to make each of them threefold coordinated. The remaining atom, however, is unable to make any bond contributing a net magnetic moment in the system. While for SV-induced silicene, the three twofold undercoordinated atoms come close to each other to form three Si–Si bond. This makes the whole system non-magnetic. In the case of DV-induced graphene and silicene, there are four twofold coordinated atoms surrounding the vacancy. During bond reconstruction, four dangling sp2 bonds are saturated in pairs to form two C–C or Si–Si bonds. In this way, both the systems possess non-magnetic ground state. For TV, the ground state of graphene is found to be magnetic, while for silicene it is non-magnetic. We have depicted in Figure 6 the magnetic properties of Al and P doped disordered silicene systems. It can be seen from Figure 6 that for Al doped systems, when the defect concentration reaches 9.37% and 15.62%, the net magnetic moment of the system reaches nearly 1 μB. While for P doped systems, the magnetic moment is respectively 0.27, 1 and 0.42 μB for 6.25, 9.37 and 12.50% doping concentrations. It will be interesting to study in future the stability and tuning of this magnetism under the combined influence of strain and external electric field. When TM atoms are embedded in vacancy-induced silicene sheet, magnetism can be observed [110]. In Figure 7, we have illustrated the magnetic moments of both SV- and DV-induced silicene sheet embedded with TM atoms. From the figure, it is clear that Sc and Ti embedded systems are non-magnetic. It is because, in the valence shell Sc has three electrons, which is lesser than that of Si, so the d orbital gets saturated by forming three Sc–Si bonds in SV and four Sc–Si bonds in DV and possesses non-magnetic ground state. Ti has four valence electrons, which is the same as Si, so non-magnetic states have been observed both for single and for double vacancy. However, for V, Cr, Mn, Fe and Co, magnetic states are observed for both kinds of vacancies. Again for Ni, Cu and Zn, the systems possess no net magnetic moment because of the saturation of all the 3d electrons. To study the magnetic property further, SV-induced systems have been further doped with N and C atoms before embedding with TM atoms. The results are depicted in Figure 8. Here also, Sc, Ti, Cu and Zn produce non-magnetic ground state. But here interestingly Ni, which earlier produces non-magnetic ground state, now produces magnetic moment in the system. Ghosh et al. [111] have explored the electronic and magnetic properties of silicene with extended line defects. The defect induces 0.2 μB amount of magnetic moment to the system. This defect-induced system also exhibits spin localization, whose origin can be explained on the basis of buckling in silicene. The intrinsic buckling in silicene localizes the 3pz electrons more strongly than that in graphene.

Figure 7:

Magnetic moments of silicene with embedded TM atoms in SV (red) and DV (black) defects (Reprinted with permission from Sun et al. [110]. Copyright @ American Institute of Physics (2015)).

Figure 8:

Bar diagram of magnetic moments of TM atoms embedded into N or C doped SV defect in silicene. The green bars indicate the magnetic moments of transition metal atoms embedded into SV without doping for comparison. The inset shows the configurations of TM atoms in C-doped SV (Reprinted with permission from Sun et al. [110]. Copyright @ American Institute of Physics (2015)).

The use of superhalogen (MnCl3) over conventional halogen to tune the electronic as well as magnetic properties of silicene has been explored extensively by Zhao et al. [112]. The advantage of MnCl3 over conventional halogen is that it has even larger electron affinity than that of Cl atom. Isolated MnCl3 is intrinsically magnetic. So, it would be interesting to study the effect of MnCl3 on silicene; both of them have been experimentally realized in laboratory. Different adsorption sites have been considered. For the convenience of the readers, in Figure 9, we have schematically illustrated different adsorption sites for monolayer silicene. It has been exhibited that MnCl3 prefers to occupy the hollow site (HS) of silicene. But its magnetic behaviour is quite complex. The magnetic moment of isolated MnCl3 is 4 μB. But when one MnCl3 molecule is adsorbed into a supercell, the total moment of the supercell remains 4 μB. This magnetic moment changes to 3 and 5 μB if a Cl atom is introduced on the same side or on the opposite side of MnCl3. However, two MnCl3 superhalogens adsorbed onto the same side result in a total magnetic moment of 8 μB per supercell. It will be interesting to study the effect of this superhalogen on an anisotropic and isotropic strained silicene.

Figure 9:

Top view of the schematic representation of adatom adsorbed silicene on different adsorption sites: (i) on hollow site (HS), (ii) on bridge site (BS) and (iii) on top site (TS) (Reprinted with permission from Nath et al. [53]. Copyright @ Elsevier (2015)).

The optical properties of four differently shaped silicene nanodisks [67] have been described in the previous section. It has been found that among the four nanodisks, only ZT silicene nanodisk is magnetic in nature having ∼4 μB of magnetic moment. Earlier studies [113, 114, 115] have also found similar results for graphene and silicene nanodisks. The origin of this magnetism in ZT nanodisk can be traced back due to the occurrence of zero-energy states [113, 114, 115] which are nothing but singly occupied molecular orbital [116]. It is now believed that Coulomb exchange interaction plays a very important role to align the spins at the edge atoms in ZT nanodisk which gives rise to magnetism. AT nanodisk also has zero-energy states, but at the edge it has even number of atoms which force it to be non-magnetic. Due to this large amount of magnetic moment, one may think ZT nanodisk as a potential candidate for future spintronic devices.

Electronic and magnetic properties of FS germanene

Germanene exhibits similar electronic band structure like graphene with linear band dispersion relation and zero bandgap near Dirac K points if effect of SOC is not taken into account [118, 130, 131].

However, the inclusion of SOC introduces a non-zero value of bandgap at Dirac K point along with Dirac cone-like signatures, which again ensures about more fascinating properties and hence effective applications of germanene in optoelectronics, photo-voltaics, etc. Liu et al. have predicted [27] through a systematic first-principles investigations that an appreciable gap of 4 meV and 23.9 meV can be opened at Dirac points for planar and LB germanium, respectively, by SOC. These features are clearly depicted in Figure 10(a) and (b).

Figure 10:

Relativistic band structure of germanium with honeycomb structure. (Left panel) the relativistic band structure of germanium with planar and (right panel) low-buckled honeycomb structure. Inset: Gap induced by SOC at Dirac K point (zooming).

Large amount of SOC indicates that germanene can exhibit QSHE in an experimentally accessible temperature regime, even at near room temperature [132]. Zhang et al. have studied the electronic properties of germanene sheets that are found on Ge2Pt crystals after deposition of Pt on Ge(110) substrates. They have observed a V-shaped DOS which is indicative for a 2D Dirac system [133]. Besides, Behera et al. have also reported that energy dispersion relation near Dirac K point is linear for germanene by first-principles calculations [134]. Houssa et al. [135] have predicted germanene as metallic in nature with a low DOS at Fermi energy (EF) by DFT calculations using LDA. Moreover, Lebegue et al. have also strongly confirmed germanene as a poor metal possessing small non-zero DOS at EF by using both LDA and GGA [136]. Walhout et al. by an experimental study investigated about temperature dependence of the DOS of germanene grown on Ge/Pt crystals. They have successfully found V-shaped DOS, the hallmark of a 2D Dirac system [137]. They have explored about the fact that substrate effect can modify the original DOS of FS germanene significantly. Li et al. through a very recent DFT calculation [138] have proposed a new approach to fabricate germanene via dehydrogenating H-Ge, having the same Dirac electronic properties.

Hydrogenated silicene and germanene, termed as silicane and germanane respectively, also possess very interesting electronic properties. M. Houssa et al. [139] have predicted that in both silicane and germanane, there is a finite band opening in the band structure. The type of gap in silicane is direct or indirect depending on its atomic configuration (chairlike or boatlike). But in the case of germanene, there is always a direct bandgap opening of about 3.2 eV independent of atomic configuration which makes this material potentially interesting for application in optoelectronic devices (shown in Figure 11(a), (b), (c) and (d)).

Figure 11:

Band structures, calculated using LDA functional, (a) chairlike silicane, (b) boatlike silicane, (c) chairlike germanane and (d) boatlike germanane. Reference zero-energy level is set to the top of the valence band.

Rupp et al. [140] have reported about modifications in stability and electronic properties of germanene in the presence of impurity atoms using DFT. They have observed that the adsorption of one hydrogen (H) atom by boron (B) or nitrogen (N) impurities leads to p- and n-type semiconducting properties, respectively. Germanene, like all other 2D honeycomb structures of III–V binary compounds, is non-magnetic in nature [141]. Electronic and magnetic properties of germanene can be tuned by doping or adsorption of foreign elements in pristine system [141] or introducing vacancy or applying some strain engineering. Li et al. have proposed [142] an efficient technique of band opening by nanopatterning germanene into super-lattices using DFT. They have indicated that the broken sublattice symmetry in nanopattterning germanene is the reason for opening of bandgap. Recently, Liang et al. have explored [143] about the fact that surface functionalization and strain will modify electronic and magnetic properties of hydrogenated, fluorinated and chlorinated germanene by employing DFT. They have highlighted that fluorinated germanenes are energetically more stable than hydrogenated and chlorinated germanenes because fluorine atoms possess stronger electronegativity. Pang et al. [144] have studied the effect of alkali metal (AM) atoms adsorbed in germanene and indicated that it is possible to tailor both the gap and the concentration of charge carries of AM/germanene systems by controlling the coverage of AM. From their analysis, it is also evident that AM/germanene could be of great interest in nanoindustry, like FET applications due to strong binding of AM atoms with germanene and exceptional interesting properties of AM/germanene systems. In another work, Pang et al. have analysed [145] about the structural, electronic and magnetic properties of 3d-TM adsorbed germanene. It was concluded that in most TM/germanene structures TM–Ge bonds exhibit mostly covalent chemical bonding character. Electronic and thermal properties of germanene has been studied recently by Zaveh et al. [146] employing DFT and density functional perturbation theory (DFPT). They have observed that specific heat at constant volume (Cv) varies with temperature (T) as T2 at low temperature from their calculations, which is consistent with the general argument that Cv∝ωd/s, for excitations obeying the dispersion ω∝ks in an arbitrary spatial dimension d. Li et al. have examined systematically [147] about alkali, alkali-earth, group III and TM adatom adsorption on bare germanene sheet employing first-principles theory. They have elucidated the fact that interaction between metal adatoms and germanene is quite stronger than that of graphene, due to large buckling in germanene. Moreover, Kaloni has predicted that HS is energetically more favourable for 3d-TM adsorption and TM adsorbed germanenes possess magnetic moment ranges from 0.97 to 4.95 μB [120]. So, it is possible to induce and tailor magnetism in non-magnetic germanene by incorporation of suitable adatoms in pristine system. Xia et al. have predicted by a DFT-based study on gas adsorption of germanene that germanene can be used as a gas sensing element efficiently [148]. So one can think to utilize germanene in the implementation of sensor device-based technology. Ni et al. have explored [23] that semi-metallic LB germanene is more suitable than semi-metallic planar graphene for inducing a finite value bandgap. This suggests that germanene is more efficient than graphene for practical applications. Ozcelik et al. have predicted some new extraordinary phases in Ge adatom adsorbed on germanene by employing first-principles calculations [149]. They have revealed that, through an exothermic and spontaneous process, Ge adatom constructs a dumbbell (DB) structure on germanene. These stable DB-based phases exhibit unique electronic and magnetic properties, which can be modified by controlling the coverage of DBs. For example, at high coverage metallic state is always maintained by germanene + DB phases , but semiconductor state can also be achieved by changing different parameters like DB–DB distances or size of unit cell. Besides, Gurel et al. have investigated about the modifications of electronic, magnetic and chemical properties of germanene by charging and applying perpendicular electric field. It can be concluded from their study that charging maintains hexagonal honeycomb symmetry in germanene, but the symmetry is broken by electric field. For this, a band splitting occurs, and the value of bandgap depends linearly on applied electric field [150]. Ye et al. have concluded [151] from an ab-initio study that, it is possible to open a bandgap ∼0.02–0.31 eV at the Dirac point in germanene, by the adsorption of AM atoms. Interestingly, they have also confirmed by effective mass calculation that carrier mobility in germanene remains unaffected by AM adsorption which is necessary for device application. Band structure of FS germanene can be reconstructed by inducing suitable n- or p-type adatoms like arsenic (As) and gallium (Ga) [152]. Doping has been incorporated at same or different sublattice positions of same hexagonal unit cell, which is at equivalent sites or non-equivalent sites, respectively. Different structures including pristine have shown in Figure 12(a)– (i), namely, pristine, S1, S2, S3, S4, S5, S6, S7 and S8, respectively.

Figure 12:

Structures (a) pristine, (b) S1, (c) S2, (d) S3, (e) S4, (f) S5, (g) S6, (h) S7, (i) S8. Largest and light-green coloured atoms are Ga, medium and black coloured atoms are Ge, smallest and deep-yellow coloured atoms are As.

It is predicted from DFT calculations that semi-metallic germanene is transformed to metallic nature by incorporating single or double doping of As and Ga, whereas semi-metallic property is preserved for AsGa codoped configurations, which can be justified by observing the position of Fermi level (EF), number of CB and VB which cross EF and analysing DOS. Thus, electronic and magnetic properties of FS germanene can be tailored by different ways like doping, adsorption, strain engineering and applying an external uniform electric field.

Optical properties of FS germanene

Now, before going for the analysis of optical properties of germanene, we would like to review the optical properties of bulk Ge briefly. Values of static dielectric constant ϵ1(0) and IR refractive index for bulk Ge have been reported as ∼15.9 [153] and ∼4 [154], respectively. Wei et al. have investigated that many-body effect plays a crucial role in the different properties of germanene. From absorption spectra analysis, they have concluded that, for germanene, π→π∗ resonant excitation appears at an energy value 1.10 eV with a binding energy 0.82 eV. They have further indicated that excitonic effect in 2D germanene is much stronger than that of graphene [155]. Bechstedt et al. have explored about IR absorbance of germanene by GGA–DFT method. They have demonstrated that value of IR absorbance for germanene is 0.02292, which is nearly equal to πα = 0.022925, where α= 1/137.036, the Sommerfeld fine structure constant [81]. Interestingly, it has also been concluded that this value of IR absorbance is a universal characteristic feature of studied group IV 2D materials, independent of materials and values of buckling[61, 80] . Optical conductivity of 2D germanene sheet has been studied by Matthes et al. using DFT. They have reported that in optical conductivity, most intensity peaks with phonon energy can be described by three damped harmonic oscillators in the region 0 to 10 eV [62]. Optical properties of germanene have been investigated by Pulci et al. using many-body perturbation theory [83]. Exciton binding energy and oscillator strengths in germanene have been found to be very strong, which make it a novel promising material in 2D nanoindustry. Imaginary part of dielectric constant (ϵ2(ω)) for silicene and germanene are shown in Figure 13. It can be depicted from this figure that, π plasmon peak appears at 1.2 eV and 1.4 eV and π+σ plasmon peak appears at 3.93 eV and 3.1 eV for silicene and germanene, respectively [156, 157].

Figure 13:

Imaginary part of dielectric constant for FS 2D materials silicene (black-dashed) and germanene (red-dotted) calculated using DFT.

Optical properties of FS germanene can also be modified by incorporating doping of foreign elements significantly. It has been elucidated that the value of universal IR absorbance can be enhanced or reduced than pristine germanene by incorporating suitable combinations of doping elements, site of doping and concentration of doping [152]. Different optical properties in terms of the real part of dielectric constant have been investigated (ϵ1(ω)) using first-principles DFT methodology. The optical properties of AsGa codoped structures were also studied schematically.

Figure 14:

Real part of dielectric constant for perpendicular polarization. (a) As doped structures, (b) Ga doped structures, (c) AsGa codoped structures.

To analyse the optical properties, the electric field is applied for perpendicular polarization, that is along the axis (Z axis) perpendicular to the plane of germanene sheet. Value of ϵ1(0) is 1.61 and there are no prominent modifications in ϵ1(0) due to doping as depicted from Figure 14(a)–(c). Plasma frequency (ωP) is defined by the energy position where ϵ1(ω) is equal to zero. Number of ωP for pristine layer is six, whereas the same is reduced to four, three, two and one in case of structures S4, S7, S6 and S8, respectively. As doped structures exhibit red-shifting nature of peaks and Ga doped structures possess blue-shifting nature of peaks as doping concentration is increasing, whereas peak positions remain almost unchanged in the case of codoped structures with respect to pristine layer. Moreover, it is also possible to tune other optical properties like EELS, optical absorption spectra, refractive index, reflectivity and optical conductivity by doping of impurity atoms in germanene. So it is noteworthy to conclude that the optical properties of germanene can be tailored in the presence of foreign elements, which may be helpful in the application of 2D optoelectronic industry, based on beyond graphene materials.

Electronic properties of pristine and functionalized TG sheet

The electronic properties of planar TG was first reported by Enyashin et al. [39]. Their calculated band structure has revealed that EF lies below the top of the VB, which confirms its metallic nature. This metallic nature is also supported by its appreciable (non-zero finite) DOS at EF. The distribution of electronic states near Fermi level for HOMO and LUMO indicates they are formed by contributions from states of the same C atoms. Long et al. [163] and Liu et al. [159] also substantiated the metallic nature of this structure in their respective studies. However, unlike planar TG, the ambipolar 2D structure buckled TG shows Dirac-like Fermions attributed to π and π∗ bands. This dissimilarity in the electronic properties can be described as follows. Planar TG consists of only one type of sublattice (one type of bonds) in the unit cell, whereas the buckled structure has two types of non-equivalent bonds. Nonetheless, Huang et al. [164] expressed their doubt on the existence of Dirac-like Fermions of buckled TG. They argued, however, that buckled TG is also normal metal and the band crossing near the Fermi level in buckled TG (as shown in Ref. [159]) can be derived from planar TG in terms of a different supercell by band folding (BF). Yet they pointed out that the carriers of buckled TG also show high Fermi velocity (vF) of the order of 106m/s, which is a property of Dirac-like Fermions. Keeping all these in mind, a search of reversible hydrogen storage media leads to the investigation on Li-decorated planar TG [165]. This was modelled by one Li atom adsorbed on each (2×2) cell. For single adsorption, there are many possible choices like HS, BS and TS [53] as described in Figure 9. However, Li prefers to be adsorbed above the centre of the octagon (Figure 15), i.e. the HS of TG. Nearest Li–Li distance for planar TG (6.9 Å) is larger than that of graphene (4.92 Å) and Li doped B2C sheet (5.12 Å). It is shown that each Li atom can adsorb hydrogen molecules. The underlying reason behind such adsorption can be described as follows. Li atoms get positively charged because of a charge transfer between Li and the sheet, which polarizes the H2 molecule. So it is clear that, under such a process, TG can be functionalized into a feasible hydrogen storage medium. In addition, Liu et al. [161] explored that Li-decorated TG exhibits a high sensitivity to CO and the Li−CO adsorption strength can be manipulated by external electrical field. Majidi [166] also has investigated the electronic properties of planar TG-like CBN and BN sheets and indicated these sheets are semiconductors in spite of the metallic nature of pristine planar TG.

TG nanoribbons (NRs) and clusters

Along with two types of TG sheets, Liu et al. [159] also investigated the variation of their properties with structural modifications, i.e. confinements. For buckled TG, armchair-like ribbon with width of one lattice constant (square-octagon periodic repetition along one direction) has the strongest quantum confinement and explores the linear dispersion relation. Whereas, periodic repetition of octagons, i.e. zigzag-like TGNR, exhibits metallic properties. Furthermore, spin-polarized first-principles calculations by them have indicated that zigzag-like TGNR prefers ferromagnetic state, while the former one is diamagnetic. Motivated by these intriguing properties of square-octagon repeated nets, Wang et al. [167] have employed first-principles study to investigate the structural and electronic properties of another tetra-symmetrical planar structure with space group P4/mmm. Lattice constant of the structure is 3.447 Å and its unit cell consists of 4 C atoms. Proper relaxation shows that, unlike graphene and the TG proposed by Liu et al., it consists of two distinct bonds (with bond lengths of 1.372 Å and 1.467 Å) and two different angles (π2) and (3π4). Absence of negative phonon modes and first-principles MD simulation at room temperature confirms its dynamical stability. Two types of charge density around two different bonds indicates a non-uniform charge distribution, which helps to designate these bonds as non sp2 type. Band structure and finite DOS at Fermi level prefer this material as metallic and predict its sound potential for future nanoelectronics. They have also extended their work by calculating the width-dependent electronic properties of armchair and zigzag nanoribbons. They have mentioned the zigzag ribbon as uniformly metallic regardless of its width whereas armchair structure shows width-dependent odd–even metal–semiconductor oscillating behaviour. Those bandgaps also decrease with increasing width and are expected to vanish at infinite width (for sheet).

Later, Dai et al. [168] have investigated the transport properties of the TGNRs. In that work, they have also supported the width-dependent metal–semiconductor oscillations by calculating a general expression for the bandgap.

For further details about these tetrasymmetrical structures, first-principles based Raman and IR spectra have been investigated for different cluster sizes [169]. The study has revealed that a phonon Raman mode appearing near 1,711–1,713 cm−1 (shown in Figure 16) survives for different cluster sizes. The position (in cm−1) agrees well with the previous reported phonon mode of planar TG sheet at 1,732 cm−1 by Wang et al. [167]. Therefore, this mode (shown in the inset(a) of Figure 16) must be a characterizing Raman mode for pure planar TG. This method is also applied by Das et al. [170] to identify the G band of graphene in graphene quantum dots of variable sizes, which agrees well with the experimental descriptions. The characterizing mode for TG is a stretching mode which arises form the planar stretching of the bonds connecting the squares. Besides, the study of electronic properties has indicated that these pure TG clusters (dangling bonds are saturated by H atoms) do not show any dipole moment and possess zero DOS at EF. Absence of DOS at EF proves its semi-conducting nature. This gap decreases with increasing cluster size, which is well expected as for infinite size one must obtain the metallic property of planar TG. Furthermore, for a particular cluster size effect of B and N doping is also explored as B and N doping significantly tune the properties of graphene [171]. To start with, preferred doping cites (minimum energy position) for B and N are identified and are kept unchanged for rest of the study. Doping introduces asymmetry in charge distribution which results in finite dipole moment. B (N) doped structure shows relatively low (high) intense peak compared to pristine structure. Formation energy calculation indicates N doped structure is more stable than pristine and B doped structure. Vibrational details corresponding to prominent Raman modes are also reported in the study [169], for characterization purpose. Among those vibrational modes, low-intensity (low frequency as well) breathing like modes have been observed for both pristine and doped structures. IR spectra for these 3 × 3 TG clusters are shown in Figure 17. It is revealed that larger absorptions are caused by both in-plane and out-of-plane vibration. Corresponding wave numbers are 1,297.78 cm−1 (for in-plane vibration) and 48.90 cm−1 and 237.90 cm−1 (for out-of-plane vibration).

Other allotropes beyond TG

Apart from the above-mentioned TG allotrope, there are other competitors with overwhelming properties. Some of them are briefly discussed in this section.

Partial replacement of sp2 hybridized aromatic bonds in graphene by carbyne chains forms a new family of graphene allotropes called graphynes. Baughman et al. [172] showed that different replacement portions of this acetylene linkage result in different types of graphynes. Among them, α, β and γ graphynes are well studied because of their high symmetric forms [173, 174]. Presence of Dirac cone in the energy band structure indicates semi-metallic behaviour of α and β graphynes. Whereas, γ graphyne is semiconducting in nature because of Kekule-distortion effect. Huang et al. [164] demonstrated the role of TB hopping parameters for graphynes in determining the condition for which Dirac cones can exist. Recently, successful syntheses of few graphynes are also reported [175].

Graphdiyne [176], first designed by Haley et al. [177], is composed of two acetylenic linkages between nearest-neighbour hexagonal rings and belongs to the space group P6mm. Later, many interesting properties like high degree of stability against temperature, semiconducting behaviour with silicon-like conductivity [178] and applicability in nanoscale devices [179] were reported. Interestingly, graphdiyne nanoribbons and nanotubes are experimentally synthesized and found to be useful in optoelectronic and spintronics [179, 180].

Another proposed allotrope is penta-graphene [181] which is composed of pentagons with C at its vertices, which closely resembles MacMohon’s net, a semiregular tiling of the Euclidean plane similar to Cairo pentagonal tiling. Total thickness of this 2D multidecker sandwich sheet is found to be 1.2 Å. The structure possesses P−421m symmetry and its tetragonal unit cell of lattice constant 3.64 Å consists of six atoms. Furthermore, the electronic properties confirm its indirect bandgap of 3.25 eV having semiconducting behaviour. Between rolled-up and staged penta-graphene, it is observed that penta-tubes are semiconducting regardless of its chirality. Despite all intriguing properties reported above, Ewels et al. [182] have strongly suggested that it will be impossible to construct penta-graphene experimentally. This allotrope cannot be isolated easily from some of its isomers because of their similar energies. Along with this, even a few catalytic impurities and environment effect however can force this structure to reconstruct rapidly towards graphene by continuous energy loss. In a recent study, Rajbanshi et al. have studied the properties of penta-graphene nanoribbons [183] and indicated that these are direct bandgap semiconductors with width-dependent tunable bandgap.

Stone–Wales (SW) defects are formed by a π2 rotation of a carbon dimer with respect to the midpoint of the bond. Such SW defects in graphene lead towards the transformation of four hexagonal rings into pairs of pentagons and heptagons. This transformed structure is also well studied and known as pentaheptites [184]. These structures are, in general, metallic in nature.

Another 2D allotropes can be achieved by systematic tessellations of octagons and pentagons (OP). Such alignments are of two types as predicted by Su et al. [185]. This can be viewed as colligating of ribbons formed by five-five-eight membered rings along a straight line path (for OPG-L) or along a zigzag path (OPG-Z). These structures are energetically more favourable than recently synthesized graphdiyne. Further studies on electronic properties indicate that OPG-L is a metal while OPG-Z is a gapless semi-metal.

Replacement of one-third sp2 bonds of graphene by acetylenic linkages forms new set of energetically stable allotropes popularly known as Haeckelites as proposed by Terrones et al. [186]. This structure is a mixture of pentagons, hexagons or/and heptagons. There are three different members in this family: (a) symmetric arrangement of only heptagons and pentagons, (b) repeated structure of three connected heptagons, with alternating pentagons and hexagons, are in its surroundings and (c) tiling by connecting pentalene and heptalene and surrounded by hexagonal rings. In a later study, Enyashin et al. [39] have also introduced another periodic arrangement of pentagons, hexagons and octagons in this Haeckelites family and claimed its semiconducting behaviour.

Some other possible allotropes are also drawing keen attention of the researchers because of their novel exotic properties as follows. S-graphene is a periodic arrangement of six- and four-member rings with eight atoms in its unit cell. Similarly, D-graphene and E-graphenes are composed of sp – sp2 and sp3 – sp2 hybridized C atoms, respectively [188]. In a review, Wang et al. [187] have explored that these tree 2D rectangular systems also exhibit Dirac cones. Another allotrope with tetra-rings and acetylenic linkages, known as rectangular graphyne (R-graphyne) [189], shows metallic behaviour. It is found that R-graphene nanoribbons can show both metallic, semiconducting and unexpected semi-mettalic properties depending on the shape and size of the ribbons. Two similarly proposed structures supergraphene and squarographenes are investigated to be metallic and show that relaxed squarographenes can be found in two forms, either arranged by distorted hexagons and regular squares [190] or by undistorted hexagons and rhombuses [186] while graphene-like material supergraphene is gapless semi-metal. A pictorial representation of the mentioned structures is given in Figure 18. Recently researchers, i.e. Kotakoski et al. and Lahiri et al., have used advanced experimental techniques like electron beam irradiation, defect engineering, etc. to form quasi-1D carbon structures consisting of repeated tetrarings-octarings and repeated octarings-pentarings [191, 192].

Although mostly electronic and structural properties in these materials are explored, however, studies involving the optical properties including Raman are still lacking. We hope to get a good direction if some of these materials are synthesized.

Figure 15:

Favourable site for Li adsorption on planar TG. The arrows denote Li diffusion paths from the octagonal site to a neighbouring one.

Figure 16

Raman and IR spectra of 4×4 TG cluster, characterizing mode of vibration for TG is shown in the inset (a), H passivated TG cluster is shown in inset (b).