Electronic and optical properties of carbon nanodisks and nanocones
EElectronic and optical properties of carbon nanodisks and nanocones
P. Ulloa ∗ and M. Pacheco Departamento de F´ısica, Universidad T´ecnica Federico Santa Mar´ıa, Casilla 110-V, Valpara´ıso, Chile
L. E. Oliveira
Instituto de F´ısica, Universidade Estadual de Campinas-UNICAMP, Campinas-SP, 13083-859, Brazil
A. Latg´e
Instituto de F´ısica, Universidade Federal Fluminense, 24210-340, Niter´oi-RJ, Brazil (Dated: December 2, 2018)A theoretical study of the electronic properties of nanodisks and nanocones is presented within theframework of a tight-binding scheme. The electronic densities of states and absorption coefficientsare calculated for such structures with different sizes and topologies. A discrete position approxi-mation is used to describe the electronic states taking into account the effect of the overlap integralto first order. For small finite systems, both total and local densities of states depend sensitively onthe number of atoms and characteristic geometry of the structures. Results for the local densitiesof charge reveal a finite charge distribution around some atoms at the apices and borders of thecone structures. For structures with more than 5000 atoms, the contribution to the total density ofstates near the Fermi level essentially comes from states localized at the edges. For other energiesthe average density of states exhibits similar features to the case of a graphene lattice. Results forthe absorption spectra of nanocones show a peculiar dependence on the photon polarization in theinfrared range for all investigated structures.
I. INTRODUCTION
Carbon nanocones (CNCs) have been observed by mi-croscopy techniques[1, 2], showing an ordered atomicstructure. Large progress has been made on synthe-sis, characterization and manipulation of CNCs and car-bon nanodisks (CNDs) [3–8]. From the technologicalpoint of view, applications such as microscopy probesand electron-emitter devices may be envisaged by con-sidering the electric current established through the coneapex when electric fields are applied[9].There are different theoretical schemes to describethe electronic properties of cone-like structures. Mod-els based on the Dirac equation[10, 11] give a conve-nient insight of properties in the long wavelength limit.However, for finite-size graphenes, the longest stationarywavelength occurs in the border and a correct descrip-tion of the states near the Fermi level is given in termsof edge states[12, 13]. Ab initio models [14–16] are ableto predict detailed features, but they are restricted tostructures composed of a few hundred atoms due to theirconsiderable computational costs. Calculations based ona single π orbital are able to describe the relevant elec-tronic properties[17, 18]. In that spirit, we calculatethe electronic structure and optical spectra of CNDs andCNCs within a tight binding approach. CNC structuredsystems generated by pentagonal and heptagonal defectswere previously studied using a Green function recursivemethod [18, 19]. It was shown that, for cones generatedwith an odd disclination number n w , it is not possible ∗ Electronic address: [email protected] to define A and B graphene sublattices. In this case,therefore, the electron-hole symmetry is broken.The total number N C of carbon atoms in a cone struc-ture may be estimated by dividing the cone surface areaby half of the hexagonal cell’s surface, N C = [4 π/ (3 √ − n w / r D /a CC ) , (1)where the disclination number n w corresponds to the in-teger number of π/ r D is the cone generatrix [see Figure1]. The nanocone disclination angle is given by n w π/ n w = 1 and r D = 1 µ m, the CNC has ≈ atoms. By extracting an integer number n w of π/ n w = 1, thecone angle is 2 θ = 112 . ◦ , corresponding to the flattestpossible cone. In this case, h/r C = 0 .
66 and h/r D = 0 . s is considered different from zero.As we will show later, this has important effects on thecone energy spectrum.Although some of the graphene electronic propertiesare present in the CNCs, deviations are always mani-fested as a consequence of the different atomic arrange-ments, the finite-size of the nanocones, and also the pos-sible point symmetry of the distinct cones. In the ab-sence of external fields, the calculated density of states(DOS) shows a peak at the Fermi energy and the localdensity of states (LDOS) shows that electron states are a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Figure 1: (Color online) Pictorial view of (a) a carbon diskcomposed of six wedge-section of π/ n w = { , . . . , } wedge-sections from thedisk, and (c) by folding it is constructed a cone. Geometricalelements: generatrix r D , height h c , base radius r c , and apexopening angle 2 θ , where sin θ = 1 − n w / localized at the cone base. On the other hand, the sym-metries observed in the LDOS at different energies allowa systematic description of the electronic structure andselection rules of optical transitions driven by polarizedradiation. Unlike the nanodisk, the presence of topolog-ical disorders in nanocones involve a deviation from theelectrical neutrality at the apex and at the zigzag edges. II. THEORETICAL FRAMEWORK
In what follows, we present results for n w = { , , } ,corresponding to CND and CNCs whose disclination an-gles are 60 ◦ and 120 ◦ . For those systems, the sp hy-bridization may be neglected. The electronic wave func-tion may be written as | Ψ (cid:105) = N C (cid:88) j =1 C j | π j (cid:105) , (2)where the | π j (cid:105) denote the atomic orbitals 2 p at site (cid:126)R i .Note that the overlapping between neighbouring orbitalsprevents the set consisting of the ket | π j (cid:105) to be an or-thogonal basis. Only in the ideal case of zero overlap s = 0, the coefficients C j in | Ψ (cid:105) = (cid:80) N C j =1 C j | π j (cid:105) mightbe considered equal to the discrete amplitude probability (cid:104) π j | Ψ (cid:105) to find an electron at the j -th atom (described bythe one electron state | Ψ (cid:105) ). We use the s (cid:54) = 0 basis, | π j (cid:105) ,to construct the eigenvalue equation and the | π j (cid:105) baseto calculate the properties related to discrete positions.Of course, to relate both bases it is required to know the (cid:104) π j | π j (cid:105) projection.We define a N C × N C matrix ∆ (1) relating the nearestneighbouring atomic sites i, j ,∆ (1) ij = (cid:26) , ( i, j ) are n.n.0 , otherwise (cid:27) . (3)Similarly, ∆ (0) ij = (cid:26) , ( i = j )0 , otherwise (cid:27) . (4) The S overlap matrix elements are then given by S ij = (cid:104) π i | π j (cid:105) = ∆ (0) ij + s ∆ (1) ij . (5)The hopping matrix elements of the tight-binding Hamil-tonian ˆ H = ˆ H ( at ) + ˆ V are (cid:104) π i | ˆ V | π j (cid:105) = t ∆ (1) ij , (6)where t is the hopping energy parameter. Assuming theeigenvalue equation ˆ H ( at ) | π j (cid:105) = ε p | π j (cid:105) , the atomic ma-trix elements are (cid:104) π i | ˆ H ( at ) | π j (cid:105) = ε p (cid:104) ∆ (0) ij + s ∆ (1) ij (cid:105) , (7)and (cid:104) π i | ˆ H | Ψ (cid:105) = ε (cid:104) π i | Ψ (cid:105) , ≤ i ≤ N C . (8)The resulting equation system may be written as ageneralized eigenvalue problem H (cid:126)C = εS (cid:126)C , where thecolumn vector (cid:126)C contains the coefficient C j , (cid:104) ε p (∆ (0) + s ∆ (1) ) + t ∆ (1) (cid:105) (cid:126)C = ε (cid:104) ∆ (0) + s ∆ (1) (cid:105) (cid:126)C. (9)The general solution may be expressed in terms of theauxiliar variables (cid:126)C (0) and ε (0), which satisfy t ∆ (1) (cid:126)C (0) = ε (0) (cid:126)C (0) . (10)As (cid:126)C (0) also satisfies Eq. (9), we obtain ε = [ ε p + (1 + sε p /t ) ε (0)] / [1 + sε (0) /t ] . (11)The orthogonality condition for the electronic states (cid:104) Ψ k | Ψ l (cid:105) = (cid:126)C k † S (cid:126)C l = δ k,l (12)implies that (cid:126)C = (cid:126)C (0) (cid:113) (cid:126)C (0) † S (cid:126)C (0) . (13)For the calculation of the DOS we use a Lorentzian dis-tributionDOS( ε ) = 2 N C (cid:88) j =1 δ ( ε − ε j ) ≈ N C (cid:88) j =1 Γ /π ( ε − ε j ) + Γ . (14)The LDOS is calculated in terms of the discrete am-plitude probability, (cid:104) π i | Ψ j (cid:105) ,LDOS( (cid:126)R i , ε ) = 2 N C (cid:88) j =1 |(cid:104) π i | Ψ j (cid:105)| δ ( ε − ε j ) , (15)where (cid:104) π i | Ψ j (cid:105) = (cid:104) ∆ (0) + s (1) (cid:105) (cid:126)C j , (16)as it is shown in Appendix A.The local density of charge (LDOC) related to the π electrons is calculated by assuming that the other 5 elec-trons and the 6 protons of the carbon atom act as a netcharge + e . Assuming zero temperature and the indepen-dent electron approximation, only the states 1 ≤ j ≤ n F will be occupied, where n F = (cid:26) N C / , N C is even( N C + 1) / , N C is odd . (17)Taking into account that the states below n F con-tribute with − e and the fact that the n F state con-tribution depends on the parity of the number of atomsin the system, the LDOC is written asLDOC( (cid:126)R i ) = e − n F − (cid:88) j =1 |(cid:104) π i | Ψ j (cid:105)| − γ |(cid:104) π i | Ψ n F (cid:105)| (18)with γ =0 and 1, for N C even and odd, respectively.Optical absorption coefficients α (cid:15) ( ω ) are calculated byconsidering perpendicular ( ˆ (cid:15) ⊥ = ˆ (cid:15) x , ˆ (cid:15) y ) and parallel ( ˆ (cid:15) (cid:107) =ˆ (cid:15) z ) polarizations, in relation to the cone axis, α ˆ (cid:15) ( ω ) ∝ ω (cid:88) i,j (cid:12)(cid:12) (cid:104) Ψ i | ˆ (cid:15) · (cid:126)p | Ψ j (cid:105) (cid:12)(cid:12) δ (cid:2) ε j − ε i − (cid:126) ω (cid:3) , (19)with ε i,j corresponding to the energies of occupied andunoccupied states, respectively.The oscillator strength may be written in terms of thespatial operators (ˆ x , ˆ y and ˆ z ) [20], i.e., (cid:104) Ψ i | ˆ p z | Ψ j (cid:105) = m e i (cid:126) ( ε j − ε i ) (cid:104) Ψ i | ˆ z | Ψ j (cid:105) , (20)where (cid:104) Ψ i | ˆ z | Ψ j (cid:105) is calculated to first order in s, using(A9) of Appendix A, (cid:104) Ψ i | ˆ z | Ψ j (cid:105) = (cid:126)C i † (cid:104) z + s (cid:16) ∆ (1) z + z ∆ (1) (cid:17)(cid:105) (cid:126)C j . (21) III. RESULTS AND DISCUSSIONA. Electronic Density of States
In what follows, we present numerical results for sys-tems composed of up to 5000 atoms. In the limitingcase of N C → ∞ , the energy spectrum is in the rangefrom ε min = − | t | / (1 + 3 s ) to ε max = +3 | t | / (1 − s ),the van Hove singularities occur at ε v vH = −| t | / (1 + s ), ε c vH = + | t | / (1 − s ), and the Fermi energy is at ε F = 0.A Γ = | t | /
100 broadening and an overlap s = 0 .
13 areassumed. Different plots in Figure 2 show the densityof states averaged over the N C atoms and the LDOS fora CND [Figures 2(a) and 2(d)], a single-pentagon CNC[Figures 2(b) and 2(e)], and for a 2-pentagon CNC [Fig-ures 2(c) and 2(f)], for N C = 258 , ε = 0. Figure 2: (color online) DOS and LDOS for a N C = 258nanodisk [(a) and (d)], a N C = 245 1-pentagon nanocone [(b)and (e)], and a N C = 246 2-pentagon nanocone [(c) and (f)].LDOS curves (black-1, red-2, blue-3) match, for each system,the corresponding atoms shown in Figure 3. Vertical lines ineach panel indicate the position of the Fermi energy.Figure 3: Pictorial view of (a) a nanodisk center, (b) a1-pentagon nanocone apex, and (c) a 2-pentagon nanoconeapex. Atoms with different colors-numbers indicate differentpoint symmetries for each system.Figure 4: (color online) DOS and LDOS for a N C = 5016nanodisk [(a) and (d)], a N C = 5005 1-pentagon nanocone[(b) and (e)], and a N C = 5002 2-pentagon nanocone [(c) and(f)]. LDOS curves (black-1, red-2, blue-3) match, for eachsystem, the corresponding atoms shown in Figure 3. Verticallines in each panel indicate the position of the Fermi energy. Figure 5: (Color online) LDOS in arbitrary units for a 5016-atom nanodisk [panels (a) to (e)], a 5005-atom nanocone with onepentagon at the apex [panels (f) to (j)], and a 5002-atom nanocone with 2 pentagons at apex [panels (k) to (o)]. The consideredenergies are (a,f,k) ε min , (b,g,l) ε v vH , (c,h,m) ε F , (d,i,n) ε c vH , and (e,j,o) ε max . The LDOS is measured with respect to the meanLDOS which is equal to the DOS at the considered energy. As expected for small finite systems, the DOS, LDOS,and the position of the Fermi energy depend on the num-ber of atoms considered in the numerical calculation andon their characteristic geometries [21–23]. A remarkabledifference between CND and CNCs structures is the exis-tence of a finite DOS above the Fermi level for nanocones.This clear metallic character of the DOS for nanoconesis more robust for the 2-pentagon CNC [22, 24]. Onemay remark that this feature comes out from a symme-try break induced by the presence of topological defectsin the CNC lattices, which generates new states abovethe Fermi energy, not present in the CND structure. Thecontributions to the DOS coming from the apex atomsstates are apparent in the LDOS of Figures 2(e) and 2(f).Also notice that for the 2-pentagon case, in which thereis a large topological disorder, the LDOS spectra exhibitsignificant differences depending on the point symmetryof the considered atom (cf. Figure 3).For increasing number of atoms, the total DOS forthe different nanostructures are very similar to the corre-sponding DOS of a graphene layer, except for the edgesstates which show up as a peak at the Fermi energy, asshown in Figures 4(a), 4(b), and 4(c). It is interestingto note that the apex atomic states do not contribute tothe total DOS near the Fermi energy but mainly near thegraphene-like van Hove peaks. Notice that in the case oftwo-pentagon nanocones the LDOS at the tip exhibits arobust metallic character.To analyse the finite-size effects and the role played bythe different symmetries of the cone-tip sites, we depictLDOS contour plots for the three studied structures byconsidering some characteristic energies: the minimumenergy, the resonant peak below the Fermi energy, the Fermi energy, the resonant peak above the Fermi energy,and the maximum energy. Figure 5 illustrates the ex-ample of a CND with 5016 atoms (top row), a single-pentagon CNC with 5005 atoms (middle row), and a2-pentagon CNC with 5002 atoms (bottom row). Theelectronic states corresponding to energies at the bandextrema have the largest wavelength compared to thecharacteristic size of the system. In this way, the de-tails of the lattice become less important and the statesexhibit azimuthal symmetry. An interesting feature forthe nanocones is that at these energies the apex corre-sponds to a node for the maximum energy and an antin-ode for the minimum energy, respectively. On the otherhand, the states at the Fermi energy are localized at thecone border, mainly at the zigzag edges as it is clearlyshown in Figures 5(c), 5(h), and 5(m). For the stateswhose energy is near to the van Hove peaks, the LDOSreflect the symmetries of each system, i.e., for CND the2 π/ π/ π/ B. Electric Charge Distribution
The electric charge per site, in terms of the fundamen-tal charge e , was obtained using Eq. (18). Results forthe electric charge distribution for CNDs indicate thatall the atomic sites preserve the charge neutrality, i.e.,LDOC=0. For the CNCs, however, the atoms at the sites 1 2 3 max1-pentagon -0.071e +0.014e -0.059e +0.042e2-pentagon -0.055e -0.067e -0.066e +0.076eTable I: LDOC (fundamental charge units) at some relevantatoms in the cone apices shown in Figure 3(b) and (c). Themaximum (max) value occurs at the zigzag edge of each sys-tem.Figure 6: (Color online) Electric charge distribution in neutralCNCs, for a single-pentagon structure with 245 atoms (a) andtwo-pentagon system with 246 atoms (b). apex acquire negative charge and the atoms around thecone base exhibit positive charges at the zigzag edges.As N C increases, the LDOCs at the apices, for the twostudied CNC structures, tend to the asymptotic valuesshown in table I, which are in good agreement with thevalues reported by Green method calculations [18, 19].Figure 6 depicts the LDOC for the two types of CNCstructures, showing that the nonequilibrium of the chargedistribution is restricted to the apex and edge regions:electric neutrality is found at all the other surface sites.The values found for the LDOC at the apex regions arefound to be independent of the size of the cones whereasthis is not true for the edge states. When the numberof atoms of the CNC structure is even, the edge-statesLDOC exhibits the same symmetry of the cone. For odd N C , the Fermi level is occupied by a single electron andthen the LDOC at the edge states reflect the broken sym-metry. C. Absorption Spectra
We have also calculated the absorption coefficient forthe CND and CNC structures, for different photon po-larizations. Figure 7 shows the results for the absorptioncoefficients α x and α y , for polarization perpendicular tothe cone axis, and α z for parallel polarization. Calcu-lated results are shown for a nanodisk composed of 5016atoms, a single-pentagon nanocone with 5005 atoms, anda 2-pentagon nanocone with 5002 atoms. For the case oflarge CNDs, the spectra present the general features ob-served for the absorption of a graphene monolayer. Inthe infrared region the absorption coefficient of a gra-phene monolayer is expected to be strictly constant [25],whereas for higher energies the spectrum shows a stronginterband absorption peak coming from transitions nearthe M point of the Brillouin zone of graphene [26]. The Figure 7: (Color online) Absorption coefficient for x (blackcurves), y (red curves) and z (blue lines) polarizations for (a)a nanodisk with 5016 atoms, (b) a single-pentagon nanoconecomposed of 5005 atoms, and (c) a two-pentagons nanoconewith 5002 atoms. The photon energies are given in units of (cid:126) ω/t . main difference for a finite CND is a departure from acompletely frequency-independent behavior for low ener-gies, where the absorption coefficient shows oscillationsas a function of the photon energy instead of a constantvalue. This is a consequence of the border states thatare manifested as a peak in the total DOS at the Fermienergy [27, 28]. For CNCs, the general behaviour is thesame as for nanodisks, except for the dependence of theabsorption on the photon polarization, in particular forlow energies. Furthermore, the main absorption peaksfor different polarizations occur when the photon energyis equal to the energy between the two DOS van Hove-like peaks (cf. Figure 4). Notice that the overlap integral s (cid:54) = 0 leads to an energy shift of the main resonant ab-sorption peak given by δ ≈ s | t | / (1 − s ) ≈
100 meV.This is a significant value for actual experimental mea-surements.Concerning the different polarization directions, oneshould notice that, as occurs in C v symmetric systems, α z = 0 and α x = α y for the nanodisk. On the otherhand, the absorption coefficients for the different conesstudied (single and two pentagons) are finite for parallelpolarization, and it depends on the structure details: as α z increases for a two-pentagon CNC structure, α x,y de-creases. Due to the lack of π/
2- rotation symmetry, one
Figure 8: (Color online) Absorption coefficient for x (blackcurves), y (red curves) and z (blue lines) polarizations for (a)a nanodisk with 258 atoms, (b) a single-pentagon nanoconecomposed of 245 atoms, and (c) a two-pentagon nanoconewith 246 atoms. The photon energies are given in units of (cid:126) ω/t . Curves in each panel are vertically shifted, for bettervisualization of different polarization results. should expect, in principle, different results for x- andy-polarizations for any nanocone. However, such differ-ence is observable just for the absorption coefficient ofthe two-pentagon CNC system, mainly in the range oflow photon energies. The fact that α x = α y , for the caseof 1-pentagon CNC structure, may be explained usingsimilar symmetry arguments applied to C v symmetrydots[28], extended to the C v symmetric cones. In thecase of a 2-pentagon CNC, the apex exibits a C v sym-metry, preventing the cone to be a C v symmetric sys-tem. As the apex plays a minor role, α x and α y will beslightly different. A large difference between the α z andthe α x,y CNC absorption spectra occurs in the limit oflow radiation energy. The α z coefficient goes to zero as (cid:126) ω → α x,y shows oscillatory features. The be-haviour of the absorption for parallel polarization is dueto the localization of the electronic states at the atomicsites around the cone border. As the spatial distributionof those states are restricted to a narrow extension alongthe z coordinate, the z degree of freedom is frozen for lowexcitation energies.The dependence of the absorption spectra on the ge-ometrical details of the different structures is more no-ticeable for finite-size nanostructures. This can be seen in Figure 8 which depicts the absorption coefficients forthe CND composed of 258 atoms, the single-pentagonCNC with 245 atoms, and the 2-pentagon CNC with246 atoms. The degeneracy of the x- and y- polariza-tion spectra is apparent for the smaller one-pentagon na-nocone, as expected due to symmetry issues. On theother hand, the symmetry reduction for the 2-pentagonstructure leads to a rich absorption spectra, exhibitingpeaks at different energies and with comparable weightsfor distinct polarizations. In that sense, one may pro-pose absorption experiments as an alternative route todistinguish between different nanocone geometries. IV. CONCLUSIONS
Here we have presented a theoretical study on theelectronic properties of nanodisks and nanocones in theframework of a tight-binding approach. We have pro-posed a discrete position approximation to describe theelectronic states which takes into account the effect ofthe overlap integral to first order. While the | π (cid:105) basekeeps the phenomenology of the overlap between neigh-boring atomic orbitals, the | π (cid:105) base allows the construc-tion of diagonal matrices of position-dependent opera-tors. A transformation rule was set up to take advantageof these two bases scenarios.We have investigated the effects on the DOS andLDOS, of the size and topology of CND and CNC stru-tures. We have found that both total and local densityof states sensitively depend on the number of atoms andcharacteristic geometry of the structures. One impor-tant aspect is the fact that cone and disk borders playa relevant role on the LDOS at the Fermi energy. Forsmall finite systems the presence of states localized in thecone apices determines the form of the DOS close to theFermi energy. The observed features indicate that smallnanocones could present good field-emission properties.This is corroborated by the calculation of the LDOC,that indicates the existence of finite charges at the apexregion of the nanocones. For large systems, the contri-bution to the DOS near the Fermi level is mainly due tostates localized in the edges of the structures wheres forother energies, the DOS exhibits similar features to thecase of a graphene lattice.The absorption coefficient for different CNC structuresare calculated and we have found a peculiar dependenceon the photon polarization in the infrared range for theinvestigated systems. The symmetry reduction of the 2-pentagon nanocones causes the formation of highly struc-tured absorption spectra, with comparable weights fordistinct polarizations. The breaking of the degeneracy fordifferent polarizations is found to be more pronounced forsmall nanocones. Absorption experiments may be usedas natural measurements to distinguish between differentnanocone geometries. Acknowledgements
This work was supported by Fondecyt grant 1100672and USM internal grant 11.11.62. P. Ulloa thanks DGIPand Mecesup PhD scholarships and the warm hospital-ity of the Departamento de Fisica da Materia Conden-sada da Universidade Estadual de Campinas and Insti-tuto de F´ısica da Universidade Federal Fluminense. Spe-cial thanks to Professor Patricio Vargas for his helpfuladvices.
Appendix A
A discrete position scheme in terms of the | π j (cid:105) stateswas used to represent functions of the position given interms of the atomic base, since they satisfy the sameproperties of the position states, i.e., orthogonality (cid:104) π i | π j (cid:105) = ∆ (0) ij , (A1)and completeness N C (cid:88) k =1 | π k (cid:105) (cid:104) π k | = ˆ1 (A2)in a N C -dimensional subspace. The identity operatormay also be constructed using the s (cid:54) = 0 base asˆ1 = (cid:88) k,l | π k (cid:105) ( S − ) kl (cid:104) π l | , (A3)with the S − ≈ ∆ (0) − s ∆ (1) + O ( s ) matrix being dif-ferent from the N C × N C identity matrix ∆ (0) .We take | π (cid:105) as the discrete position state and assumethat the matrix elements f Rij of position-dependent func- tions f ( (cid:126)R ) are known in the s = 0 representation, f Rij = (cid:104) π i | ˆ f | π j (cid:105) = f ( (cid:126)R j ) ∆ (0) ij . (A4)Differently from the f R matrices, f matrices in the s (cid:54) = 0representation f ij = (cid:104) π i | ˆ f | π j (cid:105) (A5)are not diagonal. However, by performing the similaritytransformation (cid:104) π i | ˆ f | π j (cid:105) = (cid:88) k,l (cid:104) π i | π k (cid:105) (cid:104) π k | ˆ f | π l (cid:105) (cid:104) π l | π j (cid:105) , (A6)we may obtain the unknown f matrix in terms of theknown f R matrix, provided the transformation rule be-tween the π and π bases is known. By assuming (cid:104) π i | π j (cid:105) = ¯ α ∆ (0) ij + ¯ β ∆ (1) ij , the s (cid:54) = 0 representation maybe found. The coefficients ¯ α and ¯ β are obtained by usingthe identity (A2) into Eq. (5), N C (cid:88) k =1 (cid:104) π i | π k (cid:105) (cid:104) π k | π j (cid:105) = ∆ (0) ij + s ∆ (1) ij + O ( s ) , (A7)and, to first order in s , (¯ α = 1 and ¯ β = s/
2) we have (cid:104) π i | π j (cid:105) = ∆ (0) ij + ( s/
2) ∆ (1) ij . (A8)By replacing (A8) in (A6), one obtains f ij = (cid:104) f ( R ) + s (cid:16) f ( R ) ∆ (1) + ∆ (1) f ( R ) (cid:17)(cid:105) i,j + O ( s )(A9)as the matrix elements of a position-dependent functionin the π -base. [1] K. Sattler, Scanning tunneling microscopy of carbon nan-otubes and nanocones, Carbon 33 (7) (1995) 915–920.[2] T. Garberg, S. N. Naess, G. Helgesen, K. D. Knudsen,G. Kopstad, A. Elgsaeter, A transmission electron micro-scope and electron diffraction study of carbon nanodisks,Carbon 46 (12) (2008) 1535–1543.[3] S. N. Naess, A. Elgsaeter, G. Helgesen, K. D. Knudsen,Carbon nanocones: wall structure and morphology, Sci.Technol. Adv. Mater. 10 (6) (2009) 065002.[4] C.-T. Lin, C.-Y. Lee, H.-T. Chiu, T.-S. Chin, Graphenestructure in carbon nanocones and nanodiscs, Langmuir23 (26) (2007) 12806–12810.[5] A. Krishnan, E. Dujardin, M. M. J. Treacy, J. Hugdahl,S. Lynum, T. W. Ebbesen, Graphitic cones and the nu-cleation of curved carbon surfaces, Nature 388 (6641)(1997) 451–454.[6] W. Zhang, M. Dubois, K. Guer´ın, P. Bonnet, E. Petit,N. Delpuech, D. Albertini, F. Masin, A. Hamwi, Effectof graphitization on fluorination of carbon nanocones and nanodiscs, Carbon 47 (12) (2009) 2763–2775.[7] V. del Campo, R. Henr´ıquez, P. H¨aberle, Effects of sur-face impurities on epitaxial graphene growth, App. Surf.Sci. 264 (0) (2013) 727–731.[8] K. A. Ritter, J. W. Lyding, The influence of edge struc-ture on the electronic properties of graphene quantumdots and nanoribbons, Nature Mater. 8 (3) (2009) 235–242.[9] F. Houdellier, A. Masseboeuf, M. Monthioux, M. J.H¨ytch, New carbon cone nanotip for use in a highly co-herent cold field emission electron microscope, Carbon50 (5) (2012) 2037–2044.[10] Lammert2004, Graphene cones: Classification by ficti-tious flux and electronic properties, Phys. Rev. B 69 (3)(2004) 035406.[11] Y. A. Sitenko, N. D. Vlasii, On the possible inducedcharge on a graphitic nanocone at finite temperature, J.Phys. A: Math. Theor. 41 (16) (2008) 164034.[12] M. Wimmer, A. R. Akhmerov, F. Guinea, Robustness[1] K. Sattler, Scanning tunneling microscopy of carbon nan-otubes and nanocones, Carbon 33 (7) (1995) 915–920.[2] T. Garberg, S. N. Naess, G. Helgesen, K. D. Knudsen,G. Kopstad, A. Elgsaeter, A transmission electron micro-scope and electron diffraction study of carbon nanodisks,Carbon 46 (12) (2008) 1535–1543.[3] S. N. Naess, A. Elgsaeter, G. Helgesen, K. D. Knudsen,Carbon nanocones: wall structure and morphology, Sci.Technol. Adv. Mater. 10 (6) (2009) 065002.[4] C.-T. Lin, C.-Y. Lee, H.-T. Chiu, T.-S. Chin, Graphenestructure in carbon nanocones and nanodiscs, Langmuir23 (26) (2007) 12806–12810.[5] A. Krishnan, E. Dujardin, M. M. J. Treacy, J. Hugdahl,S. Lynum, T. W. Ebbesen, Graphitic cones and the nu-cleation of curved carbon surfaces, Nature 388 (6641)(1997) 451–454.[6] W. Zhang, M. Dubois, K. Guer´ın, P. Bonnet, E. Petit,N. Delpuech, D. Albertini, F. Masin, A. Hamwi, Effectof graphitization on fluorination of carbon nanocones and nanodiscs, Carbon 47 (12) (2009) 2763–2775.[7] V. del Campo, R. Henr´ıquez, P. H¨aberle, Effects of sur-face impurities on epitaxial graphene growth, App. Surf.Sci. 264 (0) (2013) 727–731.[8] K. A. Ritter, J. W. Lyding, The influence of edge struc-ture on the electronic properties of graphene quantumdots and nanoribbons, Nature Mater. 8 (3) (2009) 235–242.[9] F. Houdellier, A. Masseboeuf, M. Monthioux, M. J.H¨ytch, New carbon cone nanotip for use in a highly co-herent cold field emission electron microscope, Carbon50 (5) (2012) 2037–2044.[10] Lammert2004, Graphene cones: Classification by ficti-tious flux and electronic properties, Phys. Rev. B 69 (3)(2004) 035406.[11] Y. A. Sitenko, N. D. Vlasii, On the possible inducedcharge on a graphitic nanocone at finite temperature, J.Phys. A: Math. Theor. 41 (16) (2008) 164034.[12] M. Wimmer, A. R. Akhmerov, F. Guinea, Robustness