High-order harmonic generation in hexagonal nanoribbons
EEPJ manuscript No. (will be inserted by the editor)
High-order harmonic generation in hexagonalnanoribbons
Christoph J¨urß a and Dieter Bauer b Institute of Physics, University of Rostock, 18051 Rostock, Germany
Abstract.
The generation of high-order harmonics in finite, hexagonalnanoribbons is simulated. Ribbons with armchair and zig-zag edgesare investigated by using a tight-binding approach with only nearestneighbor hopping. By turning an alternating on-site potential off or on,the system describes for example graphene or hexagonal boron nitride,respectively. The incoming laser pulse is linearly polarized along theribbons. The emitted light has a polarization component parallel tothe polarization of the incoming field. The presence or absence of a po-larization component perpendicular to the polarization of the incomingfield can be explained by the symmetry of the ribbons. Characteristicfeatures in the harmonic spectra for the finite ribbons are analyzed withthe help of the band structure for the corresponding periodic systems.
Ultrafast dynamics in condensed matter systems have been studied intensively inrecent years [1–8]. In particular, high-order harmonic generation (HHG) has provento be a powerful tool as it is able to probe static and dynamic properties of the solidtarget by all-optical means [9–13].HHG was initially observed for atoms and molecules in the gas phase. For non-perturbative laser intensities and photon energies well below the ionization potential,the energy of the emitted photons can be large multiples of the incident photon’senergy, the high-order harmonics.The mechanisms underlying HHG in solids are similar to those in the gas phase.For instance, the celebrated semi-classical three-step model [14, 15] introduced forisolated atoms, where, in the first step, the electron is excited into the continuum. Inthe second step, the electron propagates in the presence of the laser-field, and, in thethird step, it recombines with the ion upon generating a photon with an energy givenby the kinetic energy of the electron at the time of recombination and the ionizationpotential. A similar model exists for solids [16] where, first, the electron is excitedfrom the valence band to the conduction band, second, the electron in the conductionband and the hole in the valence band propagate in the presence of the laser field,and, third, the electron and hole recombine upon generating a harmonic photon.If the solid is an insulator or semi-conductor, it has a non-vanishing band gapbetween the valence and the conduction band. Harmonics explained by the three- a e-mail: [email protected] b e-mail: [email protected] a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Will be inserted by the editor step model are due to transitions between valence and conduction band. Hence, theyare called interband harmonics. Their minimal energy has to be larger than the bandgap. Harmonics smaller than the band gap are generated by intraband harmonics,i.e., by the movement of the electron (and hole) within bands.Many studies focus on the bulk of a solid. In reality, solids are finite and have edges.Edge states might cause interesting effects in high-harmonic spectra, in particularwhen they are topological in nature [17–19]. In this paper, we focus on the high-harmonic spectra from finite systems and compare with the corresponding result forthe bulk. We restrict ourselves to the topologically trivial phase in this work.Graphene is one particularly interesting two-dimensional solid because of its rela-tivistic Dirac cones. In graphene, the atoms form a hexagonal lattice structure. Hexag-onal boron nitride (h-BN) is a different example with the same lattice structure. HHGin hexagonal lattice structures has been studied for the bulk and for ribbons for topo-logically trivial graphene and h-BN [20–23] and the topologically nontrivial Haldanemodel [24–26].In this work, we investigate the generation of high-harmonics in hexagonal nanorib-bons for two different edge configurations: zig-zag and armchair. Ribbons with andwithout alternating on-site potentials are investigated in the topologically trivialphase only. The system without alternating on-site potential contains one atomicelement (as, e.g., in graphene) whereas two different elements are contained in thecase of alternating on-site potential (e.g., h-BN).The outline of this work is as follows. In Chapter 2, the basic theory is summa-rized, starting with the finite system without external field in Sec. 2.1. We show thecalculation and the results for the band structures for the armchair (Sec. 2.1.1) andthe zig-zag (Sec. 2.1.2) ribbon with periodic boundary conditions. In Sec. 2.2 thecoupling to an external field is presented. HHG spectra for the finite armchair (Sec.3.1) and finite zig-zag (Sec. 3.2) ribbons are discussed.
In this work, we investigate hexagonal ribbons in two different configurations, arm-chair (Fig. 1 (a)) and zig-zag (Fig. 1 (b)). We consider two different types of sites: Aand B. The on-site potential is M ( − M ) on lattice sites A (B). In Fig. 1, the latticesites A and B are indicated by unfilled and filled circles, respectively. The lattice con-stant for the armchair ribbon is given by d a = 3 a , where a is the distance between twoneighboring sites. For the zig-zag ribbon, the lattice constant is d zz = √ a . Atomicunits (a.u.), ¯ h = | e | = m e = 4 π(cid:15) = 1, are used if not stated otherwise. The systems have N atomic sites. The atomic orbital at site i is denoted as | i (cid:105) . Ageneral single-electron wavefunction is given by | ψ (cid:105) = N (cid:88) i =1 g i | i (cid:105) . (1)The Hamiltonian in position space and tight-binding approximation readsˆ H = t (cid:88) ( | j (cid:105) (cid:104) i | + h . c . ) + M (cid:32)(cid:88) i ∈ A | i (cid:105) (cid:104) i | − (cid:88) i ∈ B | i (cid:105) (cid:104) i | (cid:33) (2) ill be inserted by the editor 3 Fig. 1.
Sketch of ribbons built from hexagons, for finite (a,b) and infinite (c,d) ribbons. (a,c)armchair and (b,d) zig-zag configuration. The distance between nearest neighbors (indicatedby solid lines) is a , the hopping amplitude between them is t ∈ R . An alternating on-sitepotential M ( − M ) at sites A (B), indicated by unfilled (filled) circles, is included. The unitcells are marked by dotted rectangles. The lattice constant is d a for the armchair and d zz forthe zig-zag ribbon. Here, the finite armchair ribbon contains N hex = 4, the zig-zag N hex = 6hexagons (unit cells). For the infinte ribbon (c,d), the hoppings inside a unit cell m = n andto neighboring unit cells m = n ± α (red). where the sum (cid:80) runs over nearest neighbors i and j and the sums (cid:80) i ∈ A,B over sites A or B , respectively. The parameter t is the hopping amplitude betweenadjacent sites. Hopping between next-nearest neighbors is not considered in this work.The eigenstates | ψ i (cid:105) fulfill the time-independent Schr¨odinger equation (TISE)ˆ H | ψ i (cid:105) = E i | ψ i (cid:105) . (3)In the following, we describe the propagation of states | ψ (cid:105) in position space forribbons with N hex hexagons (Fig. 1 (a,b)) in time. However, one aim of this work is torelate features in the harmonic spectrum of the finite ribbons with energy differencesin the band structure of the corresponding ribbon bulk. Hence, band structures arecalculated for the ribbons with periodic boundary conditions in x -direction, see Fig. 1(c,d). The resulting Hamiltonian will be given in crystal-momentum space ( k -space).For the distance between adjacent sites we take the value for graphene [27], i.e., a = 2 .
68 a.u. (cid:39) .
42 ˚A. For the nearest-neighbor hopping amplitude we use data fromsimulations without tight-binding approximation [28] with which we want to compare.To that end the energies of an armchair ribbon with four unit cells were calculated,and the nearest-neighbor hopping was adjusted till the band gap was identical forboth methods. As a result we find t = − . ≈ − .
116 eV. The negative sign of t is chosen to obey the node rule of quantum mechanics, as in this case the coefficients g i have all the same sign for the state with the lowest energy. The calculation leading to the bulk-Hamiltonian for the armchair ribbon is in Ap-pendix A. The unit cell contains six sites, see Fig. 1 (c). Hence, there are six bands,see Fig. 2 (a,b). The energies are calculated numerically.The armchair ribbon has a band gap of ∆E gap = 0 . M = 0 (Fig. 2 (a)). The band gap ∆E gap increases with the on-sitepotential M (Fig. 2 (b)). The band structure is symmetric around E = 0. Two of thebands are flat, their energy is constant over the whole Brillouin zone. It is given by E flat = ± (cid:112) t + M , see appendix A. Will be inserted by the editor c d
Fig. 2.
Band structure for ribbon with armchair (a,b) and zig-zag (c,d) edges for M = 0(a,c) and M = 0 .
12 (b,d) in the first Brillouin zone.
In the finite ribbon with N hex hexagons, there are also N hex states that have thesame energy. Their energy is identical to the energy of the flat bands in the bulk. The bulk-Hamiltonian for the zig-zag ribbon was calculated in [26]. Here, we do notconsider hopping between next-nearest neighbors as in [26] (i.e., t = 0). The bulk-Hamiltonian reads ˆ H bulk , zz ( k ) = M T ( k ) 0 0 T ( k i ) − M t t M T ( k )0 0 T ( k ) − M (4)with T = 2 t cos( k i d zz / H bulk , zz u ( k ) = E ( k ) u ( k ) (5)with the periodic factor u ( k ) = ( u ( k ) , u ( k ) , u ( k ) , u ( k )) (cid:62) in the Bloch-like ansatz.Other than for the armchair ribbon, the energies of the zig-zag ribbon can bewritten in a compact, analytical form E ( k ) = ± (cid:114) M + t / (cid:16)(cid:112)
16 cos ( k d zz /
2) + 1 ± (cid:17) , (6)where both ± are independent, leading to four bands. Results with and without M are shown in Fig. 2 (c,d). For a vanishing on-site potential (c) there is no band gapbetween the bands with a negative energy (valance bands) and the bands with apositive energy (conduction bands). However, only transitions between the two black,solid or the two red, dashed bands are allowed for a linearly polarized laser field indipole approximation [29, 30]. This creates an effective band gap, ∆E gap , .A band gap centered at E = 0 appears for non-vanishing on-site potential. It isgiven by ∆E gap = 2 | M | , an example is shown in Fig. 2 (d). Transitions between allbands are allowed for M (cid:54) = 0. The coupling of the systems to an external field and the propagation of an electronicwavefunction in time is described in Ref. [26]. ill be inserted by the editor 5
The vector potential is linearly polarized along the x -direction (i.e., along theribbons). For times 0 ≤ t ≤ πn cyc /ω , the vector potential is given by A ( t ) = A sin (cid:18) ω t n cyc (cid:19) sin( ω t ) e x , (7)and it is zero otherwise. The following laser parameters are used if not stated oth-erwise: amplitude A = 0 . (cid:39) . × Wcm − ), angular frequency ω = 7 . · − (i.e., wavelength λ = 6 . µ m), and the pulse comprises n cyc = 4cycles.The total current is given by J ( t ) = (cid:88) l (cid:104) Ψ l ( t ) | ˆ j ( t ) | Ψ l ( t ) (cid:105) , (8)i.e., the sum over all currents arising from the occupied states | Ψ l ( t ) (cid:105) , propagated intime. It is assumed that at the beginning of the pulse, all eigenstates with negativeenergy (i.e., below the Fermi level) are occupied.The intensity of the emitted light is proportional to (cid:12)(cid:12) P (cid:107) , ⊥ ( ω ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) FFT (cid:104) ˙ J x,y ( t ) (cid:105)(cid:12)(cid:12)(cid:12) . (9)The symbols (cid:107) and ⊥ denote the parallel ( x -direction) and perpendicular ( y -direction)polarization direction, respectively. In this paper, we discuss the high-order harmonic spectra for an armchair ribbonconsisting of N hex = 4 hexagons ( N = 24) and a zig-zag ribbon built of N hex = 6hexagons ( N = 26). The results are compared with simulations without the tight-binding approximation for systems of the same size [28]. We briefly discuss the size-dependence of the zig-zag ribbon at the end of this Section. The high-order harmonic spectra for parallel polarization direction as function ofthe on-site potential for the armchair ribbon are shown in Fig. 3 (a). In addition,the spectra for M = 0, M = 0 .
05, and M = 0 .
12 are shown in Fig. 3 (b). Theenergy is given in units of the laser frequency ω = 0 . y -direction) even with a non-vanishing on-site potential(see Fig. 1 (a)). In both plots, the minimal band gaps of the periodic system betweenvalence and conduction band ∆E gap are indicated. The band gap increases with theon-site potential M . The line ∆E max (Fig. 3 (a)) shows the maximal energy differencebetween valence and conduction band. It also increases with M . The horizontal linesin Fig. 3 (a) mark those M s for which spectra are shown in Fig. 3 (b).The band gap of the periodic system with vanishing on-site potential is E gap =0 . ≈ . ω . For the finite system with N hex = 4 one finds a band gap of0 . ≈ . ω . The band gap of the finite system is larger because there areonly 24 eigenstates of the Hamiltonian. Due to the restricted number of states, the Will be inserted by the editor Δ E gap Δ E gap Δ E gap (a) (b) Fig. 3. (a) High-order harmonic spectra for a finite armchair ribbon with N hex = 4 hexagonsas function of the on-site potential M . The spectra show the emission polarized parallel tothe polarization direction of the incoming laser field. The line representing ∆E gap ( ∆E max )indicates the minimal band gap (maximal energy difference) between the valence and conduc-tion bands of the respective periodic system. The horizontal lines mark the on-site potentialof the spectra shown in (b). sampling of the energy spectrum is not sufficient to capture the minimal band gap ofthe periodic ribbon. A larger finite chain would resolve it but is not part of this study.We refer to Ref. [31], where the size dependency of a one-dimensional, linear chainwas studied. For an on-site potential of M = 0 .
12, the band gap of the finite system isgiven by 0 . ≈ . ω and that of the periodic system is E gap = 0 .
248 = 33 . ω .Here, the band gap of the finite and the band gap of the periodic system are similar.In the harmonic spectra, one can see that the harmonic yield for small energiesdrops exponentially. Up to the energy of the band gap, the harmonic yield is relativelylow. For energies larger than the band gap, one observes a plateau where the yieldis almost constant. An ultimate cut-off is observed at an energy that correspondsto the maximum energy difference between valence and conduction bands ∆E max .Transitions with larger energies are not possible in the tight-binding model. Hence,no harmonics are emitted at larger energies.The harmonics below the band gap are due to movement of electrons inside thebands, known as intraband harmonics. Due to the fully occupied valence bands, thereare electrons that move in opposite directions because of opposite band curvature,and therefore the emitted radiation by ”individual” electrons destructively interferes,leading to the drop in the harmonic yield [17]. The plateau for larger energies is dueto transitions between the valence and conduction bands, called interband harmonics.The band gap increases with the on-site potential M . As a consequence, the smallestenergy of the plateau-region shifts to higher harmonic orders. The ultimate cut-off ofthe plateau also increases, because the maximal energy difference of the bands alsobecomes larger with increasing M . The onset and the cut-off of the plateau can beestimated by the periodic system. Its minimal band gap E gap and maximal energydifference E max is plotted in Fig. 3 (a). The colored-contour plot in Fig. 3 (a) mightnot be able to show the starting and beginning of the plateau properly. It is bettervisible in Fig. 3 (b). The colored vertical arrows indicate the band gap of the respectiveperiodic system. It agrees with the onset of the plateau if the on-site potential M isnon-zero. The cut-off of the plateau for M = 0 is at around harmonic order 50 andfor M = 0 .
12 at around 60 harmonic orders. The maximal energy difference of theperiodic system is given by ∆E max = 0 . ≈ . ω and ∆E max = 0 . ≈ . ω for M = 0 and M = 0 .
12, respectively, agreeing with the cut-offs. The plateau isrestricted to the region between E gap and E max , as expected.The flat bands of the band structure (see Fig. 2 (a,b)) are separated by an energy of ∆E flat = 2 | t | = 0 . ≈ . ω for M = 0. The harmonic spectrum shows a peakat this energy. With increasing on-site potential M , the energy difference between ill be inserted by the editor 7 those bands increases, indicated by the dotted line ∆E flat in Fig. 3 (a). There are asmany states with the same energy inside the flat bands as there are hexagons in theribbon. Therefore, many possible transitions have the same transition energy. Thislarge number of transitions with identical energies causes the peak in the spectrum.The results presented so far are qualitatively the same as the results in Ref. [28], inwhich no tight-binding approximation is used. The system with an on-site potential of M = 0 .
12 has approximitly the same band gap as the system with an on-site potentialof V os = 0 . ∆E max . These harmonics are absentwhen using the tight-binding approximation. However their yield is several orders ofmagnitude below the yield of the smallest interband harmonic orders.Note that the hopping parameter t is chosen in order to fit the band gap of thesystems for both methods. However, the maximal energy difference E max is different.One reason for that is the symmetry of the tight-binding bulk Hamiltonian thatenforces mirror-symmetric valence and conduction bands about the zero-energy axis.This symmetry is absent in the continuous description of Ref. [28]. Harmonic spectra for the finite zig-zag ribbon with N hex = 6 hexagons as functionof M are shown in Fig. 4 in parallel (4 (a)) and perpendicular (4 (b)) polarizationdirection with respect to the polarization of the incoming laser field. In addition,Fig. 5 shows spectra for three different on-site potentials M (Fig. 5 (a) in paralleland Fig. 5 (b) in perpendicular polarization direction). The corresponding M valuesare indicated by horizontal lines in Fig. 4. The marked energies ∆E gap and ∆E max indicate the minimal band gap and the maximal energy difference between valenceand conduction band of the periodic system, respectively.In both figures, one can see that without on-site potential, the zig-zag ribbondoes not emit light perpendicular to the polarization direction of the incoming laserfield (i.e., the y -direction). As the on-site potential becomes finite, the symmetryof the system in y -direction is broken. This can be seen in Fig. 1 (b): the on-sitepotential at the lowest sites ( α = 1) is M , on the topmost sites ( α = 4) it is − M . Asa consequence, the electrons are attracted more towards the upper sites than to thelower sites. Hence, light polarized in y -direction is now also emitted, i.e., perpendicularto the polarization of the incoming laser field, see Figs. 4 (b) and 5 (b).The band gap increases almost linearly with the on-site potential for larger M and vanishes for M = 0. The spectra without on-site potential show an exponentialdecrease of the harmonic yield. A typical drop similar to the one for the armchairribbon can be observed below harmonic order 11, best seen in Fig. 5 (a). This dropof the harmonic yield is explainable by the destructive interference of the intrabandemission. One could expect that the interband harmonics should compensate the dropin the yield because of the vanishing band gap. However, as was shown in Refs. [29,30],transitions between certain bands are forbidden in the graphene zig-zag ribbon. Thisfact was already indicated in Fig. 2 (c): transitions are only allowed between the lowestvalence and the lowest conduction band (black, solid lines) and between the highestvalence and conduction band (red, dashed lines). Therefore, the effective minimalband gap of the periodic system is given by ∆E gap , = | t | = 0 . ≈ . ω . Thisis in good agreement with the onset of the plateau. The maximal energy differencebetween the bands where transitions are allowed is given by ∆E max , = 0 . ≈ . ω , which agrees well with the cut-off.Further, as the band gap increases, we can see the typical drop of the harmonicyield for energies below ∆E gap . The plateau lies in an energy region between ∆E gap Will be inserted by the editor
Fig. 4.
Harmonic spectra for the finite zig-zag ribbon containing N hex = 6 hexagons asfunction of the on-site potential M . Spectra in parallel (a) and perpendicular (b) polarizationdirection are shown. The line ∆E gap indicates the minimal band gap, the line ∆E max themaximal gap between valence and conduction band as function of M for the periodic system. Δ E gap ,0 Δ E gap Δ E gap Fig. 5.
High-order harmonic spectra of a finite zig-zag ribbon with N hex = 6 hexagonsfor different on-site potentials M for parallel (a) and perpendicular (b) polarization. Thevertical arrows mark the energy of the band gap for the respective periodic system. and ∆E max for both polarization directions. This shows that the overall qualitativefeatures in the harmonic spectra of this small zig-zag nanoribbon can be alreadyunderstood with the help of the band structure of the periodic system. The results aresimilar to simulations without tight-binding approximation [28]. The only differenceis the presence of harmonics above ∆E max without tight-binding approximation dueto higher lying states, similar to the armchair ribbon.As the on-site potential increases, the selection rule for M = 0 does not applyanymore due to the broken symmetry in y -direction. However, we observe for thesmall system with N hex = 6 and a small on-site potential that the spectra still show adrop in the harmonic yield for small energies, indicating the destructive interferenceof the intraband harmonics and an onset of the interband plateau only at higherharmonic order than expected from ∆E gap . Hence, transitions between the highestvalence band and the lowest conduction band are still very unlikely in such a small,finite zig-zag ribbon, otherwise the interband harmonics would fill up the drop of theintraband harmonic yield.The approximation of a finite system by a periodic system fits better for largerfinite systems. In Fig. 6, the harmonic spectrum in parallel polarization direction of afinite chain containing N hex = 15 hexagons is shown. For a vanishing on-site potential M = 0 .
0, the drop in the harmonic yield up to order 11 can be observed.For a small on-site potential of M = 0 .
001 the harmonic yield below the bandgap is increased by several orders. Still, both valence bands are fully occupied, whichmeans that the intraband harmonics interfere destructively as for M = 0 . ill be inserted by the editor 9 | P () | M = 0.0 M = 0.001 Fig. 6.
High-harmonic spectrum in parallel polarization direction for a zig-zag ribbon with N hex = 15 hexagons for on-site potential M = 0 and M = 0 . A = 0 . between all bands are allowed, showing that for longer, finite zig-zag ribbons the M = 0 selection rule breaks more abruptly for slightly non-vanishing M than forshorter ribbons.We note that in the calculation for N hex = 15 we chose the amplitude of the vectorpotential A = 0 .
05 (i.e., intensity (cid:39) . × Wcm − ). For the same intensity asbefore ( A = 0 .
2) the yield drop for M = 0 is not clearly visible. With increasinglaser intensity the excursion of the electrons along the crystal momentum increases,diminishing the destructive interference of the intraband emission. As a result, thedrop in the harmonic yield is not as pronounced as for smaller intensities. The factthat in the small ribbon with N hex = 6 the drop is more pronounced might be due tothe smaller number of states that do not resemble well continuous bands. In this work, we simulated high-harmonic generation in finite hexagonal nanoribbonswith armchair and zig-zag edges. In an intense laser field polarized linearly along theribbon, the armchair ribbon emits linearly polarized light parallel to the polarizationof the incoming field. The zig-zag ribbon emits light parallel and perpendicular to thepolarization of the incoming laser pulse if an alternating on-site potential is included.Both ribbons show a suppressed harmonic yield for energies below the band gap. Theband gap itself is determined by the on-site potential. Characteristic features in theharmonic spectra, such as onset and cut-off of the interband-harmonics plateau, can beunderstood with the help of the band structure for the corresponding periodic systems.The results for the finite ribbons are similar to those from simulations without thetight-binding approximation.
Acknowledgment
C.J. acknowledges financial support by the doctoral fellowship program of the Uni-versity of Rostock.
A Derivation of armchair bulk-Hamiltonian
For an armchair ribbon with N unit cells and periodic boundary conditions, theHamiltonian readsˆ H arm0 = N (cid:88) m =1 (cid:34) (cid:88) α =1 (cid:18) t | m, α (cid:105) (cid:104) m, ( α + 1) mod 6 | + ( − α +1 M | m, α (cid:105) (cid:104) m, α | (cid:19) + t | m, (cid:105) (cid:104) m + 1 , | ] + h . c ., (10)where we now write the state | i (cid:105) at site i as | m, α (cid:105) where α indicates the site withinunit cell m . In order to obtain the bulk-Hamiltonian, we make a Bloch-like ansatz,taking the relative position within a unit cell into account, | ψ ( k ) (cid:105) = 1 √ N N (cid:88) m =1 e i mkd a | m (cid:105) ⊗ (cid:16) u ( k )e i kd a / | (cid:105) + u ( k ) | (cid:105) + u ( k )e i kd a / | (cid:105) + u ( k )e i kd a / | (cid:105) + u ( k )e kd a / | (cid:105) + u ( k )e i kd a / | (cid:105) (cid:17) . (11)Here, d a = 3 a is the lattice constant for the armchair ribbon. We plug this ansatz intothe time-independent Schr¨odinger equation, ˆ H | ψ ( k ) (cid:105) = E ( k ) | ψ ( k ) (cid:105) , and multiplyby √ N e − i m (cid:48) kd a (cid:104) m (cid:48) | from the left, leading toˆ H bulk , arm u ( k ) = E ( k ) u ( k ) , (12)withˆ H bulk , arm = M t e − i kd a / t e i kd a / t e i kd a / − M t e i kd a / t e − i kd a / t e − i kd a / M t e i kd a / t e − i kd a / − M t e i kd a / t e i kd a / t e − i kd a / M t e − i kd a / t e − i kd a / t e i kd a / − M , (13)where u ( k ) = ( u ( k ) , u ( k ) , u ( k ) , u ( k ) , u ( k ) , u ( k )). For given k , one obtains sixeigenstates u j ( k ) and energies E j ( k ) ( j = 1 , , . . . , k . However, we show the calculation of one band analytically. For the periodicpart of the Bloch-state we make the ansatz u flat ( k ) = ( β, , − β, − γ, , γ ) (cid:62) , (14)where the values of β and γ are unknown and k -dependent. Note that this state iszero at the connection points α = 2 and 5 to the neighboring hexagons (see Fig. 1(c)).We obtainˆ H bulk , arm u flat ( k ) = M β + t e i kd a / γ − M β − t e i kd a / γM γ − t e − i kd a / β − M γ + t e − i kd a / β ! = E flat ( β, , − β, − γ, , γ ) (cid:62) . (15) ill be inserted by the editor 11 This relation holds if
M β + t e i kd a / γ = E flat β and M γ − t e − i kd a / β = E flat γ, (16)resulting in the energy E flat = ± (cid:113) t + M . (17)As this energy is independent of k , the corresponding two bands are flat.One can also show that the same energies are obtained for a finite armchair ribboncontaining N hex hexagons. The ansatz is that the state is zero everywhere except atone hexagon, where it is given by eq. (14). The same energy (17) is obtained. Thedegeneracy is given by the number of hexagons in the ribbon N hex . References
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