Electronic spectrum of Kekule patterned graphene considering second neighbor-interactions
EElectronic spectrum of Kekul´e patterned graphene consideringsecond neighbor-interactions
El´ıas Andrade, Gerardo G. Naumis, and R. Carrillo-Bastos Departamento de Sistemas Complejos, Instituto de Fisica, Universidad NacionalAut´onoma de M´exico, Apartado Postal 20-364,01000, Ciudad de M´exico, M´exico. Facultad de Ciencias, Universidad Aut´onoma de Baja California,Apartado Postal 1880, 22800 Ensenada, Baja California, M´exico (Dated: January 25, 2021)
Abstract
The effects of second-neighbor interactions in Kekul´e patterned graphene electronic properties arestudied starting from a tight-binding Hamiltonian. Thereafter, a low-energy effective Hamiltonianis obtained by projecting the high energy bands at the Γ point into the subspace defined by theKekul´e wave vector. The spectrum of the low energy Hamiltonian is in excellent agreement withthe one obtained from a numerical diagonalization of the full tight-binding Hamiltonian. Themain effect of the second-neighbour interaction is that a set of bands gains an effective mass anda shift in energy, thus lifting the degeneracy of the conduction bands at the Dirac point. Thisband structure is akin to a “spin-one Dirac cone”, a result expected for honeycomb lattices witha distinction between one third of the atoms in one sublattice. Finally, we present a study ofKekul´e patterned graphene nanoribbons. This shows that the previous effects are enhanced as thewidth decreases. Moreover, edge states become dispersive, as expected due to second neighborsinteraction, but here the Kek-Y bond texture results in an hybridization of both edge states. Thepresent study shows the importance of second neighbors in realistic models of Kekul´e patternedgraphene, specially at surfaces. a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n . INTODUCTION The space-modulation of two-dimensional materials has opened avenues for new excit-ing physical phenomena and applications . Several mechanisms allow to perform suchmodulations, these include interactions with substrates as in moire patterns , strain ,adatoms , magnetic fields , and time dependent electromagnetic fields .Among such modulated systems, in graphene it has been experimentally observed that va-cancies in a Cu substrate induce a spatial frequency modulation with the size of an hexagonalring of carbon atoms . This modulated system is known as kekul´e-distorted graphene .As an example of its interest and importance, such kekul´e distortion has been proposed as apossible mechanism behind superconductivity in magic-angle twisted bilayer graphene .Also, strain in kekul´e distorted graphene can be used to perform valleytronics , a featurethat recently has been experimentally confirmed . Multiflavor Dirac fermions were pre-dicted to emerge in kekul´e graphene bilayers , and it is even possible to produce such mod-ulation in non-atomic systems, as with mechanical waves in solids and acoustical lattices .Kekul´e modulations are also reacheable via photonic , polaronic and atomic systems .Gamayun et. al demostrated the absence of a gap for a Kek-Y distortion and deducedthe low energy Hamiltonian for Kekul´e distortions by using a first-neighbor tight-bindingHamiltonian . They show how the two Dirac cones merge at the center of the Brillouinzone, producing either a gap (Kek-O) or the superposition of two cones with different Fermivelocities (Kek-Y) .Several works have been made using such first-neighbor tight-binding Hamiltonian, forexample to study uniaxial strain and the electronic transport properties .However, its is known that second-neighbors interactions in graphene are very important .They are fundamental to explain the electronic properties at graphene surface as in graphenenanoribbons . This leads to the natural question of what are the effects of second neigh-bors interactions in a Kekulk´e patterns. Although Density Functional Calculations alreadycontains such effects, due to the involved energies and the low resolution of the mesh calcula-tions near the Dirac cones, its is difficult to assert a detailed picture of the energy dispersion.2 x a) δ δ δ b) c) FIG. 1. Kek-Y distorted graphene lattice. The A and B sublattices are denoted by the black andwhite circles respectively. a) Deformed lattice, not in scale for illustration purposes. b) Nearestneighbors hoppings. The bold black lines indicate a stronger bond than the gray ones. c) Next-nearest neighbors interactions scheme. They form two independent triangular lattices for eachsublattice, here shown in blue for the one corresponding to the A sublattice. These A-A bondsremain unaltered by the Kek-Y pattern. For the B sublattice, a B-B bond texture is induced insuch a way that the bonds near the Y deformation (bold red lines) are stronger than the others(pink lines).
In that sense, a tight-binding calculation can be very useful. Here we tackle this questionby producing a low-energy Hamiltonian for a Kekul´e patterned graphene which includessecond-neighbor interactions. The resulting model is validated through a comparison withthe numerical calculations.The paper is organized as follows. In Sec. I we introduce the Hamiltonian for a honeycomblattice with a Kek-Y distortion up to next-nearest neighbors, and in Sec. I we calculate aneffective four band low-energy Hamiltonian, finally in II we present our conclusions andremarks.
FIRST AND SECOND NEIGHBOR KEKUL´E-Y GRAPHENE HAMILTONIAN
We can consider the Kek-Y bond modulation as a periodic strain which reduces thedistance between one third of the atoms in one sublattice with its three nearest neighbors as3hown in Fig. 1a. Thus the hopping integral gets modified by the change in the interatomicdistances accordingly with strain theory . However, this would also mean a change in thenext-nearest neighbors hoppings. In Fig. 1c we illustrate this point, by showing in red (pink)the stronger (weaker) bonds for the B sublattice. In blue we show the hoppings between theA sublattice atoms, which remain unaltered.Assuming the Gruneissen parameter to be equal for both first and second neighbors wecan consider that the bond changes with the same proportionality. Thus the Hamiltonianfor graphene with Kek-Y distortion considering hopping up to next-nearest neighbors is, H = − (cid:88) r (cid:88) j =1 t (0) r a † r b r + δ j + h.c. + (cid:88) r (cid:88) m (cid:54) = n t a † r a r + δ m − δ n + (cid:88) r (cid:88) m (cid:54) = n t (2) r b † r + δ m b r + δ n , (1)where r is the position vector that runs over the atomic positions of sites in sublattice A andits given by r = n a + n a where n , n are integers, a = a (cid:16) − √ , (cid:17) , a = a (cid:16) √ , (cid:17) are the lattice vectors and a is the distance between carbon atoms 1 .
42 ˚A. The vectors δ = a (cid:16) √ , − (cid:17) , δ = a (cid:16) − √ , − (cid:17) and δ = a (0 ,
1) go from a site in the A sublatticetowards its three nearest neighbors in the B sublattice as shown in Fig. 1. The spacedependent hopping parameters t (0) r and t (2) r between nearest neighbors and next-nearestneighbors respectively, are periodically modulated as t (0) r /t = t (2) r /t = 1 + 2∆cos( G · r ) , (2)here t = 2 . t for second neighbors which will be taken as t = 0 . t unless otherwise indicated. ∆ is theKekul´e coupling amplitude and G = π √ ,
0) is the Kekul´e wave vector. The Fouriertransform of our tight-binding Hamiltonian is then, H ( k ) = − (cid:15) ( k ) a † k b k − ∆ (cid:15) ( k + G ) a † k + G b k − ∆ (cid:15) ( k − G ) a † k − G b k + h.c. + f ( k ) a † k a k + f ( k ) b † k b k + ∆ f + ( k ) b † k + G b k + ∆ f − ( k ) b † k − G b k , (3)where (cid:15) ( k ) = t (cid:88) j =1 e i k · δ j , (4a) f ( k ) = t (cid:88) m (cid:54) = n e i k · ( δ m − δ n ) , (4b)4 ± ( k ) = t (cid:88) m (cid:54) = n e i k · ( δ m − δ n ) e ∓ i G · δ n , (4c)here (cid:15) ( k ) and f ( k ) are the dispersion relations for a honeycomb and a triangular latticerespectively. Some properties of f ± ( k ) are: f ± ∗ ( k ± G ) = f ∓ ( k ∓ G ) , f ∓ ∗ ( k ± G ) = f ± ( k ) . (5)By defining the column vector c k = ( a k , b k , a k + G , b k + G , a k − G , b k − G ) we can rewrite theHamiltonian as a 6 x 6 matrix: H ( k ) = c † k H Γ TT † H G c k , (6a)made from the original Γ point graphene Hamiltonian, H Γ = f ( k ) − (cid:15) ( k ) − (cid:15) ( k ) f ( k ) , (6b)the G and − G points Hamiltonian, H G = f ( k + G ) − (cid:15) ( k + G ) 0 − ∆ (cid:15) ( k − G ) − (cid:15) ∗ ( k + G ) f ( k + G ) − ∆ (cid:15) ∗ ( k + G ) ∆ f − ( k − G )0 − ∆ (cid:15) ( k + G ) f ( k − G ) − (cid:15) ( k − G ) − ∆ (cid:15) ∗ ( k − G ) ∆ f + ( k + G ) − (cid:15) ∗ ( k − G ) f ( k − G ) , (6c)and the interaction between them, T = − ∆ (cid:15) ( k + G ) 0 − ∆ (cid:15) ( k − G ) − ∆ (cid:15) ∗ ( k ) ∆ f − ( k + G ) − ∆ (cid:15) ∗ ( k ) ∆ f + ( k − G ) . (6d)In Fig. 2 we present the density of states (DOS) obtained from a numerical diagonaliza-tion of Eq. (6a). As expected, the main effect is the breaking of the electron-hole symmetryreflected in changes of the band widths. Also, around the Fermi energy there are changesthat we explore in the following section, as we will develop a low-energy approximation andcompare it with the numerical diagonalization of the Hamiltonian given in (6a).5 D ( E ) E-E [t ] t =0 t =0.1t FIG. 2. Comparison of the DOS for Kekul´e patterned graphene with and without the second-neighbor interaction, where the zero energy corresponds to the Fermi energy. Notice the electron-hole asymmetry which is specially clear for the band widths.
LOW-ENERGY HAMILTONIAN
Let us now build a low energy Hamiltonian starting with the full 6 × k the functions that appear in Eq. (6a). We obtain the following results, (cid:15) ( k ) ≈ t , (cid:15) ( k ± G ) ≈ t ( ∓ k x + ik y ) , (7a) f ( k ) ≈ t , f ( k ± G ) ≈ − t , (7b) f ± ( k ± G ) ≈ , f ± ( k ∓ G ) ≈ t ( ± k x + ik y ) , f ± ( k ) ≈ t ( ∓ k x − ik y ) . (7c)6sing the previous approximations, the linearized components of the Hamiltonian Eq. (6a)are given by, H Γ ≈ t − t − t t , (8a) H G ≈ − t v F (cid:126) ( k x − ik y ) 0 − ∆ v F (cid:126) ( k x + ik y ) v F (cid:126) ( k x + ik y ) − t ∆ v F (cid:126) ( k x + ik y ) 00 ∆ v F (cid:126) ( k x − ik y ) − t − v F (cid:126) ( k x + ik y ) − ∆ v F (cid:126) ( k x − ik y )) 0 − v F (cid:126) ( k x − ik y ) − t , (8b) T ≈ v F (cid:126) ( k x − ik y ) 0 − ∆ v F (cid:126) ( k x + ik y ) − t − ∆ v (cid:126) ( k x − ik y ) − t ∆ v (cid:126) ( k x + ik y ) , (8c)where we defined two velocities, one is the usual Fermi velocity in pristine graphene, v F = 3 at (cid:126) (9)and the other is due to second neighbors, v = 9 at (cid:126) = 3 (cid:18) t t (cid:19) v F . (10)As we are interested in the spectrum at low energies, the relevant part is the one associatedwith the valleys K and K (cid:48) which corresponds to the block matrix H G , and then we canadd the effects from higher energy bands in the Γ point as a perturbation. We have verifiedthat this is an essential step in order to recover the spectrum obtained from a direct diago-nalization. We can obtain the effective Hamiltonian by projecting H Γ into the subspace of H G . To do this, consider the Schrodinger equation applied to Eq. (6a), H Γ Ψ Γ + T Ψ G = E Ψ Γ (11)and T † Ψ Γ + H G Ψ G = E Ψ G (12)where Ψ Γ and Ψ G are the components of the solution Ψ = (Ψ Γ , Ψ G ) on each subspace.From the first equation we can obtain Ψ Γ and use it on the second to obtain an effectiveHamiltonian for the Ψ G component resulting in an effective Hamiltonian, H Eff = H G + T † ( E − H Γ ) − T (13)7his Hamiltonian is exact but needs a self-consistent procedure to find E . However, if weexpand the term ( E − H Γ ) − and keep the first order term. we can make the approximation E ≈ E = − t which is the original energy dispersion in the Γ point. Now we write theDirac-like equation for this system, H Ψ K (cid:48) Ψ K = E Ψ K (cid:48) Ψ K , (14a)Ψ K (cid:48) = − ψ B, K (cid:48) ψ A, K (cid:48) , Ψ K = ψ A, K ψ K, K , (14b)where the explicit form of the low-energy Hamiltonian is finally given by, H = E v F (1 − ∆ )( p x − ip y ) v F ∆(1 − ∆)( p x − ip y ) 0 v F (1 − ∆ )( p x + ip y ) E + µ µ v F ∆(1 − ∆)( p x − ip y ) v F ∆(1 − ∆ )( p x + ip y ) µ E + µ v F (1 − ∆ )( p x − ip y )0 v F ∆(1 − ∆ )( p x + ip y ) v F (1 − ∆ )( p x + ip y ) E , which can be compactly written as, H = E σ ⊗ τ + v f (1 − ∆ )( p · σ ) ⊗ τ + v f ∆(1 − ∆) σ ⊗ ( p · τ )+ µ σ ⊗ τ + σ x ⊗ τ x + σ y ⊗ τ y − σ z ⊗ τ z ) , (14c)with µ defined as, µ = 9∆ t t t − t . (15)The four low-energy bands are E ± D = E ± v D | p | , (16a) E ± M = E + µ ± (cid:113) v M p + µ , (16b)were we defined v D = v F (1 − ∆) and v M = v F (1 − ∆)(1 + 2∆).Therefore, there is a mix of two-flavor Fermion gases. One with and effective DiracHamiltonian, H D = E σ + v D p · σ , (17)8nd the other, H M = ( E + µ ) σ + v M p · σ + µ σ z . (18)In Fig. 3 we compare our results from Eq. (16) with the numerical diagonalization of thetight-binding Hamiltonian given by Eq. (6a). First we notice an excellent agreement withinthis regime. Without second neighbors interaction, the Kek-Y bond texture couples bothvalleys in the Γ point, resulting in two concentric cones with different velocities . Turningon the interaction gives rise to two main effects, a set of bands gains an effective mass anda shift in energy. This last effect results in a particle-hole symmetry breaking, lifting thedegeneracy of the conduction bands at the Dirac point, therefore only three bands intersect.This structure is akin to a “spin-one Dirac cone”, expected for a honeycomb lattices with adistinction between one third of the atoms in one sublattice . We can see that the effectof adding next-nearest-neighbors interaction is equivalent to that of an on-site potential µ on the atom at which the Y deformation is centered .From Eq. (16) we can easily calculate the density of states per unit cell. Consideringspin degeneracy, it is given by, D ( E ) = Aπ (cid:126) (cid:20) | E − E | v D + E − ( E + µ ) v M Θ( E − E − µ ) + ( E + µ ) − Ev M Θ( E − E ) (cid:21) , (19)where A = 9 √ a / E ) is the Heaviside function. Althoughthe density of states retains its linear behavior around the Dirac point, the massive bandsproduce a discontinuity shown in Fig. 4.Second neighbors hoppings are particularly important for graphene nanoribbons (GNR).We calculated numerically the band structure for zigzag edged GNR. In Fig. 5 our resultsare shown for different values of t and width W . Due to the change in the periodicityproduced by the Kekul´e texture, the unit cell size a z is three times bigger, thus a z = 3 √ a .We can see that edge states become dispersive, which is a well known effect of secondneighbors interaction , however the combination with the Kek-Y bond texture results inan hybridization of both edge states. The velocity induced in the edge states may indirectlyclose the small gap predicted for Kek-Y zigzag GNR .9 - - - - - - - - [ / a ] E [ t ] FIG. 3. Low-energy dispersion around the Γ point with ∆ = 0 . t /t = 0 .
1. The solid linesindicate our analytic results for both the conic (orange) and the bands with an effective mass (blue).The numerical tight-binding calculations obtained by a direct diagonalization of the Hamiltonianare represented by the dots.
II. CONCLUSIONS
The effects of second-neighbor interactions in Kekul´e patterned graphene were studiedstarting from a tight-binding Hamiltonian. From there, a low-energy effective Hamilto-nian was derived using a projection technique. This Hamiltonian was validated thorough acomparison with a numerical calculation obtained from the diagonalization of the full tight-binding Hamiltonian. We found that beyond the expected electron-hole symmetry breaking,the main effect of the second-neighbor interaction is that in one of the Dirac cones, theelectron becomes massive when compared with the calculation made considering only first-neighbour interaction. As a result, the density of states near the Fermi energy contains ajump in the otherwise linear behavior. Finally, we considered the effects of second-neighborinteractions in Kekul´e patterned graphene nanoribbons, as it is known that such effects are10 a) E [t ] k [1/a] b) D(E)
FIG. 4. a) A zoom of the energy dispersion around the Fermi energy of Kekul´e patterned grapheneincluding up to second-neighbor interactions obtained from the low-energy approximation. b) Thecorresponding density of states showing how the cone with an effective mass produces two jumpsin the DOS. essential to reproduce a minimally realistic behavior at the edges. As expected, the samemass effect is seen in the nanoribbons and in fact is amplified as the width is decreased.Thus, we expect that such second-neighbor effects to be important in the electronic andoptical properties of Kekul´e bond textures.We thank UNAM DGAPA project IN102620 and CONACYT project 1564464. E. An-drade thanks an schoolarship from CONACyT. Umesha Mogera and Giridhar U. Kulkarni. A new twist in graphene research: Twisted graphene.
Carbon , 156:470 – 487, 2020. Di Wu, Yi Pan, and Tai Min. Twistronics in graphene, from transfer assembly to epitaxy. .00.5 -0.75 0.00 0.75-0.50.00.5 -0.75 0.00 0.75 -0.75 0.00 0.75 d) c)b) a) e) k [ /a z ] E - E [t ] f) FIG. 5. Dispersion for zigzag graphene nanoribbons with Kekul´e-Y bond texture and hoppings upto next nearest neighbors. The amplitude of the bond texture is ∆ = 0 . t /t = 0 , . . W = 4 .
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