Physics Letters A 376 (2012) Contents lists available at SciVerse ScienceDirect. Physics Letters A.

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1 Physis Letters A Contents lists available at SiVerse SieneDiret Physis Letters A Edge states at the interfae between monolayer and bilayer graphene Zi-xiang Hu a,b,, Wenxin Ding a Department of Physis, ChongQing University, ChongQing, , China b Department of Eletrial Engineering, Prineton University, Prineton, NJ 08544, USA NHMFL and Department of Physis, Florida State University, Tallahassee, FL 32306, USA artile info abstrat Artile history: Reeived 31 August 2011 Reeived in revised form 22 November 2011 Aepted 22 November 2011 Available online 3 Deember 2011 Communiated by R. Wu Keywords: Edge state Hybrid Dira equation The eletroni properties for monolayer bilayer hybrid graphene with zigzag interfae are studied by both the Dira equation and numerial alulation in zero field and in a magneti field. Basially there are two types of zigzag interfae dependent on the way of lattie staking at the edge. Our study shows they have different loations of the loalized edge states. Aordingly, the energy-momentum dispersion and loal density of states behave quit differently along the interfae near the Fermi energy E F = Elsevier B.V. All rights reserved. 1. Introdution In reent years, the experimental aessibility of the single and multilayered graphene samples [1 5] has attrated onsiderable theoretial and experimental attentions due to its unusual eletroni struture desribed by the Dira equation, namely eletrons in monolayer graphene have linear dispersion thus behave like massless Dira fermions at the orners of the Brillouin zone BZ [6]. In the presene of magneti field perpendiular to the graphene plane, the system shows anomalous integer quantum Hall effet [7 10] IQHE whih is also different from that of the onventional two-dimensional eletron system in semiondutor heterostrutures. The Hall ondutivity in the IQHE of graphene shows plateaus at σ xy = 4N + 1/2e 2 /h, in whih the fator 4 omes from the four-fold valley and spin degeneray and the shift 1/2 reflets partile hole symmetry in the 0 th Landau level. Edge states in graphene have been the fous of muh theoretial study beause of the important role they play in the transport [11]. It is well known that there are two basi types of edges in graphene, namely, the armhair and zigzag edges. Some theoretial works [12 26] on the eletroni struture of finite-sized systems, either as moleules or as one-dimensional systems, have shown that zigzag edged graphene has loalized state near the Fermi energy, but those with armhair edge do not have suh state. Therefore, the transport properties are very different due to the existene of the edge state. On the other hand the hybrid * Corresponding author at: Department of Eletrial Engineering, Prineton University, Prineton, NJ 08544, USA. address: zihu@prineton.edu Z.-X. Hu. edge struture, omposed of partial monolayer and partial bilayer graphene, whih is quit general in reality, hasn t reeived so muh attention. Experimentally [28] the anomalous quantum osillations in magnetoondutane were observed due to the peuliar physis along the interfae. After the experimental study, we presented our studies of the eletroni properties of the hybrid interfae of graphene in the APS meeting [29]. Reently we notied there are some new experiments and orrelated theoretial studies [32,33] on this topi. Thus we deided to present our systematial study of the edge states of the hybrid interfae. In this work, we study the eletroni properties of the hybrid interfae via both tight-binding model and its effetive theory in the ontinuum limit the Dira equation. The edge states in graphene an be studied experimentally by using a loal probe suh as sanning tunneling mirosopy STM. The STM experiments measure the differential ondutane whih is proportional to the density of states. Numerially we an study the loal density of states LDOS whih is defined as [27]: Nr, ev = Ψ α r 2 δev E α, 1 where Ψ α r is the eigenfuntion with energy E α.theldosshows the strength of the loal eletroni density whih is related to the strength of the signal in STM experimental data. Our study of the hybrid edge graphene shows that there are always zero energy states loalized near the zigzag hybrid edge although the distribution of LDOS of these edge states strongly depends on the details of how the edge staks together. The dispersion urve around the Dira ones also shows different haraters for different edge arrangements either within or without a magneti field /$ see front matter 2011 Elsevier B.V. All rights reserved. doi: /j.physleta

2 Z.-X. Hu, W.X. Ding / Physis Letters A Fig. 1. Color online. The shemati pitures for two kinds of hybrid monolayer bilayer interfaes. The atoms of the extended bottom layer layer 1 are indiated by blak dots smaller while the green dots larger represent the top layer layer 2 whih terminates at the interfae. In the left plot, the top layer is ended withb 2 sites B-type while on the right it is ended with A 2 sites A-type. This Letter is arranged as follows: In Setion 2, wesetupthe model Hamiltonian in different geometries. The zero energy solutions in zero field and in magneti field are obtained by solving the Dira equations. The numerial results in a finite system are shown in Setions 3 and 4. Some disussions and onlusions are in Setion Model and different geometries We onsider a bilayer graphene with Bernal staking, as shown in Fig. 1; its tight-binding Hamiltonian an be written as: H = t 2 a i;m,n b i;m,n + b i;m 1,n + b i;m,n 1 i=1 m,n t a 2;m,n b 1;m,n + h.., 2 m,n where a i;m,n b i;m,n is the annihilation operator at position m, n in sublattie A i B i, and i = 1, 2, indiating the two different layers. The first term is the Hamiltonian within eah layer, and the seond term desribes the interlayer oupling in whih we only onsider the hopping between the two atoms staked right on top of eah other. Let us label the bottom extended layer as layer 1, the half-plane upper layer as layer 2. In this work we will only onsider the Bernal staking. One an expand the effetive Hamiltonian near the two Dira points K and K whih are time reversal symmetri partners. In momentum spae, the Hamiltonian near K an be written as: H = k Ψ k H k Ψ k, 3 where 0 v F k k 0 0 v H k = F k 0 t 0 k = 0 γ 0 v 0 t 0 v F k F, 0 γ 0 k v F k k 0 in whih k = k x + ik y, γ = t /v F, and Ψ k = a 1;k, b 1;k, a 2;k, b 2;k. For the other Dira point, as stated before, H K = H K.Hereweonly onsider the zigzag type interfae or edge to explore the loalized edge state. Without magneti field and the interfae, it is suffiient to disuss just one Dira one in the ontinuum model due to the symmetry. But with the interfae breaking the inversion symmetry and the magneti field breaking the time reversal symmetry, the two Dira points are not equal to eah other, and both must be studied. Aording to the lattie orientation we adopt as shown in Fig. 1, the zigzag interfae is along the x diretion. For simpliity, we onsider an infinite stripe along the x diretion, therefore the system has translational symmetry along the x diretion thus k x remains a good quantum number. We then do Fourier transformation in the x diretion and redue the 2D problem to 1D. There are atually TWO distint geometries whih are physially different. i The outmost sites of the upper layer are the B 2 sites whih do not stak diretly on the lower layer atoms as shown in Fig. 1 left. The B 2 sites are the low energy degrees of freedom along with A 1 whih are kept if one further onsiders a 2 2 effetive theory on energy sale ɛ t. We label it the B-type staking aording to the type of the outmost atoms; ii A-type staking, i.e., the A 2 sites are the outmost sites on the upper layer as shown in Fig. 1 right. The A 2 sites, together with the B 1 sites whih they stak right on top of, form the dimer sites. In the 2 2 low energy effetive theory the wavefuntions have almost zero weight on those dimes sites when ɛ t. As a result these dimers are ignored in this limit. In the other words, they are oupied onsiderably only at high energy omparing to t. 3. Interfae properties in zero field For a semi-infinite sheet, it is well known that the existene of the zero energy edge modes in both monolayer and bilayer graphene. Presumably, suh modes are also expeted at the interfae between them. Let us onsider the following geometry: a half-plane of monolayer graphene and a half-plane of bilayer graphene joined along the zigzag edge; and look for solutions with zero eigen-energy by using the Dira equation. ψa x, y Ψ mono x, y = 5 ψ B x, y and ψ A1 x, y ψ Ψ bi x, y = B1 x, y 6 ψ A2 x, y ψ B2 x, y are the wavefuntions at Dira point in the monolayer and bilayer respetively. Assuming the interfae loates at y = 0, the boundary ondition for monolayer is then straightforward: both omponents of the wavefuntion must be ontinuous. For the upper layer, it terminates at y = 0 and therefore satisfies the open boundary ondition. Note that the term terminates indiates the last row of lattie sites are the high/low energy sites A 2 /B 2, however, the boundary ondition is not the wavefuntions being zero on these sites. They should be the wavefuntions of the sites one

3 612 Z.-X. Hu, W.X. Ding / Physis Letters A unit ell outside the boundary being zero. Therefore, at the edge, the wavefuntions satisfy [31]: ψ A x, 0 = ψ A1 x, 0, ψ B x, 0 = ψ B1 x, 0, { ψa2 x, 0 = 0 B-type, ψ B2 x, 0 = 0 A-type. 7 In the monolayer region with zigzag interfae, we do the substitution k k x + y in the Dira Hamiltonian H mono = 0 k v F k 0 [11]. The zero energy solution is { φa y = e yk x C 2, φ B y = e yk x 8 C 1. In analogy, we do the same substitution in the bilayer Hamiltonian Eq. 4 and obtain its zero energy solution: φ B1 y = e yk x A 1, φ B2 y = e yk x yγ A 1 + e yk x A 2, φ A1 y = e yk x A 3 + e yk x 9 yγ A 4, φ A2 y = e yk x A 4. Applying the boundary ondition, for the B-type interfae so φ A2 y = 0 = 0, one easily finds that nonzero solutions only exist for k x > 0. The solution is C 1 = C 2 = A 1 = A 3 = A 4 = 0, A 2 = onst. 10 For the A-type interfae, one finds that for k x > 0 C 1 = C 2 = A 1 = A 2 = A 3 = 0, A 4 = onst. 11 Near the other Dira one, the solutions remain the same but only exist for k x < 0. The only nonzero onstant is said to be determined by normalization. Both results are in agreement with the tight-binding analysis [30]. Even though the zero energy states exist for both types of interfae, there is an important differene between them. In the ase of B-type edge, the wavefuntion is only nonzero in the upper layer, whih is trivial as suh mode is expeted when a graphene sheet is terminated at a zigzag edge. However, the A-type is less trivial. The wavefuntion also lives on the extended layer at the interfae where no ut is present. We interpret this as following: when the dimer sites are the boundary, the interlayer oupling t imposes an energy ost for eletrons going through the interfae in the extended layer whih an be onsidered as an effetive potential barrier. The potential barrier an loalize the eletron states along the interfae. We numerially diagonalize a system with a finite width up to 600 unit ells in the y diretion. The intralayer hopping strength t is set to identity and the interlayer hopping strength t = 0.2t. The dispersion relation is shown in Fig. 2. Compare the dispersions in different geometries, the A-type edge has an obviously stronger level antirossing feature than the other. The reason is that for B-type edge, the zero energy edge state just loates on the bilayer Fig. 2. The energy-momentum dispersion relation around one Dira one for the hybrid edge graphene with a B-type and b A-type respetively. The latter one has more energy level antirossing near the Fermi level due to the effetive potential at the edge. graphene whih has quadrati dispersion and has nothing to do with the monolayer part with a linear dispersion. However, in the ase of A-type edge, the zero energy edge state also has omponent on the monolayer graphene at the interfae, the linear dispersion part for monolayer should also onnet to the edge state whih indues more energy level antirossing around the Dira points. Fig. 3 shows the LDOS for the two kinds of hybrid graphene. We label the oordinate of the extended layer by d [0, 600 and the d [900, 1200 for upper layer. Therefore, the interfae loates at d = 300 and d = 900 for layer 1 and 2 respetively. One an notie that the loalized edge state shows as a peak at zero energy along the interfae. The prominent differene is that the peak just appears on the top layer of bilayer part in the B-type ase; butforthea-typease,thezeroenergypeakappearsonbothtwo layers although still loates at the bilayer side. The numerial results are in agreement with the analysis by the Dira equations and we onlude that the two kinds of edge arrangements should have different onsequenes in experiments sine the different distributions of the zero mode. Here we notie that the other peaks in LDOS at the end of the finite system is the signal of the general monolayer or bilayer zigzag edge graphene as disussed in many others work [12 26]. 4. Interfae properties in magneti fields In the presene of magneti field, the Dira equation should be modified by doing the substitution k k + ea. Assume the magneti field is along the diretion of perpendiular to the plane B = Bẑ, B > 0. We adopt the Landau gauge A = Ax, A y = yb, 0 due to the translational invariant along x diretion. Thus the zero energy solution of a monolayer graphene is { φ A y = C 1 e k x y eb 12 φ B y = C 2 e k x y+ eb 2 y2. Fig. 3. Color online. The LDOS of the hybrid bilayer graphene with B-type left and A-type right hybrid interfae respetively. The width of monolayer is 600 unit ells of honeyomb lattie. The region d [0, 600 is the lower extended layer and region d [600, 1200 is the upper layer whih is ut in the middle.

4 Z.-X. Hu, W.X. Ding / Physis Letters A Similarly, the zero energy solution for bilayer graphene beomes: φ A1 y = A 3 γ y + A 2 e k x y eb φ B1 y = A 1 e k x y+ eb 13 φ A2 y = A 3 e k x y eb φ B2 y = A 1 γ y + A 4 e k x y+ eb 2 y2. Applying the same boundary onditions as in the zero field ase, we will have the solutions. For B-type, one gets C 2 = A 1 = A 3 = A 4 = 0, C 1 = A 2 = onst. 14 For A-type the solution is C 2 = A 1 = A 4 = 0, C 1 = A 2 = onst 1, A 3 = onst The onstants are determined by normalization onditions also. One immediately noties that the wavefuntion lives only on the A sublattie. We should note that on the other Dira point, the solutions remain the same form, but resides on the B sublattie. Another important feature of the solution is that for A-type we atually have TWO independent solutions here Dispersion relation The Shrödinger equation in the magneti field for a monolayer and bilayer graphene an be written as 1 0 ξ + ξ Ψ 2 ξ + ξ 0 mono = ɛψ mono 16 and 0 ξ + ξ ξ + ξ 0 γ 0 Ψ 2 0 γ 0 ξ + ξ bi = ɛψ bi, ξ + ξ 0 where ξ = y l B l B k x, l B = eb, ɛ = ω E, ω = 2 v F, l B γ = γ v F ω.the solution in the bulk of the monolayer is 1 Γɛ 2 D ɛ 2 1 2ξ Ψ mono = ± 1 Γɛ 2 +1 D ɛ 2 2ξ = ψɛ 2 1 ξ ±ψ ɛ 2ξ, 18 where the ɛ =± N, N = 0, 1, 2,..., and D ν s are the paraboli ylinder funtions, whih ombined with the fator 1/Γ v + 1 give us the eigen-wavefuntions of a harmoni osillator ψ ν ξ. The bulk solution to the bilayer Hamiltonian an be written in a similar way: Φ bi = ɛɛ 2 j+1 γ 2 γ j j+1 ɛ 2 j+1 γ j+1 ψ j ɛ j+1 ψ j ψ j+1 ψ j 1, γ where ɛ =± 2 +2 j± 1+ γ γ 2 j j = 0, 1, 2,... is the 2 eigen-energy. However, for the interfae problem, we need solutions on the half-plane. In this ase, we put the bilayer part on the y > 0side, and the monolayer part on the y < 0 side. Therefore, for the bilayers, the solutions on 0, take on the same form as in the bulk, but j s are no longer required to be integers. Instead, we now should replae j s by j 1,2 = ɛ ± ɛ 2 γ , 20 and ɛ now varies ontinuously. For the monolayer, the solution on, 0 anbehosenas ψɛ2 Ψ mono = 1 ξ. 21 ψ ɛ 2 ξ With the above solutions, and ombined with the boundary onditions that are already disussed in the zero field ases, we obtain the transendental equations that ditate the dispersion relations. For the B-type interfae, ɛ 2 j D j1 1 2k x D ɛ 2 2k x D j2 2k x + D ɛ 2 1 2k x D j2 +1 2k x = ɛ 2 j D j2 1 2k x D ɛ 2 2k x D j1 2k x + D ɛ 2 1 2k x D j1 +1 2k x, 22 where we have set l B = 1. For the A-type, the dispersion equation is ɛ 2 j γ 2 D j1 2k x D ɛ 2 2k x D j2 2k x + D ɛ 2 1 2k x D j2 +1 2k x = ɛ 2 j γ 2 D j2 2k x D ɛ 2 2k x D j1 2k x + D ɛ 2 1 2k x D j1 +1 2k x. 23 However, one must note that in the presene of magneti field, the time reversal symmetry is broken, therefore, the other Dira one, the time reversal partner, no long behaves the same way. So the dispersion relation must be alulated separately. By the same approah, one an get, for the B-type edge, ɛ 2 j 1 D j1 2k x j 2 D ɛ 2 2k x D j2 1 2k x + ɛ 2 D ɛ2 1 2k x D j2 2k x = ɛ 2 j 2 D j2 2k x j 1 D ɛ 2 2k x D j1 1 2k x + ɛ 2 D ɛ 2 1 2k x D j1 2k x ; 24 for the A-type edge, j ɛ 2 j 1 γ 2 D j1 2k x j 2 D ɛ 2 2k x D j2 1 2k x + ɛ 2 D ɛ2 1 2k x D j2 2k x = j ɛ 2 j 2 γ 2 D j2 2k x j 1 D ɛ 2 2k x D j1 1 2k x + ɛ 2 D ɛ 2 1 2k x D j1 2k x. 25 It is easy to see from the dispersion equation that when k x is very large, the eigen-energy should restore to either the monolayer value or the bilayer value as the paraboli ylinder funtions D ν only onverge to zero for integer ν at infinity. But what interests us is how these bulk Landau levels onnet with eah other when rossing the interfae. To study that, we solve the above equations for ɛ and k x near the interfae k x = 0 for a finite-sized system as shown in Fig. 4. It is shown that the Landau levels math in different ways at two Dira ones. Near the Dira one K K, the zero energy Landau levels in the bilayer region split, one branh rises up to beome the n = 1 Landau levels in the monolayer while the other branh remains as zero energy states. The other Landau levels ontinues from bilayer to monolayer aordingly, i.e. n = 1 bi n = 2 mono... However, near the other Dira one K K, the zero energy state remains intatly, so the higher Landau levels onnet in the way of n = 1 bi n = 1 mono...wealsosee an effet of the effetive potential barrier imposed by the A-type in the dispersive Landau levels near the interfae. For a ertain

5 614 Z.-X. Hu, W.X. Ding / Physis Letters A Fig. 4. Color online. The dispersion urve around Dira ones in the presene of magneti field. The upper two figures are for the B-type and A-type interfaes around one Dira one respetively, and the lower two figures are that for the other Dira one. Fig. 5. The LDOS of the hybrid bilayer graphene with B-type left and A-type right edge in a magneti field. The strength of the magneti field is expressed as magneti flux φ in eah unit ell φ = φ 0 /1315 φ 0 is the magneti flux quanta whih orresponds to B 60T. range of field strength the dispersion develops a loal maximum for A-type while the B-type only develops a flat plateau feature. Fig. 5 shows the LDOS for hybrid bilayer graphene in a magneti field with B-type and A-type edge respetively from a numerial alulation of a system with width of 600 unit ells in y diretion. The Landau level strutures are lear in the bulk both in monolayer and bilayer graphene. Their mismath indues dispersion along the interfae. On the other hand, aording to the analysis of the Dira equation, it also shows the existene of the density peak at Fermi energy E = 0 only on the upper layer for B-type edge and on both two layers for A-type edge. 5. Disussion and onlusion We study the edge state at the hybrid interfae between monolayer and bilayer graphene both in zero field and in a magneti field. There are two types of interfae strutures addressed by the B-type and A-type due to the type of the terminated atoms at the edge. We find the loalized zero energy edge state always exists for both two types of interfae whether in the presene of the magneti field or not, however, the distribution of the edge state and the energy-momentum dispersion near the Dira one and the LDOS shows signifiantly different features to two types of hybrid edge. For B-type interfae, the edge state only lives on the edge of the upper layer graphene whih reflets as one LDOS peak at E = 0 in the upper layer. The edge state has weights on both two layers along the A-type interfae. Thus, we observed the enhaned LDOS peak in the bottom layer also. The dispersion urve of the A-type edge develops strong antirossing feature in the absene of magneti field and loal maximum in the presene of magneti field whih we explained as the effet of the effetive potential imposed by dimer edge. Similar differenes between B-type and A-type were also disussed in the alulation of transmission oeffiients aross the interfae [31,33,34]. For the A-type edge the transmission probability is redued signifiantly for inoming eletrons with energy E t whih an be interpreted in a similar way. We also show that in the presene of a magneti field the dispersion of Landau levels ontinuously goes through the interfae in different manners at two Dira ones beause of the inversion symmetry broken. Here we just onsider the nearest interlayer interation γ in Eq. 4 for simpliity. The other interlayer terms, suh as the A 1 B 2 interation γ 3 γ whih is at the upper right orners in Eq. 4 are only relevant at weak fields. It has nontrivial effet in the bulk, suh as the shift the energy of the Landau level and splitting of the Dira point [35]. However, if we look at the zero energy solution of the Dira equation, the differenes of the relative weight of the wavefuntion between two types of interfaes are still survives. For B-type, the amplitude of φ B1 is muh smaller omparing to that of φ B2, while for A-type edge, they are quite omparable. Therefore, the edge state properties are still dominated by γ, the dimer oupling. In summary, the physial properties of hybrid graphene systems are mostly dominated by both that of the monolayer and bilayer graphene, and also how they are staked. The existene of antirossing in zero field and the dispersive Landau levels in the magneti field near the interfae ould be related to the unexpeted feature other than that of the monolayer or bilayer graphene in the magneto transport experiment [28]. Further study of the hybrid strutures for a more realisti setup, like inluding edge disorder, gate voltage, et., are needed to deeply understand the experimental data. Aknowledgements We wish to thank K. Yang and Y. Barlas for very helpful disussions. This work is supported by Fundamental Researh Funds for the Central Universities CDJRC Z.-X.H. and NSF under Grant No. DMR W.X.D..

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