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Topological confinement effect of edge potentials in zigzag-edge graphene nanoribbons under a staggered bulk potential
Current Applied Physics 17 (2017) 1244e1248
Contents lists available at ScienceDirect
Current Applied Physics journal homepage: www.elsevier.com/locate/cap
Topological confinement effect of edge potentials in zigzag-edge graphene nanoribbons under a staggered bulk potential Kyu Won Lee, Cheol Eui Lee* Department of Physics, Korea University, Seoul, 02841, South Korea
a r t i c l e i n f o
a b s t r a c t
Article history: Received 22 May 2017 Accepted 15 June 2017 Available online 17 June 2017
We have investigated topological confinement effects of edge potentials on gapless edge states in zigzagedge graphene nanoribbons (ZGNRs) under a staggered bulk potential. A variety of gapless edge states were predicted with the concept of topological confinement effect alone, which was confirmed by using tight-binding model calculations. Half-metallicity of ZGNR, which has been semiclassically described, was revealed to fundamentally result from a topological confinement effect. Edge potentials were found to allow an infinitesimal staggered bulk potential to result in gapless edge states, regardless of the ribbon width. A uniform or staggered potential applied to the boundary region narrower than a critical width was found to play a role of the edge potentials, and the critical width was estimated. © 2017 Elsevier B.V. All rights reserved.
Keywords: A. Topological confinement effect B. Edge potential C. Gapless edge states D. Tight binding model
1. Introduction Since the first prediction of quantum spin Hall (QSH) effect in graphene [1], where carriers with opposite spins flow to opposite edges, a variety of topologically nontrivial insulator phases usually classified by the quantum Hall effect has been reported [1e5]. Quantum Hall effect originates from gapless edge states formed at the interface between topologically-different phases, and the number of gapless edge states is determined by the bulk topology. The bulk-edge correspondence is true for the integer quantum Hall and the QSH effects [6,7]. In the quantum valley Hall (QVH) effect, where carriers in different valleys flow to opposite edges, the number of gapless edge states strongly depends on boundary conditions [8e10], and the bulk-edge correspondence is true only at the topological domain wall where the bulk gap changes sign [10]. A topological confinement effect that gapless edge states are bound at topological domain walls and propagate along the wall has been reported in (few-layer) graphene and silicene [11e15]. In a QVH insulator, gapless edge states sensitive to the boundary conditions can open a new route to a variety of quantum Hall effects, which cannot be expected from the bulk topology. Graphene under a staggered potential which breaks inversion symmetry is agreed to be a QVH insulator [8,16]. The gapless edge states corresponding to a QVH phase are not found at the interface
* Corresponding author. E-mail address: [email protected] (C.E. Lee). http://dx.doi.org/10.1016/j.cap.2017.06.008 1567-1739/© 2017 Elsevier B.V. All rights reserved.
between vacuum and zigzag-edge graphene nanoribbon (ZGNR), and thus no quantum Hall effect can be expected [2,8]. On the other hand, a tight-binding (TB) model study of ZGNR showed that gapless edge states can be produced by edge potentials to form topological domain walls [8]. Another TB model study of ZGNR reported that edge magnetization can lead to spin- and valleypolarized gapless edge states, whose origin was not explained [17]. Edge potential and magnetization can arise from electronelectron interactions of the edge-localized electrons, which has been intensively investigated for ferromagnetism and halfmetallicity of ZGNRs [18,19]. Under a staggered AB-sublattice potential DA ¼ DB ¼ D, graphene has a gap 2D and a spin-valley-resolved Chern number Ct;s ¼ 1t 2
sign(D) [8,16]. t ¼ ±1 and s ¼ ±1 correspond to valley and spin indices, respectively. The valley Chern number given by CV ¼ ðCþ1;þ1 þCþ1;1 Þ ðC1;þ1 þC1;1 Þ ¼ 2sign(D) corresponds to the QVH conductivity sV in units of e2 =h. Sign(D) ¼ ±1 defines two different topological domains. Under a staggered magnetization DA ¼ DB ¼ Ms with a mean exchange field M, we can obtain Ct;s ¼ 12 st sign(M) by letting D/Ms [2]. The spin-valley Chern number given by CSV ¼ ðCþ1;þ1 Cþ1;1 Þ ðC1;þ1 C1;1 Þ ¼ 2sign(M) corresponds to the quantum spin-valley Hall (QSVH) conductivity sSV in units of e2 =h. At the vacuum-ZGNR interfaces, any gapless edge states leading to a quantum Hall effect are not found because gapless edge states can exist only at topological domain walls as introduced above.
K.W. Lee, C.E. Lee / Current Applied Physics 17 (2017) 1244e1248
In this paper, we have investigated effect of edge potentials on gapless edge states in view of topological confinement effect in ZGNRs under a staggered bulk potential (magnetization). We predicted various gapless edge states with the concept of topology confinement effect alone, and thus performed TB model calculations to confirm and investigate the gapless edge states. The halfmetallicity of ZGNR, which has been semiclassically described [19e21], was revealed to fundamentally result from a topological confinement effect. Edge potentials were found to allow an infinitesimal staggered bulk potential to result in gapless edge states regardless of the ribbon width. A uniform or staggered potential applied to the boundary region narrower than a critical width was found to play a role of the edge potentials, and the critical width was estimated. 2. Methods ZGNR was considered as a one-dimensional system periodic in the x axis (zigzag direction) with a lattice constant a0 ¼ 2.46 Å, as shown in Fig. 1, and will be referred to as N-ZGNR where N is the number of C-C pairs in a unit cell. The opposite edges at y ¼ 0 and at y ¼ L consist of B- and A-sublattices, respectively. In the righthand side of Fig. 1, the plus and minus signs indicate the sign of on-site energy at each site indicated by dashed horizontal lines. Edge potentials VA and VB correspond to the on-site energies at the y ¼ L and y ¼ 0 edges, respectively. For simplicity, we assume that staggered bulk potentials DA and DB exist only for 0 < y < L. The TB Hamiltonian can be given as follows
H ¼ g0
X 〈p;q〉s
y
cps cqs þ
X p2bulk;s
Dps nps þ
X
Vps nps ;
(1)
p2edge;s
where cps is the electron annihilation operator at a site p with a spin s ¼ ±1 and nps ¼ cyps cps . g0 ¼ 2.7 eV is the nearest-neighbor hopping energy. Dps corresponds to a staggered bulk potential and is zero at edge sites. Dps ¼ DA ¼ D for A-sublattice and Dps ¼ DB ¼ D for Bsublattice. A staggered bulk magnetization can be included by letting D/Ms. Vps ¼ ±V corresponds to an edge potential. Vps ¼ VB at the y ¼ 0 edge and Vps ¼ VA at the y ¼ L edge. Edge magnetization can be included by letting V/Ms. Staggered bulk potentials can be directly extended to edge sites by letting VB /VB þDB and VA /VA þDA , which was confirmed to give essentially the same results.
Fig. 1. Atomic geometry and potential profile. The x axis corresponds to the periodic direction. The edges at y ¼ 0 and at y ¼ L consist of B- (solid circles) and A-sublattices (hollow circles), respectively. In the righthand side, the plus and minus signs indicate the sign of potential at each site indicated by dashed horizontal lines. The illustrated potential profile corresponds to the case (II) described in text.
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3. Results and discussion Whether an edge potential forms a topological domain wall is determined by the signs of the edge and bulk potentials, since Aand B-sublattices should have on-site energies with the same sign at a topological domain wall (considering that A- and B-sublattices have on-site energies with opposite signs in a single domain). For example, Fig. 1 shows that sign(VB ) ¼ sign(DA ) and sign(VA ) ¼ sign(DB ), indicating that a topological domain wall is formed only at the y ¼ 0 edge. Thus, just considering the signs of edge and bulk potentials, we can predict gapless edge states confined at a topological domain wall. From the bulk-edge correspondence true at topological domain walls, CV (¼CSV ) ¼ 2 in graphene under a staggered potential (magnetization) indicates that each domain wall accommodates two gapless edge states. We can consider three cases of edge potentials for a staggered bulk potential. (I) Topological domain walls at both edges when sign(VB ) ¼ sign(DA ) and sign(VA ) ¼ sign(DB ), indicating gapless edge states at both edges. (II) Domain wall only at one edge when sign(VB ) ¼ sign(DA ) and sign(VA ) ¼ sign(DB ), indicating gapless edge states only at one edge and thus a quantum anomalous Hall (QAH) effect. (III) No domain wall when sign(VB ) ¼ sign(DA ) and sign(VA ) ¼ sign(DB ), indicating no gapless edge state. Also, we can consider two cases of edge magnetization for a staggered bulk potential DA ¼ DB ¼ D. (IV) Opposite spin polarization at opposite edges, VA ¼ VB ¼ Ms. We assume that D and M are positive. For s ¼ 1, sign(VB ) ¼ sign(DA ) and sign(VA ) ¼ sign(DB ) indicate no domain wall and no gapless edge state. For s ¼ 1, sign(VB ) ¼ sign(DA ) and sign(VA ) ¼ sign(DB ) indicate domain walls and gapless edge states at both edges. Thus, gapless edge states can be expected only for a spin polarization, indicating that the half-metallic edge states fundamentally result from a topological confinement effect. (V) Same spin polarization at both edges, VA ¼ VB ¼ Ms. For s ¼ 1, sign(VB ) ¼ sign(DA ) and sign(VA ) ¼ sign(DB ) indicate a domain wall only at the y ¼ 0 edge. For s ¼ 1, sign(VB ) ¼ sign(DA ) and sign(VA ) ¼ sign(DB ) indicate a domain wall only at the y ¼ L edge. Thus, gapless edge states at opposite edges have opposite spin polarizations. In the same way, for a staggered bulk magnetization DA ¼ DB ¼ Ms, (VI) antisymmetric edge potentials VA ¼ VB are expected to lead to half-metallic edge states, and (VII) symmetric edge potentials VA ¼ VB are expected to lead to the gapless edge states with opposite spin polarizations at opposite edges. Halfmetallicity in the antiferromagnetic phase of ZGNRs has been semiclassically described [19e21], where energy-level shifts of opposite signs for opposite spin polarizations due to the potential difference between opposite edges lead to a gap closure only for a spin polarization. However, our analyses for the cases (IV) and (VI) indicate that the half-metallicity in ZGNRs fundamentally originates from a topological confinement effect. Next, we will discuss TB model calculations to confirm and investigate the gapless edge states. Fig. 2 shows the band structure (first column), the amplitude of the wavefunction jJj at the Fermi level EF ¼ 0 (second column), and the schematic for the propagating states at EF ¼ 0 (third column) constructed from the band velocity and jJj at EF ¼ 0 with D ¼ 0.1 eV in 100-ZGNR. The signs of the edge potentials in Fig. 2 (a)-(e) correspond to the cases (I)-(V) described above, respectively. Except that in Fig. 2(c) corresponding to the case (III), gapless edge states can be found within the bulk gap (jEj < 0:1 eV). Fig. 2(a1) and (b1), where jJj is displayed as a function of y, show gapless edge states formed at both edges and only at one edge, respectively. Fig. 2(d1) shows the half-metallic gapless edge states and Fig. 2(e1) shows the gapless edge states with opposite spin polarizations at
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K.W. Lee, C.E. Lee / Current Applied Physics 17 (2017) 1244e1248
Fig. 3. Staggered bulk magnetization D ¼ Ms with M ¼ 0.1 eV in 100-ZGNR. (a) VB ¼ VA ¼ V and (b) VB ¼ VA ¼ V with V ¼ 0.1 eV. The first, second, and third columns respectively correspond to the band structure, jJj at EF ¼ 0, and the schematic for the propagating states at EF ¼ 0 constructed from the first and second columns. The same conventions as in Fig. 2 are used.
Fig. 2. Staggered bulk potential D ¼ 0.1 eV in 100-ZGNR. (a) VB ¼ VA ¼ V, (b) VB ¼ VA ¼ V and (c) VB ¼ VA ¼ V with V ¼ 0.2 eV. (d) VB ¼ VA ¼ Ms and (e) VB ¼ VA ¼ Ms with M ¼ 0.1 eV. The first, second, and third columns respectively correspond to the band structure, the amplitude of wave function jJj at the Fermi level EF ¼ 0, and the schematic for the propagating states at EF ¼ 0 constructed from the first and second columns. Red and blue colors correspond to the opposite spins. Black color corresponds to spin-degenerated states. K and Kþ correspond to opposite valleys. In the third column, the up and down arrows indicate opposite propagating directions, and the solid and dashed arrows correspond to opposite valleys. gs ¼ 2 is the spin degeneracy. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
opposite edges. The gapless edge states in Fig. 2(a1)-(e1) are fully consistent with the predictions made with topology confinement effect alone. Since each state contributes e2 =h to the conductivity, the quantized Hall conductivity can be estimated in units of e2 =h by counting the number of the spin-, valley-, and edge-resolved states propagating in a given direction. sH and sS are the (charge) Hall conductivity and the spin Hall conductivity, respectively. Fig. 2(a2) shows a typical QVH effect with sV ¼ 2gs , where backscattering is forbidden by the valley separation in the Brillouin zone. gs ¼ 2 is the spin degeneracy. Fig. 2(b2) shows that sH ¼ sV ¼ 1gs, indicating a QAH effect as well as a QVH effect. Fig. 2(d2) shows the halfmetallic QVH effect with sV ¼ 2. Fig. 2(e2) shows that sS ¼ sV ¼ 2, indicating a QSH effect as well as the QVH effect. The gapless edge states in Fig. 2(d2) and (e2) are consistent with a previous work [17]. Despite non-zero sH and sS in Fig. 2(b2) and (e2), backscattering is still forbidden by the valley separation in the Brillouin zone, indicating that the gapless edge states correspond to a QVH insulator.
Fig. 3 shows the band structure (first column), jJj at EF ¼ 0 (second column), and the schematic for the propagating states at EF ¼ 0 (third column) with a staggered bulk magnetization M ¼ 0.1 eV in 100-ZGNR. The signs of edge potentials in Fig. 3(a) and (b) correspond to the cases (VI) and (VII), respectively. Fig. 3(a1) and (b1) respectively show the half-metallic edge states and the gapless edge states with opposite spin polarizations at opposite edges, which are quite consistent with the predictions made with topology confinement effect alone. Fig. 3(a2) shows the halfmetallic QVH effect with sSV ¼ sV ¼ 2. In the half-metallic phase, sSV is always equal to sV . Fig. 3(b2) shows that sSV ¼ sS ¼ 2, where backscattering is forbidden by the valley separation in the Brillouin zone. Because inversion of the spin-valley degree of freedom characterized by the product of spin and valley indices (st) is equivalent to inversion of the valley for a given spin, the gapless edge states correspond to a QSVH insulator. Finally, we investigated the critical width of the boundary region where the applied potentials can play the role of edge potentials, and we found an interesting property of the edge potential. We will discuss staggered and uniform potentials applied to boundary regions, respectively, in Figs. 4 and 5. Fig. 4(a) shows the potential profile of N-ZGNR divided into three regions of widths (N1 ,N0 ,N1 ) with N ¼ N0 þ2N1 under staggered potentials (V,D,V). DA ¼ DB ¼ D and VA ¼ VB ¼ V. When sign(V) ¼ sign(D), the boundary and central regions have opposite topologies to each other, and topological domain walls corresponding to the case (I) are formed at the interfaces between the regions. Fig. 4(b)-(d) show the band gap Eg under various conditions as a function of 3=4 R ¼ ðN1 =N0 ÞV=D , which is a scaled form of N1 and was introduced to obtain a universal form of the critical width. Fig. 4(b) shows Eg as a function of R for N ¼ 50, 100, 150, and 200 with D ¼ 0.05 eV and V ¼ 2D. Fig. 4(c) shows Eg as a function of R for D ¼ 0.05, 0.1, 0.15, and 0.2 eV with N ¼ 100 and V ¼ 2D. Fig. 4(d) shows Eg as a function of R for jV=Dj ¼ 0.5, 1, 2, 3, and 4 with N ¼ 100 3=4 when jVj is and D ¼ 0.1 eV. Eg can be roughly scaled with V=D not so different from jDj. Fig. 4 shows that, below Rc 0.06, band gap closes and gapless edge states exist regardless of N, D, and jV=D. Thus, the critical 3=4 width of boundary region is N1C Rc N0 D=V . When jD=Vj ¼ 1,
K.W. Lee, C.E. Lee / Current Applied Physics 17 (2017) 1244e1248
Fig. 4. Boundary region under a staggered potential. (a) Potential profile of N-ZGNR divided into three regions of widths (N1 ,N0 ,N1 ) with N ¼ N0 þ2N1 under staggered potentials (V,D,V). DA ¼ DB ¼ D and VA ¼ VB ¼ V. When sign(V) ¼ sign(D), topological domain walls corresponding to the case (I) are formed at the interfaces between the regions (shaded regions). Red lines indicate potentials at each site (black 3=4 dots). (b) Band gap Eg as a function of R ¼ ðN1 =N0 ÞV=D for N ¼ 50, 100, 150, and 200 with D ¼ 0.05 eV and V ¼ 2D. (c) Eg as a function of R for D ¼ 0.05, 0.1, 0.15, and 0.2 eV with N ¼ 100 and V ¼ 2D. (d) Eg as a function of R for jV=Dj ¼ 0.5, 1, 2, 3, and 4 with N ¼ 100 and D ¼ 0.1 eV. Below Rc 0.06, the band gap closes. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
the critical width is about 5% of the ribbon width, N1C 0.05N. It should be noticed that gapless edge states can exist irrespective of D for N1 < N1C , while a finite D is required for gapless edge states for N1 > N1C . Thus, edge potentials allow an infinitely-small staggered bulk potential to result in gapless edge states. A staggered bulk potential can be generated from a substrate. If boundary potentials are generated by the gate electrodes, the boundary potential is not staggered, but is uniform. Thus, we investigated uniform boundary potentials with a staggered center potential as shown in Fig. 5. Fig. 5(a) and (b) show the potential profiles of N-ZGNR divided into three regions of widths (N1 ,N0 ,N1 ) under uniform boundary potentials and a staggered center potential. DA ¼ DB ¼ D, VA ¼ VB ¼ V around y ¼ 0 and VA ¼ VB ¼ V around y ¼ L. The boundary regions are topologically trivial. In Fig. 5(a), sign(V) ¼ sign(D), which corresponds to an extension of the case (I) with gapless edge states. In Fig. 5(b), sign(V) ¼ sign(D), which corresponds to an extension of the case (III) with no gapless edge state. Boundary potentials with the same signs at opposite boundaries (not shown), VA ¼ VB ¼ V around y ¼ 0 and around y ¼ L, correspond to an extension of the case (II) with gapless edge states only at one interface. Fig. 5(c) and (d) show the results obtained by using the potential profile in Fig. 5(a). Fig. 5(c) shows the band structure of 100-ZGNR with N1 ¼ 3, D ¼ 0.05 eV, and V ¼ 2D, where gapless edge states are found. Fig. 5(d) shows Eg as a function of R for N ¼ 50, 100, 150, and 200 with D ¼ 0.05 eV and V ¼ 2D. As is the case in Fig. 4, below Rc 0.06, band gap closes and gapless edge states exist regardless of N. Thus, when N1 < N1C , boundary
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Fig. 5. Boundary region under a uniform potential. Panels (a) and (b) show potential profiles of N-ZGNR divided into three regions of widths (N1 ,N0 ,N1 ) under uniform boundary potentials and a staggered center potential. DA ¼ DB ¼ D, VA ¼ VB ¼ V around y ¼ 0 and VA ¼ VB ¼ V around y ¼ L. The boundary regions are topologically trivial. Red lines indicate potentials at each site (black dots). (a) Sign(V) ¼ sign(D), which is an extension of the case (I) and has gapless edge states. (b) Sign(V) ¼ sign(D), which is an extension of the case (III) and has no gapless edge state. Panels (c) and (d) show the results obtained by using the potential profile in panel (a). (c) Band structure of 100-ZGNR with N1 ¼ 3, D ¼ 0.05 eV, and V ¼ 2D. (d) Eg as a function of R for N ¼ 50, 100, 150, and 200 with D ¼ 0.05 eV and V ¼ 2D. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
potentials whether staggered or uniform can play the role of the edge potentials. To summarize, we have investigated the effect of edge potentials on gapless edge states in view of a topological confinement effect in ZGNRs under a staggered bulk potential. With the concept of a topological confinement effect alone, a variety of gapless edge states can be predicted, which are then confirmed by TB model calculations. Half-metallicity of ZGNR was revealed to fundamentally result from a topological confinement effect, while the halfmetallicity has been semiclassically described. Edge potentials were revealed to allow an infinitesimal staggered bulk potential to result in gapless edge states. A uniform or staggered potential applied to the boundary region narrower than a critical width was found to play a role of the edge potentials, and the critical width was estimated. Acknowledgements This work was supported by the National Research Foundation of Korea (Project No. 2016R1D1A1A09917003, No. 2016R1D1A1B03931144, No. 2015M1A7A1A01002234, and No. NRF-2010-0027963). K.W.L. gratefully acknowledges a Korea University research grant. References [1] C.L. Kane, E.J. Mele, Z2 topological order and the quantum spin Hall effect,
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Phys. Rev. Lett. 95 (2005) 146802. [2] F. Zhang, J. Jung, G.A. Fiete, Q. Niu, A.H. MacDonald, Spontaneous quantum Hall states in chirally stacked few-layer graphene systems, Phys. Rev. Lett. 106 (2011) 156801. [3] L. Xu, Y. Zhou, C.-D. Gong, Topological phase transition induced by spin-orbit coupling in bilayer graphene, J. Phys. Condens. Matter 25 (2013) 335503. [4] M. Ezawa, Spin valleytronics in silicene: quantum spin Hall-quantum anomalous Hall insulators and single-valley semimetals, Phys. Rev. B 87 (2013) 155415. [5] H. Pan, Z. Li, C.-C. Liu, G. Zhu, Z. Qiao, Y. Yao, Valley-polarized quantum anomalous Hall effect in silicene, Phys. Rev. Lett. 112 (2014) 106802. [6] Y. Hatsugai, Chern number and edge states in the integer quantum Hall effect, Phys. Rev. Lett. 71 (1993) 3697e3700. [7] X.-L. Qi, Y.-S. Wu, S.-C. Zhang, General theorem relating the bulk topological number to edge states in two-dimensional insulators, Phys. Rev. B 74 (2006) 045125. [8] W. Yao, S.A. Yang, Q. Niu, Edge states in graphene: from gapped flat-band to gapless chiral modes, Phys. Rev. Lett. 102 (2009) 096801. [9] S. Bhowmick, V.B. Shenoy, Weber-Fechner type nonlinear behavior in zigzag edge graphene nanoribbons, Phys. Rev. B 82 (2010) 155448. [10] J. Li, A.F. Morpurgo, M. Büttiker, I. Martin, Marginality of bulk-edge correspondence for single-valley Hamiltonians, Phys. Rev. B 82 (2010) 245404. [11] N.M.R. Peres, A.H. Castro Neto, F. Guinea, Dirac fermion confinement in
graphene, Phys. Rev. B 73 (2006) 241403(R). [12] G.W. Semenoff, V. Semenoff, F. Zhou, Phys. Domain walls in gapped graphene, Rev. Lett. 101 (2008) 087204. [13] I. Martin, Ya.M. Blanter, A.F. Morpurgo, Topological confinement in bilayer graphene, Phys. Rev. Lett. 100 (2008) 036804. [14] S.H.R. de Sena, J.M. Pereira Jr., F.M. Peeters, G.A. Farias, Topological confinement in trilayer graphene, Phys. Rev. B 89 (2014) 035420. [15] Y. Kim, K. Choi, J. Ihm, Topological domain walls and quantum valley Hall effects in silicene, Phys. Rev. B 89 (2014) 085429. [16] D. Xiao, W. Yao, Q. Niu, Valley-contrasting physics in graphene: magnetic moment and topological transport, Phys. Rev. Lett. 99 (2007) 236809. [17] Z. Qiao, S.A. Yang, B. Wang, Y. Yao, Q. Niu, Spin-polarized and valley helical edge modes in graphene nanoribbons, Phys. Rev. B 84 (2011) 035431. , Z Osva th, P. Nemes-Incze, C. Hwang, [18] G.Z. Magda, X. Jin, I. Hagym asi, P. Vancso , L. Tapaszto , Room-temperature magnetic order on zigzag edges of L.P. Biro narrow graphene nanoribbons, Nature 514 (2014) 608e611. [19] Y.-W. Son, M.L. Cohen, S.G. Louie, Half-metallic graphene nanoribbons, Nature 444 (2006) 347e349. [20] K.W. Lee, C.E. Lee, Half-metallic carbon nanotubes, Adv. Mater 24 (2012) 2019e2023. [21] G.W. Jeon, K.W. Lee, C.E. Lee, Layer-selective half-metallicity in bilayer graphene nanoribbons, Sci. Rep. 5 (2015) 9825.
Contents lists available at ScienceDirect
Current Applied Physics journal homepage: www.elsevier.com/locate/cap
Topological confinement effect of edge potentials in zigzag-edge graphene nanoribbons under a staggered bulk potential Kyu Won Lee, Cheol Eui Lee* Department of Physics, Korea University, Seoul, 02841, South Korea
a r t i c l e i n f o
a b s t r a c t
Article history: Received 22 May 2017 Accepted 15 June 2017 Available online 17 June 2017
We have investigated topological confinement effects of edge potentials on gapless edge states in zigzagedge graphene nanoribbons (ZGNRs) under a staggered bulk potential. A variety of gapless edge states were predicted with the concept of topological confinement effect alone, which was confirmed by using tight-binding model calculations. Half-metallicity of ZGNR, which has been semiclassically described, was revealed to fundamentally result from a topological confinement effect. Edge potentials were found to allow an infinitesimal staggered bulk potential to result in gapless edge states, regardless of the ribbon width. A uniform or staggered potential applied to the boundary region narrower than a critical width was found to play a role of the edge potentials, and the critical width was estimated. © 2017 Elsevier B.V. All rights reserved.
Keywords: A. Topological confinement effect B. Edge potential C. Gapless edge states D. Tight binding model
1. Introduction Since the first prediction of quantum spin Hall (QSH) effect in graphene [1], where carriers with opposite spins flow to opposite edges, a variety of topologically nontrivial insulator phases usually classified by the quantum Hall effect has been reported [1e5]. Quantum Hall effect originates from gapless edge states formed at the interface between topologically-different phases, and the number of gapless edge states is determined by the bulk topology. The bulk-edge correspondence is true for the integer quantum Hall and the QSH effects [6,7]. In the quantum valley Hall (QVH) effect, where carriers in different valleys flow to opposite edges, the number of gapless edge states strongly depends on boundary conditions [8e10], and the bulk-edge correspondence is true only at the topological domain wall where the bulk gap changes sign [10]. A topological confinement effect that gapless edge states are bound at topological domain walls and propagate along the wall has been reported in (few-layer) graphene and silicene [11e15]. In a QVH insulator, gapless edge states sensitive to the boundary conditions can open a new route to a variety of quantum Hall effects, which cannot be expected from the bulk topology. Graphene under a staggered potential which breaks inversion symmetry is agreed to be a QVH insulator [8,16]. The gapless edge states corresponding to a QVH phase are not found at the interface
* Corresponding author. E-mail address: [email protected] (C.E. Lee). http://dx.doi.org/10.1016/j.cap.2017.06.008 1567-1739/© 2017 Elsevier B.V. All rights reserved.
between vacuum and zigzag-edge graphene nanoribbon (ZGNR), and thus no quantum Hall effect can be expected [2,8]. On the other hand, a tight-binding (TB) model study of ZGNR showed that gapless edge states can be produced by edge potentials to form topological domain walls [8]. Another TB model study of ZGNR reported that edge magnetization can lead to spin- and valleypolarized gapless edge states, whose origin was not explained [17]. Edge potential and magnetization can arise from electronelectron interactions of the edge-localized electrons, which has been intensively investigated for ferromagnetism and halfmetallicity of ZGNRs [18,19]. Under a staggered AB-sublattice potential DA ¼ DB ¼ D, graphene has a gap 2D and a spin-valley-resolved Chern number Ct;s ¼ 1t 2
sign(D) [8,16]. t ¼ ±1 and s ¼ ±1 correspond to valley and spin indices, respectively. The valley Chern number given by CV ¼ ðCþ1;þ1 þCþ1;1 Þ ðC1;þ1 þC1;1 Þ ¼ 2sign(D) corresponds to the QVH conductivity sV in units of e2 =h. Sign(D) ¼ ±1 defines two different topological domains. Under a staggered magnetization DA ¼ DB ¼ Ms with a mean exchange field M, we can obtain Ct;s ¼ 12 st sign(M) by letting D/Ms [2]. The spin-valley Chern number given by CSV ¼ ðCþ1;þ1 Cþ1;1 Þ ðC1;þ1 C1;1 Þ ¼ 2sign(M) corresponds to the quantum spin-valley Hall (QSVH) conductivity sSV in units of e2 =h. At the vacuum-ZGNR interfaces, any gapless edge states leading to a quantum Hall effect are not found because gapless edge states can exist only at topological domain walls as introduced above.
K.W. Lee, C.E. Lee / Current Applied Physics 17 (2017) 1244e1248
In this paper, we have investigated effect of edge potentials on gapless edge states in view of topological confinement effect in ZGNRs under a staggered bulk potential (magnetization). We predicted various gapless edge states with the concept of topology confinement effect alone, and thus performed TB model calculations to confirm and investigate the gapless edge states. The halfmetallicity of ZGNR, which has been semiclassically described [19e21], was revealed to fundamentally result from a topological confinement effect. Edge potentials were found to allow an infinitesimal staggered bulk potential to result in gapless edge states regardless of the ribbon width. A uniform or staggered potential applied to the boundary region narrower than a critical width was found to play a role of the edge potentials, and the critical width was estimated. 2. Methods ZGNR was considered as a one-dimensional system periodic in the x axis (zigzag direction) with a lattice constant a0 ¼ 2.46 Å, as shown in Fig. 1, and will be referred to as N-ZGNR where N is the number of C-C pairs in a unit cell. The opposite edges at y ¼ 0 and at y ¼ L consist of B- and A-sublattices, respectively. In the righthand side of Fig. 1, the plus and minus signs indicate the sign of on-site energy at each site indicated by dashed horizontal lines. Edge potentials VA and VB correspond to the on-site energies at the y ¼ L and y ¼ 0 edges, respectively. For simplicity, we assume that staggered bulk potentials DA and DB exist only for 0 < y < L. The TB Hamiltonian can be given as follows
H ¼ g0
X 〈p;q〉s
y
cps cqs þ
X p2bulk;s
Dps nps þ
X
Vps nps ;
(1)
p2edge;s
where cps is the electron annihilation operator at a site p with a spin s ¼ ±1 and nps ¼ cyps cps . g0 ¼ 2.7 eV is the nearest-neighbor hopping energy. Dps corresponds to a staggered bulk potential and is zero at edge sites. Dps ¼ DA ¼ D for A-sublattice and Dps ¼ DB ¼ D for Bsublattice. A staggered bulk magnetization can be included by letting D/Ms. Vps ¼ ±V corresponds to an edge potential. Vps ¼ VB at the y ¼ 0 edge and Vps ¼ VA at the y ¼ L edge. Edge magnetization can be included by letting V/Ms. Staggered bulk potentials can be directly extended to edge sites by letting VB /VB þDB and VA /VA þDA , which was confirmed to give essentially the same results.
Fig. 1. Atomic geometry and potential profile. The x axis corresponds to the periodic direction. The edges at y ¼ 0 and at y ¼ L consist of B- (solid circles) and A-sublattices (hollow circles), respectively. In the righthand side, the plus and minus signs indicate the sign of potential at each site indicated by dashed horizontal lines. The illustrated potential profile corresponds to the case (II) described in text.
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3. Results and discussion Whether an edge potential forms a topological domain wall is determined by the signs of the edge and bulk potentials, since Aand B-sublattices should have on-site energies with the same sign at a topological domain wall (considering that A- and B-sublattices have on-site energies with opposite signs in a single domain). For example, Fig. 1 shows that sign(VB ) ¼ sign(DA ) and sign(VA ) ¼ sign(DB ), indicating that a topological domain wall is formed only at the y ¼ 0 edge. Thus, just considering the signs of edge and bulk potentials, we can predict gapless edge states confined at a topological domain wall. From the bulk-edge correspondence true at topological domain walls, CV (¼CSV ) ¼ 2 in graphene under a staggered potential (magnetization) indicates that each domain wall accommodates two gapless edge states. We can consider three cases of edge potentials for a staggered bulk potential. (I) Topological domain walls at both edges when sign(VB ) ¼ sign(DA ) and sign(VA ) ¼ sign(DB ), indicating gapless edge states at both edges. (II) Domain wall only at one edge when sign(VB ) ¼ sign(DA ) and sign(VA ) ¼ sign(DB ), indicating gapless edge states only at one edge and thus a quantum anomalous Hall (QAH) effect. (III) No domain wall when sign(VB ) ¼ sign(DA ) and sign(VA ) ¼ sign(DB ), indicating no gapless edge state. Also, we can consider two cases of edge magnetization for a staggered bulk potential DA ¼ DB ¼ D. (IV) Opposite spin polarization at opposite edges, VA ¼ VB ¼ Ms. We assume that D and M are positive. For s ¼ 1, sign(VB ) ¼ sign(DA ) and sign(VA ) ¼ sign(DB ) indicate no domain wall and no gapless edge state. For s ¼ 1, sign(VB ) ¼ sign(DA ) and sign(VA ) ¼ sign(DB ) indicate domain walls and gapless edge states at both edges. Thus, gapless edge states can be expected only for a spin polarization, indicating that the half-metallic edge states fundamentally result from a topological confinement effect. (V) Same spin polarization at both edges, VA ¼ VB ¼ Ms. For s ¼ 1, sign(VB ) ¼ sign(DA ) and sign(VA ) ¼ sign(DB ) indicate a domain wall only at the y ¼ 0 edge. For s ¼ 1, sign(VB ) ¼ sign(DA ) and sign(VA ) ¼ sign(DB ) indicate a domain wall only at the y ¼ L edge. Thus, gapless edge states at opposite edges have opposite spin polarizations. In the same way, for a staggered bulk magnetization DA ¼ DB ¼ Ms, (VI) antisymmetric edge potentials VA ¼ VB are expected to lead to half-metallic edge states, and (VII) symmetric edge potentials VA ¼ VB are expected to lead to the gapless edge states with opposite spin polarizations at opposite edges. Halfmetallicity in the antiferromagnetic phase of ZGNRs has been semiclassically described [19e21], where energy-level shifts of opposite signs for opposite spin polarizations due to the potential difference between opposite edges lead to a gap closure only for a spin polarization. However, our analyses for the cases (IV) and (VI) indicate that the half-metallicity in ZGNRs fundamentally originates from a topological confinement effect. Next, we will discuss TB model calculations to confirm and investigate the gapless edge states. Fig. 2 shows the band structure (first column), the amplitude of the wavefunction jJj at the Fermi level EF ¼ 0 (second column), and the schematic for the propagating states at EF ¼ 0 (third column) constructed from the band velocity and jJj at EF ¼ 0 with D ¼ 0.1 eV in 100-ZGNR. The signs of the edge potentials in Fig. 2 (a)-(e) correspond to the cases (I)-(V) described above, respectively. Except that in Fig. 2(c) corresponding to the case (III), gapless edge states can be found within the bulk gap (jEj < 0:1 eV). Fig. 2(a1) and (b1), where jJj is displayed as a function of y, show gapless edge states formed at both edges and only at one edge, respectively. Fig. 2(d1) shows the half-metallic gapless edge states and Fig. 2(e1) shows the gapless edge states with opposite spin polarizations at
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Fig. 3. Staggered bulk magnetization D ¼ Ms with M ¼ 0.1 eV in 100-ZGNR. (a) VB ¼ VA ¼ V and (b) VB ¼ VA ¼ V with V ¼ 0.1 eV. The first, second, and third columns respectively correspond to the band structure, jJj at EF ¼ 0, and the schematic for the propagating states at EF ¼ 0 constructed from the first and second columns. The same conventions as in Fig. 2 are used.
Fig. 2. Staggered bulk potential D ¼ 0.1 eV in 100-ZGNR. (a) VB ¼ VA ¼ V, (b) VB ¼ VA ¼ V and (c) VB ¼ VA ¼ V with V ¼ 0.2 eV. (d) VB ¼ VA ¼ Ms and (e) VB ¼ VA ¼ Ms with M ¼ 0.1 eV. The first, second, and third columns respectively correspond to the band structure, the amplitude of wave function jJj at the Fermi level EF ¼ 0, and the schematic for the propagating states at EF ¼ 0 constructed from the first and second columns. Red and blue colors correspond to the opposite spins. Black color corresponds to spin-degenerated states. K and Kþ correspond to opposite valleys. In the third column, the up and down arrows indicate opposite propagating directions, and the solid and dashed arrows correspond to opposite valleys. gs ¼ 2 is the spin degeneracy. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
opposite edges. The gapless edge states in Fig. 2(a1)-(e1) are fully consistent with the predictions made with topology confinement effect alone. Since each state contributes e2 =h to the conductivity, the quantized Hall conductivity can be estimated in units of e2 =h by counting the number of the spin-, valley-, and edge-resolved states propagating in a given direction. sH and sS are the (charge) Hall conductivity and the spin Hall conductivity, respectively. Fig. 2(a2) shows a typical QVH effect with sV ¼ 2gs , where backscattering is forbidden by the valley separation in the Brillouin zone. gs ¼ 2 is the spin degeneracy. Fig. 2(b2) shows that sH ¼ sV ¼ 1gs, indicating a QAH effect as well as a QVH effect. Fig. 2(d2) shows the halfmetallic QVH effect with sV ¼ 2. Fig. 2(e2) shows that sS ¼ sV ¼ 2, indicating a QSH effect as well as the QVH effect. The gapless edge states in Fig. 2(d2) and (e2) are consistent with a previous work [17]. Despite non-zero sH and sS in Fig. 2(b2) and (e2), backscattering is still forbidden by the valley separation in the Brillouin zone, indicating that the gapless edge states correspond to a QVH insulator.
Fig. 3 shows the band structure (first column), jJj at EF ¼ 0 (second column), and the schematic for the propagating states at EF ¼ 0 (third column) with a staggered bulk magnetization M ¼ 0.1 eV in 100-ZGNR. The signs of edge potentials in Fig. 3(a) and (b) correspond to the cases (VI) and (VII), respectively. Fig. 3(a1) and (b1) respectively show the half-metallic edge states and the gapless edge states with opposite spin polarizations at opposite edges, which are quite consistent with the predictions made with topology confinement effect alone. Fig. 3(a2) shows the halfmetallic QVH effect with sSV ¼ sV ¼ 2. In the half-metallic phase, sSV is always equal to sV . Fig. 3(b2) shows that sSV ¼ sS ¼ 2, where backscattering is forbidden by the valley separation in the Brillouin zone. Because inversion of the spin-valley degree of freedom characterized by the product of spin and valley indices (st) is equivalent to inversion of the valley for a given spin, the gapless edge states correspond to a QSVH insulator. Finally, we investigated the critical width of the boundary region where the applied potentials can play the role of edge potentials, and we found an interesting property of the edge potential. We will discuss staggered and uniform potentials applied to boundary regions, respectively, in Figs. 4 and 5. Fig. 4(a) shows the potential profile of N-ZGNR divided into three regions of widths (N1 ,N0 ,N1 ) with N ¼ N0 þ2N1 under staggered potentials (V,D,V). DA ¼ DB ¼ D and VA ¼ VB ¼ V. When sign(V) ¼ sign(D), the boundary and central regions have opposite topologies to each other, and topological domain walls corresponding to the case (I) are formed at the interfaces between the regions. Fig. 4(b)-(d) show the band gap Eg under various conditions as a function of 3=4 R ¼ ðN1 =N0 ÞV=D , which is a scaled form of N1 and was introduced to obtain a universal form of the critical width. Fig. 4(b) shows Eg as a function of R for N ¼ 50, 100, 150, and 200 with D ¼ 0.05 eV and V ¼ 2D. Fig. 4(c) shows Eg as a function of R for D ¼ 0.05, 0.1, 0.15, and 0.2 eV with N ¼ 100 and V ¼ 2D. Fig. 4(d) shows Eg as a function of R for jV=Dj ¼ 0.5, 1, 2, 3, and 4 with N ¼ 100 3=4 when jVj is and D ¼ 0.1 eV. Eg can be roughly scaled with V=D not so different from jDj. Fig. 4 shows that, below Rc 0.06, band gap closes and gapless edge states exist regardless of N, D, and jV=D. Thus, the critical 3=4 width of boundary region is N1C Rc N0 D=V . When jD=Vj ¼ 1,
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Fig. 4. Boundary region under a staggered potential. (a) Potential profile of N-ZGNR divided into three regions of widths (N1 ,N0 ,N1 ) with N ¼ N0 þ2N1 under staggered potentials (V,D,V). DA ¼ DB ¼ D and VA ¼ VB ¼ V. When sign(V) ¼ sign(D), topological domain walls corresponding to the case (I) are formed at the interfaces between the regions (shaded regions). Red lines indicate potentials at each site (black 3=4 dots). (b) Band gap Eg as a function of R ¼ ðN1 =N0 ÞV=D for N ¼ 50, 100, 150, and 200 with D ¼ 0.05 eV and V ¼ 2D. (c) Eg as a function of R for D ¼ 0.05, 0.1, 0.15, and 0.2 eV with N ¼ 100 and V ¼ 2D. (d) Eg as a function of R for jV=Dj ¼ 0.5, 1, 2, 3, and 4 with N ¼ 100 and D ¼ 0.1 eV. Below Rc 0.06, the band gap closes. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
the critical width is about 5% of the ribbon width, N1C 0.05N. It should be noticed that gapless edge states can exist irrespective of D for N1 < N1C , while a finite D is required for gapless edge states for N1 > N1C . Thus, edge potentials allow an infinitely-small staggered bulk potential to result in gapless edge states. A staggered bulk potential can be generated from a substrate. If boundary potentials are generated by the gate electrodes, the boundary potential is not staggered, but is uniform. Thus, we investigated uniform boundary potentials with a staggered center potential as shown in Fig. 5. Fig. 5(a) and (b) show the potential profiles of N-ZGNR divided into three regions of widths (N1 ,N0 ,N1 ) under uniform boundary potentials and a staggered center potential. DA ¼ DB ¼ D, VA ¼ VB ¼ V around y ¼ 0 and VA ¼ VB ¼ V around y ¼ L. The boundary regions are topologically trivial. In Fig. 5(a), sign(V) ¼ sign(D), which corresponds to an extension of the case (I) with gapless edge states. In Fig. 5(b), sign(V) ¼ sign(D), which corresponds to an extension of the case (III) with no gapless edge state. Boundary potentials with the same signs at opposite boundaries (not shown), VA ¼ VB ¼ V around y ¼ 0 and around y ¼ L, correspond to an extension of the case (II) with gapless edge states only at one interface. Fig. 5(c) and (d) show the results obtained by using the potential profile in Fig. 5(a). Fig. 5(c) shows the band structure of 100-ZGNR with N1 ¼ 3, D ¼ 0.05 eV, and V ¼ 2D, where gapless edge states are found. Fig. 5(d) shows Eg as a function of R for N ¼ 50, 100, 150, and 200 with D ¼ 0.05 eV and V ¼ 2D. As is the case in Fig. 4, below Rc 0.06, band gap closes and gapless edge states exist regardless of N. Thus, when N1 < N1C , boundary
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Fig. 5. Boundary region under a uniform potential. Panels (a) and (b) show potential profiles of N-ZGNR divided into three regions of widths (N1 ,N0 ,N1 ) under uniform boundary potentials and a staggered center potential. DA ¼ DB ¼ D, VA ¼ VB ¼ V around y ¼ 0 and VA ¼ VB ¼ V around y ¼ L. The boundary regions are topologically trivial. Red lines indicate potentials at each site (black dots). (a) Sign(V) ¼ sign(D), which is an extension of the case (I) and has gapless edge states. (b) Sign(V) ¼ sign(D), which is an extension of the case (III) and has no gapless edge state. Panels (c) and (d) show the results obtained by using the potential profile in panel (a). (c) Band structure of 100-ZGNR with N1 ¼ 3, D ¼ 0.05 eV, and V ¼ 2D. (d) Eg as a function of R for N ¼ 50, 100, 150, and 200 with D ¼ 0.05 eV and V ¼ 2D. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
potentials whether staggered or uniform can play the role of the edge potentials. To summarize, we have investigated the effect of edge potentials on gapless edge states in view of a topological confinement effect in ZGNRs under a staggered bulk potential. With the concept of a topological confinement effect alone, a variety of gapless edge states can be predicted, which are then confirmed by TB model calculations. Half-metallicity of ZGNR was revealed to fundamentally result from a topological confinement effect, while the halfmetallicity has been semiclassically described. Edge potentials were revealed to allow an infinitesimal staggered bulk potential to result in gapless edge states. A uniform or staggered potential applied to the boundary region narrower than a critical width was found to play a role of the edge potentials, and the critical width was estimated. Acknowledgements This work was supported by the National Research Foundation of Korea (Project No. 2016R1D1A1A09917003, No. 2016R1D1A1B03931144, No. 2015M1A7A1A01002234, and No. NRF-2010-0027963). K.W.L. gratefully acknowledges a Korea University research grant. References [1] C.L. Kane, E.J. Mele, Z2 topological order and the quantum spin Hall effect,
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