PHYSICAL REVIEW B 89, 115429 (2014)
Functionalized germanene as a prototype of large-gap two-dimensional topological insulators
Chen Si,1Junwei Liu,1Yong Xu,1,2Jian Wu,1Bing-Lin Gu,1,2and Wenhui Duan1,2,*
1Department of Physics and State Key Laboratory of Low-Dimensional Quantum Physics,
Tsinghua University, Beijing 100084, People’s Republic of China
2Institute for Advanced Study, Tsinghua University, Beijing 100084, People’s Republic of China
(Received 26 October 2013; revised manuscript received 10 March 2014; published 24 March 2014)
We propose two-dimensional (2D) topological insulators (TIs) in functionalized germanenes (GeX, X=H,
F, Cl, Br, or I) using first-principles calculations. We find GeI is a 2D TI with a bulk gap of about 0.3 eV,
while GeH, GeF, GeCl, and GeBr can be transformed into TIs with sizable gaps under achievable tensile strains.
A unique mechanism is revealed to be responsible for the large topologically nontrivial gap obtained: due to
the functionalization, the σ orbitals with stronger spin-orbit coupling (SOC) dominate the states around the
Fermi level, instead of original π orbitals with weaker SOC. Thereinto, the coupling of the pxyorbitals of Ge
and heavy halogens in forming the σ orbitals also plays a key role in the further enlargement of the gaps in
halogenated germanenes. Our results suggest a realistic possibility for the utilization of topological effects at
DOI: 10.1103/PhysRevB.89.115429 PACS number(s): 73.43.Cd,73.43.Nq,73.22.−f,73.61.−r
Recent years have witnessed many breakthroughs in the
study of topological insulators (TIs), a new class of materials
with a bulk band gap and topologically protected boundary
states [1,2]. Based on TIs, many intriguing phenomena, such
as giant magnetoelectric effects  and the appearance of Ma-
jorana fermions , are predicted, which would result in new
device paradigms for spintronics and quantum computation.
In particular, two-dimensional (2D) TIs have some unique
advantages over three-dimensional (3D) TIs in some respects:
all the scatterings of electrons are totally forbidden, leading to
dissipationless charge or spin current carried by edge states,
and the charge carriers can be easily controlled by gating.
Although many materials are theoretically predicted to be 2D
quantum wells are verified by transport experiments, which,
however, still face some challenges: very small bulk gap
and incompatibility with conventional semiconductor devices.
Therefore search and design of 2D TIs with larger gaps
from the commonly used materials is indispensable for their
Graphene, with many superior properties from mechanical
[15,16] to electronic [17,18], has made remarkable progress in
numerous applications. This has triggered extensive research
on other 2D materials, such as silicene, germanene, tin
monolayer, boron nitride layers, MoS2, and many others
[19–26]. Among them, graphene and silicene could be well
produced [22,27]; however, their practical applications as
2D TIs are substantially hindered by their extremely small
bulk gaps (10−3meV for graphene  and 1.55 meV
for silicene ). Germanene and tin monolayer have a
larger topologically nontrivial gap [20,23] but have not been
fabricated experimentally yet. Very recently, germanane, a
one-atom-thick sheet of hydrogenated germanene with the
formula GeH, structurally similar to graphane, has been
*Corresponding author: firstname.lastname@example.org
and easier integrability into the current electronics industry
, it is considered as a promising new star in the field
of 2D nanomaterials . At the same time, the success of
production of germanane has also stimulate the synthesis of
its counterparts, such as halogenated germanenes.
In this work, using first-principles calculations, we inves-
tigate the electronic and topological properties of a single
layer of hydrogenated/halogenated germanene (GeH, GeF,
GeCl, GeBr, and GeI). We find GeI is a 2D TI with an
extraordinarily large bulk gap of about 0.3 eV, and GeH, GeF,
GeCl, and GeBr are trivial insulators but can be driven into
nontrivial topological phases with sizable gaps larger than
0.1 eV under tensile strains. We clearly reveal the physical
mechanism for such large topologically nontrivial gaps: due
to the functionalization of germanene, the σ orbitals dominate
the electronic states near the Fermi level, instead of original
π orbitals, and consequently, the strong spin-orbit coupling
this, the coupling of the pxyorbitals of Ge and heavy halogens
in forming the σ orbitals plays a key role in the further
enlargement of the gaps in halogenated germanenes. The Z2
topological order is due to the s-p band inversion at the
? point driven by the external strain or different chemical
II. MODELS AND METHODS
Our calculations are performed within the framework of
density functional theory with ab initio pseudopotentials and
plane-wave formalism as implemented in the Vienna ab
initio simulation package (VASP) . The Brillouin zone is
integrated with a 18 × 18 × 1 k mesh. The plane-wave cut-off
energy is set as 400 eV. The system is modeled by a single
hydrogenated or halogenated germanene layer and a vacuum
region more than 10˚A thick to eliminate the spurious interac-
within the generalized gradient approximation (GGA) with
the Perdew-Burke-Ernzerhof (PBE) functional. Because GGA
usually underestimates the band gap of germanide severely
1098-0121/2014/89(11)/115429(5)115429-1 ©2014 American Physical Society
SI, LIU, XU, WU, GU, AND DUANPHYSICAL REVIEW B 89, 115429 (2014)
FIG. 1. (Color online) Top (a) and side (b) views of an optimized
structure of GeI displaying the primitive cell with Bravais lattice
vectors a1and a2and the buckling of Ge plane h. Green and magenta
balls represent Ge and I atoms, respectively.
, we then use Heyd-Scuseria-Ernzerhof (HSE) screened
Coulombic hybrid density functionals  to calculate the
electronic structures and Z2topological invariant. The HSE
calculations yield a band gap of 1.5 eV for the bulk GeH
in a layered crystal structure, in good agreement with recent
diffuse reflectance absorption spectroscopy measurement and
theoretical calculation .
III. RESULTS AND DISCUSSION
Figure 1 shows the optimized 2D GeI lattice structure,
which is a fully iodinated germanene single layer. All the
germanium (Ge) atoms are in sp3hybridization forming a
hexagonal network, and the iodine (I) atoms are bonded to the
Ge atoms on both sides of the plane in an alternating manner.
The equilibrium lattice constant is 4.32˚A, with the buckling
of the germanium plane (h), the Ge-Ge and Ge-I bond length
being 0.69˚A, 2.59˚A, and 2.57˚A, respectively.
Without the SOC, GeI is gapless with the valence band
maximum and the conduction band minimum degenerate at
the Fermi level (EF), as shown in Fig. 2(a). Including the
FIG. 2. (Color online) (a,b)BandstructuresforGeIwithoutSOC
(blue line) and with SOC (black line) with zooming in the energy
dispersion near the Fermi level. The red circles and green squares
represent the weights of the Ge-s and Ge-pxycharacter, respectively.
(c) The parities of eleven occupied bands at ? and three M points for
GeI. The product of the parities at each k point is given in brackets
on the right.
SOC, a gap of 0.54 eV is opened at the ? point, along with an
indirect gap of 0.3 eV [Fig. 2(b)].
To identify the 2D TI phase, a topological invariant ν is
employed as “order parameter”: ν = 0 characterizes a trivial
phase, while ν = 1 means a nontrivial phase. Following the
method proposed by Fu and Kane , ν for GeI is calculated
momenta (ki), one ? and three M points, as
δ(ki) = δ(?)δ(M)3,
whereξ = ±1denotesparityeigenvaluesandN isthenumber
of the occupied bands. Figure 2(c) shows the parities of eleven
occupied bands at ? and M. It readily yields ν = 1, indicating
a quantum spin Hall effect can be realized in the single GeI
Dirac-like edge states connecting the conduction and valence
bands. Thus we have also checked the existence of the edge
states in GeI. We use an armchair GeI nanoribbon with all
the edge atoms passivated by hydrogen atoms to eliminate the
dangling bonds. A large ribbon width of 9.3 nm is selected to
the calculated electronic structure of the GeI nanoribbon. One
can clearly see the topological edge states (red lines) that form
a single Dirac point at the ? point. Figure 3(b) displays the
real-space charge distribution of edge states at the ? point.
It is visualized that these states are located at the two edges
and distributed on not only Ge but also I atoms. The existence
of edge states further indicates that GeI is indeed a 2D TI.
computing technologies at room temperature.
The topological properties of GeI are closely related to
the Ge-Ge bond strength, which is well confirmed by the
direct comparison with GeH. GeH shares a similar geometric
structure as GeI but has a smaller equilibrium lattice constant
(see Table I). It is a normal insulator with a trivial gap of
1.60 eV [Fig. 4(a)], while tensile strain could drive it into
FIG. 3. (Color online) (a) Electronic structure for the armchair
GeI nanoribbon with a width of 9.3 nm. The helical edge states (red
lines) can be clearly seen around the ? point dispersing in the bulk
gap. (b) Real-space charge distribution of edge states at ?.
FUNCTIONALIZED GERMANENE AS A PROTOTYPE OF ...PHYSICAL REVIEW B 89, 115429 (2014)
at equilibrium, critical topological transition strain (εc) and corre-
sponding εc-strained lattice constant (ac), characteristic strain (εt)
and εt-strained lattice constant (at) where the second regime in the
TI phase appears, nontrivial gap at ? (E?
at the characteristic strain for GeH, GeF, GeCl, GeBr, and GeI.
ng) and indirect bulk gap (?)
System GeHGeFGeClGeBr GeI
a TI phase, displaying a nontrivial gap of 0.20 eV at the
? point and an indirect bulk gap of 0.13 eV [Fig. 4(b)].
Figures 4(c) and 4(d) show the band evolution at the ? point
of GeH under SOC and strain. Our calculations show that
the states near EF are mainly contributed by the s and pxy
ξ ξσ σ
=ξ ξσ σ
FIG. 4. (Color online) (a, b, e) Band structures with SOC for
unstrained GeH, 12% strained GeH, and germanene, respectively.
Splitting of the σ orbital at ? under SOC (ξσ), ≈0.2 eV. (c, d)
Schematic diagram of the evolution of energy levels at ? for GeH.
Without strain, p+
|p, ± 3/2? and |p, ± 1/2? states.
xyis lower than s−(c). After applying enough large
xyand s−are inverted (d). Under SOC, p+
xyis split into
orbitals of Ge atoms, and thus we reasonably neglect other
atomic orbitals in the following discussion. First, the chemical
bonding of Ge-Ge makes the s (pxy) orbital split into the
bonding and antibonding states, labeled with s+(p+
of corresponding states. Without strain, p+
and the trivial gap (E?
distance between them [Fig. 4(c)]. Applying tensile strain,
with the Ge-Ge bonding strength weakened, the splitting of
s+and s−(?s) is rapidly reduced, finally causing s−shifting
occupied, while the quadruply degenerate p+
if the SOC is turned off, resulting in that EFstays at the p+
level and the system becomes a semimetal. In contrast, turning
on SOC, the p+
a total angular momentum j = 3/2 and the |p, ± 1/2? state
with a total angular momentum j = 1/2, thereby forming a
nontrivial energy gap. It is noted that similar to this strain
effect, external pressure could also induce band inversion and
topological phase transition in some 3D systems . From
of electron density, which effectively induces a “quantum
electronic stress” . Based on this concept, GeI should
behave like a tensilely strained GeH, having the inverted
band structure with s−lower than p+
than this strained GeH: 0.54 eV for the former [Fig. 2(b)],
and 0.20 eV for the latter [Fig. 4(b)]. In GeH, according to
a microscopic tight-binding model with a similar basis and
a Hamiltonian as in Ref. , we get the splitting of the σ
orbital at ? under SOC (ξσ), which determines the size of E?
as shown in Fig. 4(d):
where δ = Vppσ− Vppπ; Vppσand Vppπare hopping parame-
terscorrespondingtotheσ andπ bondsformedby3p orbitals;
and λ is the SOC coefficient (HSO= λ− →
that hydrogenation-induced corrugation has little effect on ξσ
and thus was safely ignored here. For GeH the σ orbital at the
? point consists entirely of the Ge-pxyorbitals, so λ could be
approximately equal to Ge atomic SOC, ξGe. Given that λ/δ is
small enough (≈0.02 for GeH), Eq. (1) is simplified based on
the Taylor expansion:
xy) and s−
xy), where the superscripts + and − represent the parities
xyis lower than s−,
g) of the system (i.e., GeH) is just the
xy[Fig. 4(d)]. In this inverted band structure, s−is
xyis half occupied
xystate is split into the |p, ± 3/2? state with
xy, as shown in Figs. 2(a)
ξσ= (−3δ + 3λ +
9δ2+ 6δλ + 9λ2)/4,
L ·− →
S ) . It is noted
ξσ= λ +λ2
3δ+ o[λ/δ]3≈ λ ≈ ξGe.
From Eq. (2), we can see ξσ in GeH is of the order of
ξGe(0.196 eV ), in agreement with our DFT calculation
(0.20 eV). Importantly, Eq. (2) also implies that ξσis almost
independent of strain; thus the nontrivial gap in the TI phase,
which has inverted band structure as shown in Fig. 4(d), is
found to almost keep constant with the increase of strain.
In GeI, the σ orbital at the ? point is derived from the
SOC. Then the introduction of large I atomic SOC ascribed
to its heavy atomic mass further increases dramatically the
SI, LIU, XU, WU, GU, AND DUANPHYSICAL REVIEW B 89, 115429 (2014)
magnitude of ξσin GeI. Thus we now can easily understand
why the nontrivial gap of GeI is much larger than that of GeH.
or C. Figure 4(e) shows the band structure of germanene. One
can see its nontrivial gap is generated by the splitting of the π
orbital at K under SOC (ξπ), ≈36 meV, though the splitting of
the σ orbital at ? is much larger, ξσ≈ 0.2 eV. By comparing
Fig. 4(a) with Fig. 4(e), it is clearly found that an important
role of adsorbed atoms on germanene is to reduce the energy
of the π orbital at the K point and induce the dominance of the
σ orbital at the ? point near the Fermi level. Thus we can use
the larger SOC within the σ orbital to open a sizable gap.
For many 2D materials like germanene, silicene, and
graphene, the states around the Fermi level are generally
contributed by π orbitals. In order to obtain the TIs with a
visible topologically nontrivial gap, a conventional method is
to increase the weak SOC of π orbitals, such as applying
compressive strain to increase the curvature of the plane
[20,40], regularly depositing heavy transition metals (TMs)
on the surface to hybridize the π orbital with the d orbital of
TMs [41,42]. Remarkably, our work provides an alternative to
increase the nontrivial gap, i.e., by making the orbitals (such
as σ) with large effective SOC dominate the states around the
Recent theoretical work shows that SnI, iodinated tin
However, the origin of this large gap of SnI was not clearly
known. Similar to GeI, once the Sn monolayer is iodinated,
the original Sn-π orbital dominance near the Fermi level is
within the σ orbital introduces a larger nontrivial gap. Given
that Sn has a much larger atomic SOC than Ge, it is supposed
that SnI would have a larger nontrivial gap than GeI. However,
we observe that the bulk gap of GeI (0.3 eV) is unexpectedly
comparable with that of SnI. We further find it is because the
hybridization of Sn-pxyand I-pxyin forming the σ orbital is
much smaller than that of Ge-pxyand I-pxy. A simple orbital
analysis indicates that the ratio of Sn-pxyto I-pxycomponent
in the σ orbital at the ? point is about 2 : 1, while the ratio of
Ge-pxyto I-pxyis about 2 : 3.
We further studied other functionalized germanenes (GeF,
GeCl, and GeBr), structural analogs of GeI and GeH. Similar
to GeH, all of them undergo a phase transition from normal to
topological insulators under tensile strain. From the energy-
level evolution of GeH shown in Figs. 4(c) and 4(d), we
can know that the band inversion and topological phase
transition would occur once the s−state is lower than the
|p, ± 3/2?state.So,forGeH,GeF,GeCl,andGeBr,thecritical
topological transition strains (εc) are defined when s−touches
|p, ± 3/2?,andthedataofεcaresummarizedinTableI.TheTI
shift of energy levels under strain. In the first regime, s−falls
in between |p, ± 3/2? and |p, ± 1/2?, and the direct energy
gap at ? equals the energy interval of s−and |p, ± 3/2?. In
the second regimes, as shown in Fig. 4(d), s−is lower than
|p, ± 3/2? and the direct energy gap at ? is determined by
the energy intervals of |p, ± 3/2? and |p, ± 1/2?, i.e., ξσ.
Unstrained GeI was in the second regime. For GeH, GeCl, and
GeBr, when s−touches |p, ± 1/2? we define a characteristic
strain (εt) to mark where transition between the two regimes
occurs. At the characteristic strain, the indirect bulk gap (?)
is also given in Table I. It is noticed that εcfor GeF, GeCl,
and GeI is quite small, ?3%, due to their comparatively weak
Ge-Ge bond strength, indicating the experimental feasibility.
Depending on different hybridization levels of the pxyorbitals
of Ge and different adsorbed atoms in forming σ orbitals, ?
values for them are different, but they are larger than 0.1 eV,
ample for practical application at room temperature.
It is known that the nontrivial topologies of graphene,
silicene, and germanene are easily destroyed by the substrate,
which breaks their AB sublattice symmetry and introduces
the trivial gap at the K point. In contrast, although all the
topological properties of functionalized germanenes shown
in this work are obtained for freestanding sheets, their
nontrivial topologies would be quite robust when they are
on the substrate, because their band inversion occurs at the ?
point rather than the K point, and the full saturation of Ge pz
orbitals ensures a weak interaction with the substrate.
In addition, we also investigated CX, SiX, and PbX (X =
to the graphene, silicene, and silicenelike Pb monolayer 
them, PbH, PbF, PbCl, PbBr, and PbI are also 2D TIs with
a nontrivial topological invariant (ν = 1) and helical edge
states, while the others are not. Similar to GeI, on the one
hand, PbX shows s-p band inversion at the ? point, which
is the origin of the quantum spin Hall effect in these kinds
of systems. On the other hand, functionalization of the Pb
monolayer also induces the σ-orbital dominance near the
Fermi level rather than the original π-orbital dominance,
and thus the strong SOC within the σ orbital opens a large
nontrivial gap. Most remarkably, due to the strong atomic
SOC of Pb, this nontrivial gap in PbX is extremely large:
0.98, 0.96, 0.90, 0.85, and 0.70 eV for PbH, PbF, PbCl, PbBr,
and PbI, respectively. However, different from GeX and SnX,
which have stable phonon modes with all phonon frequencies
positive, PbX shows dynamical instability with soft phonon
In summary, based on first-principle calculations, we have
studied the band topologies in functionalized germanene,
germanenes. Among them, GeI is found to be a promising 2D
TI with a very large gap of about 0.3 eV, while the others
could be transformed into TIs with sizable gaps larger than
0.1 eV by applying tensile strain. These large gaps originate
from a strong SOC within the σ orbitals, which is of the
order of the Ge atomic SOC in GeH and is further magnified
in halogenated germanene due to the coupling between pxy
orbitals of Ge and heavy halogens in forming σ orbitals. The
s-p band inversion at the ? point, as the physical origin for
the Z2topological order, can be driven by different chemical
functionalizations or the external strain. Our results clearly
demonstrate the potential for utilization of topological edge
states of germanium films in low-power spintronics devices at
FUNCTIONALIZED GERMANENE AS A PROTOTYPE OF ...PHYSICAL REVIEW B 89, 115429 (2014) Download full-text
We acknowledge the support of the Ministry of Science
of China (Grant No. 11334006).
 M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010),
and references therein.
 X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011),
and references therein.
 X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B 78,
 L. Fu and C. L. Kane, Phys. Rev. Lett. 100, 096407 (2008).
 C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801
 B. A. Bernevig, T. L. Hughes, and S. C. Zhang, Science 314,
 S. Murakami, Phys. Rev. Lett. 97, 236805 (2006).
 M. Wada, S. Murakami, F. Freimuth, and G. Bihlmayer, Phys.
Rev. B 83, 121310 (2011).
Rev. Lett. 100, 236601 (2008).
 D. Xiao, W. Zhu, Y. Ran, N. Nagaosa, and S. Okamoto, Nat.
Commun. 2, 596 (2011).
 P. F. Zhang, Z. Liu, W. Duan, F. Liu, and J. Wu, Phys. Rev. B
85, 201410 (2012).
 H. Weng, X. Dai, and Z. Fang, Phys. Rev. X 4, 011002 (2014);
P. Tang, B. Yan, W. Cao, S. C. Wu, C. Felser, and W. Duan,
Phys. Rev. B 89, 041409 (2014).
 M. K¨ onig, S. Wiedmann, C. Brune, A. Roth, H. Buhmann,
L. W. Molenkamp, X. L. Qi, and S. C. Zhang, Science 318,
 C. Lee, X. Wei, J. W. Kysar, and J. Hone, Science 321, 385
 C. Si, W. Duan, Z. Liu, and F. Liu, Phys. Rev. Lett. 109, 226802
 D. S. L. Abergel, V. Apalkov, J. Berashevich, K. Ziegler, and
T. Chakraborty, Adv. Phys. 59, 261 (2010).
 C. Si, Z. Liu, W. Duan, and F. Liu, Phys. Rev. Lett. 111, 196802
(2013); Z. Li, H. Qian, J. Wu, B.-L. Gu, and W. Duan, ibid. 100,
 S. Cahangirov, M. Topsakal, E. Akt¨ urk, H. Sahin, and S. Ciraci,
Phys. Rev. Lett. 102, 236804 (2009).
 C.-C. Liu, W. Feng, and Y. Yao, Phys. Rev. Lett. 107, 076802
 W.-F. Tsai, C.-Y. Huang, T.-R. Chang, H. Lin, H.-T. Jeng, and
A. Bansil, Nat. Commun. 4, 1500 (2013).
 P. Vogt, P. De Padova, C. Quaresima, J. Avila, E. Frantzeskakis,
M. C. Asensio, A. Resta, B. Ealet, and G. Le Lay, Phys. Rev.
Lett. 108, 155501 (2012).
 Y. Xu, B. Yan, H.-J. Zhang, J. Wang, G. Xu, P. Tang, W. Duan,
and S.-C. Zhang, Phys. Rev. Lett. 111, 136804 (2013).
D. G. Kvashnin, J. Lou, B. I. Yakobson, and P. M. Ajayan, Nano
Lett. 10, 3209 (2010); W. Lei, D. Portehault, D. Liu, S. Qin, and
Y. Chen, Nature Commun. 4, 1777 (2013).
 B. Radisavljevic, A. Radenovic, J. Brivio, V. Giacometti, and
A. Kis, Nat. Nanotechnol. 6, 147 (2011).
 S. Z. Butler, S. M. Hollen, L. Cao, Y. Cui, J. A. Gupta, H. R.
Guti´ errez, T. F. Heinz, S. S. Hong, J. Huang, A. F. Ismach, E.
Johnston-Halperin, M. Kuno, V. V. Plashnitsa, R. D. Robinson,
R. S. Ruoff, S. Salahuddin, J. Shan, L. Shi, M. G. Spencer,
M. Terrones, W. Windl, and J. E. Goldberger, ACS Nano 7,
 K. Novoselov, V. Fal, L. Colombo, P. Gellert, M. Schwab, and
K. Kim, Nature (London) 490, 192 (2012).
 H. Min, J. E. Hill, N. A. Sinitsyn, B. R. Sahu, L. Kleinman, and
A. H. MacDonald, Phys. Rev. B 74, 165310 (2006).
 E. Bianco, S. Butler, S. Jiang, O. D. Restrepo, W. Windl, and
J. E. Goldberger, ACS Nano 7, 4414 (2013).
 K. J. Koski and Y. Cui, ACS Nano 7, 3739 (2013).
 G. Kresse and J. Furthm¨ uller, Comput. Mater. Sci. 6, 15 (1996).
 P. Broqvist, A. Alkauskas, and A. Pasquarello, Phys. Rev. B 78,
 J. Heyd, G. E. Scuseria, and M. Ernzerhof, J. Chem. Phys. 118,
 L. Fu and C. L. Kane, Phys. Rev. B 76, 045302 (2007).
 P. Barone, T. Rauch, D. Di Sante, J. Henk, I. Mertig, and
S. Picozzi, Phys. Rev. B 88, 045207 (2013); H. Lin, L. A. Wray,
Y. Xia, S. Xu, S. Jia, R. J. Cava, A. Bansil, and M. Z. Hasan,
Nat. Mater. 9, 546 (2010).
 H. Hu, M. Liu, Z. F. Wang, J. Zhu, D. Wu, H. Ding, Z. Liu, and
F. Liu, Phys. Rev. Lett. 109, 055501 (2012).
 A. H. Castro Neto and F. Guinea, Phys. Rev. Lett. 103, 026804
 We analytically solved the eigenvalues and the corre-
sponding eigenstates at ? for the tight-binding model
and obtained the eigenvalues of |p, ± 3/2? and |p, ±
1/2? to be E1= (−3δ + λ − 6Vppπ)/2 and E1= (−3δ − λ −
√9δ2+ 6δλ + 9λ2− 12Vppπ)/4, respectively. Then we easily
got ξσ= E1− E2= (−3δ + 3λ +√9δ2+ 6δλ + 9λ2)/4.
 D. Huertas-Hernando, F. Guinea, and A. Brataas, Phys. Rev. B
74, 155426 (2006).
 J. Hu, J. Alicea, R. Wu, and M. Franz, Phys. Rev. Lett. 109,
 Y. Li, P. Tang, P. Chen, J. Wu, B.-L. Gu, Y. Fang, S. B. Zhang,
and W. Duan, Phys. Rev. B 87, 245127 (2013); Y. Li, P. Chen,
G. Zhou, J. Li, J. Wu, B.-L. Gu, S. B. Zhang, and W. Duan,
Phys. Rev. Lett. 109, 206802 (2012).