# Functionalized Germanene as a Prototype of Large-Gap Two-Dimensional Topological Insulators

**ABSTRACT** We propose new 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 sizeable gaps under

achievable tensile strains. A unique mechanism is revealed to be responsible

for large topologically-nontrivial gap obtained: owing to the

functionalization, the $\sigma$ orbitals with stronger spin-orbit coupling

(SOC) dominate the states around the Fermi level, instead of original $\pi$

orbitals with weaker SOC; thereinto, the coupling of the $p_{xy}$ orbitals of

Ge and heavy halogens in forming the $\sigma$ 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

room temperature.

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arXiv:1401.4100v1 [cond-mat.mtrl-sci] 16 Jan 2014

Functionalized Germanene as a Prototype of Large-Gap Two-Dimensional Topological

Insulators

Chen Si1, Junwei Liu1, Yong Xu1,2, Jian Wu1, Bing-Lin Gu1,2and Wenhui Duan1,2∗

1Department of Physics and State Key Laboratory of Low-Dimensional Quantum Physics,

Tsinghua University, Beijing 100084, People’s Republic of China and

2Institue for Advanced Study, Tsinghua University,Beijing 100084, People’s Republic of China

(Dated: January 17, 2014)

We propose new 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

sizeable gaps under achievable tensile strains. A unique mechanism is revealed to be responsible

for large topologically-nontrivial gap obtained: owing 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 pxy orbitals 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

room temperature.

I.INTRODUCTION

Recent years have witnessed many breakthroughs in

the study of the topological insulators (TIs), a new class

of materials with a bulk band gap and topologically pro-

tected boundary states1,2. Based on TIs, many intriguing

phenomena, such as giant magneto-electric effects3and

the appearance of Majorana fermions4, are predicted,

which would result in new device paradigms for spin-

tronics and quantum computation. In particular, two-

dimensional (2D) TIs have some unique advantages over

three-dimensional (3D) TIs in some respects: all the scat-

terings of electrons are totally forbidden, leading to dis-

sipationless charge or spin current carried by edge states;

and the charge carriers can be easily controlled by gat-

ing. Although many materials are theoretically predicted

to be 2D TIs5–12, so far only the HgTe/CdTe13and

InAs/GaSb14quantum wells are verified by transport

experiments, which, however, still face particular chal-

lenges: very small bulk gap and incompatibility with con-

ventional semiconductor devices. Therefore search and

design of 2D TIs with larger gaps from the commonly

used materials is indispensable for their practical utiliza-

tion.

Graphene,withmany

mechanical15,16to electronic17,18, has made remarkable

progress in numerous applications. This has triggered

extensive research on other 2D materials, such as sil-

icene, germenene, tin monolayer, BN, MoS2 and many

others19–26. Among them, graphene and silicene could be

well produced22,27, however, their practical applications

as 2D TIs are substantially hindered by their extremely

small bulk gaps (10−3meV for graphene28and 1.55 meV

for silicene20). Germanene and tin monolayer have larger

topologically nontrivial gap20,23but have not been fab-

ricated experimentally yet. Very recently, germanane,

a one-atom-thick sheet of hydrogenated germanene with

the formula GeH, structurally similar to graphane, has

been synthesized successfully29. With the predicted high

superiorpropertiesfrom

mobility and easier integrability into the current electron-

ics industry29, it is considered as a promising new star

in the field of 2D nanomaterials30. At the same time,

the success of production of germanane has also stimu-

late the synthesis of its counterparts, such as halogenated

germanenes.

In this work, using first-principles calculations, we

investigate electronic and topological properties of sin-

gle 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 siz-

able gaps larger than 0.1 eV under tensile strains. We

clearly reveal the physical mechanism for such large topo-

logically 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 (SOC) within

σ orbitals opens large nontrivial gaps. Thereinto, the

coupling of the pxy orbitals of Ge and heavy halogens

in forming the σ orbitals plays a key role in the fur-

ther enlargement of the gaps in halogenated germanenes.

The Z2topological order is due to the s-p band inversion

at the Γ point driven by the external strain or different

chemical functionalizations.

II.MODELS AND METHODS

Our calculations are performed in the framework of

density functional theory with ab initio psudopotentials

and plane wave formalism as implemented in the Vienna

ab initio simulation package31. The Brillouin zone is in-

tegrated with a 18×18×1k mesh. The plane-wave cut-off

energy is set as 400 eV. The system is modeled by a sin-

gle hydrogenated or halogenated germanene layer and a

vacuum region more than 10˚ A thick to eliminate the spu-

rious interaction between neighbouring slabs. The struc-

Page 2

2

a1

a2

(a)

(b)

h

FIG. 1: (Color online) Top (a) and side (b) views of optimized

structure of GeI displaying the primitive cell with Bravais lat-

tice vectors a1 and a2 and the buckling of Ge plane h. Green

and magenta balls represent Ge and I atoms, respectively.

tures are relaxed until the remaining force acting on each

atom is less than 0.01 eV/˚ A within generalized gradient

approximation (GGA) with the Perdew-Burke-Ernzerhof

(PBE) functional. Because GGA usually underestimates

the band gap of germanide severely32, we then use Heyd-

Scuseria-Ernzerhof (HSE) screened Coulomb hybrid den-

sity functionals33to 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 crys-

tal structure, in good agreement with recent diffuse re-

flectance absorption spectroscopy measurement and the-

oretical calculation29.

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 form-

ing 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 SOC, a gap of 0.54 eV is opened at the Γ 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 “or-

der parameter”: ν = 0 characterizes a trivial phase, while

ν = 1 means a nontrivial phase. Following the method

proposed by Fu and Kane34, ν for GeI is calculated from

the parities of wave function at all time-reversal-invariant

momenta (ki), one Γ and three M points, as

δ(ki) =

N

?

n=1

ξi

2n,(−1)ν=

4?

i=1

δ(ki) = δ(Γ)δ(M)3,

(+)

( )

_

Г

3 М

_

+

_

_

+

_

+

+

+

_

_

_

+

+

_

+ +

+

_

+

_

_

(a)(b)

(c)

FIG. 2: (Color online) (a), (b) Band structures for GeI with-

out SOC (blue line) and with SOC (black line) with zooming

in the energy dispersion near the Fermi level. The red cir-

cles and green squares represent the weights of the Ge-s and

Ge-pxy character, respectively. (c) The parities of eleven oc-

cupied bands at Γ and three M points for GeI. The product of

the parities at each k point is given in brackets on the right.

where ξ = ±1 denotes parity eigenvalues and N is the

number of the occupied bands. Fig. 2(c) shows the pari-

ties of eleven occupied bands at Γ and M. It readily yields

ν = 1, indicating quantum spin Hall effect can be realized

in the single GeI layer.

For a 2D TI, a remarkable characteristic is an odd

number of Dirac-like edge states connecting the conduc-

tion and valence bands. Thus we have also checked exis-

tence of the edge states in GeI. We use an armchair GeI

nanoribbons with all the edge atoms passivated by hydro-

gen atoms to eliminate the dangling bonds. A large rib-

bon width of 9.3 nm is selected to avoid the interactions

between the two edges. Fig. 3(a) shows the calculated

electronic structure of GeI nanoribbon. One can clearly

see the topological edge states (red lines) that form a

single Dirac point at the Γ point. Fig. 3(b) displays

the real-space charge distribution of edge states at the

Γ 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 GeI

to be a 2D TI. Moreover, its large bulk gap, about 0.3

eV, could be very useful for the applications of topologi-

cal edge states in spintronic and computing technologies

at room temperature.

The topological properties of GeI is closely related to

the Ge-Ge bond strength, which is well confirmed by the

direct comparison with GeH. GeH shares a similar geo-

metric structure as GeI but has a smaller lattice constant

(see Table 1). It is a normal insulator with a trivial gap

of 1.60 eV (Fig. 4(a)), while tensile strain could drive

it into a TI phase displaying a nontrivial gap of 0.20 eV

at the Γ point and an indirect bulk gap of 0.13 eV (Fig.

4(b)). Figs. 4(c) and (d) show the band evolution at the

Γ point of GeH under SOC and strain. Our calculations

show that the states near EF are mainly contributed by

the s and pxyorbitals of Ge atoms, and thus we reason-

ably neglect other atomic orbitals in the following dis-

cussion. Firstly the chemical bonding of Ge-Ge makes

Page 3

3

(b)

(a)

FIG. 3: (Color online) (a) Electronic structure for armchair

GeI nanoribbons with the width of 9.3 nm. The helical edge

states (red lines) can be clearly seen around the Γ point dis-

persing in the bulk gap. (b) Real-space charge distribution of

edge states at Γ.

the s (pxy) orbital split into the bonding and antibond-

ing states, labeled with s+(p+

the superscripts + and − represent the parities of corre-

sponding states. Without strain, p+

and the trivial gap (EΓ

g) of the system (i.e., GeH) is just

the distance between them (Fig. 4(c)). Applying ten-

sile strain, with the Ge-Ge bonding strength weakened,

the splitting of s+and s−(∆s) is rapidly reduced, caus-

ing s−shifting below p+

xy(Fig. 4(d)). In this inverted

band structure, the s−is occupied, while the quadruply

degenerate p+

xyis half occupied if the SOC is turned off,

resulting in that EFstays at p+

comes a semimetal. In contrast, turning on SOC, p+

split into |p,±3/2? state with a total angular momentum

j = 3/2 and |p,±1/2? state with a total angular momen-

tum j = 1/2, thereby forming a nontrivial energy gap. It

is also noted similar to this strain effects external pressure

could induce band inversion and topological phase transi-

tion in some 3D systems35. From GeH to GeI, the differ-

ent functionalization introduces the variation of electron

density, which effectively induces a “quantum electronic

stress”36. Based on this concept, GeI should behave like

a tensilely strained GeH, having the inverted band struc-

ture with s−lower than p+

xy, as shown in Fig. 2(a) and

(b).

GeI, however, shows a much larger nontrivial gap at

Γ (EΓ

ng) than strained GeH: 0.54 eV for the former, 0.20

eV for the latter. In GeH, according to a microscopic

tight-binding model with similar basis and Hamiltonian

as Ref. 37, we get the splitting of σ orbital at Γ under

SOC (ξσ) which determines the size of EΓ

Fig. 4(c) and (d):

xy) and s−(p−

xy), where

xyis lower than s−,

xylevel and the system be-

xyis

ngas shown in

ξσ= (−3δ + 3λ +

?

9δ2+ 6δλ + 9λ2)/4(1)

where δ = Vppσ− Vppπ, Vppσ and Vppπ are hopping pa-

rameters corresponding to the σ and π bonds formed by

(a)

(b)

(e)

(c)

(d)

ξ ξσ σ

s

∆s

EF

s-

s+

∆s

EF

s

s-

s+

=ξ ξσ σ

FIG. 4: (Color online) (a), (b), (e) Band structures with SOC

for unstrained GeH, 12% strained GeH and germanene, re-

spectively. The splitting of σ orbital at Γ under SOC (ξσ),

≈ 0.2 eV. (c) and (d), schematic diagram of the evolution

of energy levels at Γ for GeH. Without strain, p+

than s−(c). After applying enough large strain, p+

are inverted (d). Under SOC, p+

|p,±1/2? states.

xyis lower

xyand s−

xyis split into |p,±3/2? and

3p orbitals; λ is the SOC coefficient (HSO=λ− →

It is noted that hydrogenation-induced corrugation has

little effect on ξσ and thus was ignored safely here. For

GeH the σ orbital at the Γ point consists entirely of the

Ge-pxy orbitals, 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:

L ·− →S )38.

ξσ= λ +λ2

3δ+ o[λ/δ]3≈ λ ≈ ξGe.(2)

From Eq. (2), we can see ξσ in GeH is of the order of

ξGe (0.196 eV39), in agreement with our DFT calcula-

tion (0.20 eV). Importantly, Eq. (2) also implies that ξσ

is almost independent of strain, thus the nontrivial gap

in the TI phase is found to almost keep constant with the

increase of strain. In GeI, the σ orbital at the Γ point is

derived from the hybridization of the Ge-pxy and I-pxy

orbitals. Here Eq. (1) still works but λ in it should be

the combination of Ge and I atomic SOC. Then the in-

troduction of large I atomic SOC ascribed to its heavy

Page 4

4

atomic mass further increases dramatically the magni-

tude of ξσin GeI. Thus now we could easily understand

why the nontrivial gap of GeI is much larger than that

of GeH.

Actually, Eqs. (1) and (2) also work for germanene,

silicene and graphene, where ξσis of the order of atomic

SOC of Ge, Si or C. Fig. 4(e) shows the band structure

of germanene. One can see its nontrivial gap is generated

by the splitting of π orbital at K under SOC (ξπ), ≈ 36

meV, though the splitting of σ orbital at Γ is much larger,

ξσ ≈ 0.2 eV. By comparing Fig. 4(a) with 4(e), it is

clearly found that an important role of adsorbed atoms

on germanene is to reduce the energy of π orbital at the

K point and induce the dominance of σ orbital at the Γ

point near the Fermi level. Thus we can use the larger

SOC within σ orbital to open a sizeable 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 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

plane20,40, regularly depositing heavy transition metals

(TMs) on the surface to hybridize the π orbital with the

d orbital of TMs41,42. Remarkably, our work provides

a new alternative to increase the nontrivial gap, i.e., by

making the orbitals (such as σ) with large effective SOC

dominate the states around the Fermi level.

Recent theoretical work shows that SnI, iodinated tin

monolayer, is also a 2D TI with a bulk gap of about 0.3

eV23. However, the origin of this large gap of SnI was not

clearly known. Similar to GeI, once the Sn mononlayer is

iodinated, the original Sn π orbital dominance near the

Fermi level is changed into σ orbital dominance, and then

the larger SOC with σ orbital introduces 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-pxy and I-pxy in forming 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 Γ point is about 2 : 1 while the ratio

of Ge-pxyto I-pxyis about 2 : 3.

We furtherinvestigate other functionalized

manenes (GeF, GeCl and GeBr), structural analogues

of GeI and GeH. Similar to GeH, all of them undergo

a phase transition from normal to topological insulators

under tensile strain. Table 1 summarizes their lattice

constants, Ge-Ge bond lengths, critical strains where

phase transitions occur, and the nontrivial gaps in their

TI phase.Due to the weaker Ge-Ge bond strengths

in GeF, GeCl and GeBr, their critical strains are quite

small, ≤ 3%, indicating the experimental feasibility. De-

pending on different hybridization levels of the pxy or-

bitals of Ge and different halogens in forming σ orbitals,

they show different bulk gaps, however, all of which are

ger-

TABLE I: The lattice constant(a) and Ge-Ge bond length

(dGe−Ge) at equilibrium, critical strain (εc) and strained lat-

tice constant(ac) where the topological phase transition oc-

curs, nontrivial gap at Γ (EΓ

GeH, GeF, GeCl, GeBr and GeI.

ng), indirect bulk gap (∆) for

system

a (˚ A)

dGe−Ge (˚ A)

εc

ac (˚ A)

EΓ

ng(eV)

∆ (eV)

GeH

4.09

2.47

10%

4.50

0.20

0.13

GeF

4.30

2.55

2%

4.39

0.21

0.13

GeCl

4.24

2.54

3%

4.37

0.21

0.13

GeBr

4.25

2.55

2%

4.34

0.27

0.18

GeI

4.32

2.59

0%

4.32

0.54

0.30

larger than 0.1 eV, ample for practical application at

room temperature. In addition, it is known that the non-

trivial 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 proper-

ties of functionalized germanenes shown in this work are

obtained for free-standing sheets, their nontrivial topolo-

gies would be quite robust when they are on the sub-

strate, because their band inversion occurs at the Γ point

rather than the K point and the full saturation of Ge pz

orbitals ensures a weak interaction with the substrate.

IV.CONCLUSIONS

In summary, based on first-principle calculations, we

have studied the band topologies in functionalized ger-

manene, including the recently synthesized germanene

and halogenated 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 sizeable gaps larger than 0.1 eV by applying ten-

sile strain. These large gaps are originated from strong

SOC within the σ orbitals, which is of the order of the Ge

atomic SOC in GeH and further magnified in halogenated

germanene due to the coupling between pxyorbitals of Ge

and heavy halogens in forming σ orbitals. The s-p band

inversion at the Γ point, as the physical origin for the

Z2 topological order, can be driven by different chemi-

cal functionalizations or the external strain. Our results

clearly demonstrate the potential for utilization of topo-

logical edge states of germanium films in low-power spin-

tronics devices at room temperature.

V.ACKNOWLEDGMENTS

We acknowledge the support of the Ministry of Science

and Technology of China (Grant Nos. 2011CB921901

and 2011CB606405), and the National Natural Science

Foundation of China (Grant No. 11334006).

Page 5

5

∗

Corresponding author. Email: dwh@phys.tsinghua.edu.cn

1M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045

(2010), and references therein.

2X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057

(2011), and references therein.

3X.-L. Qi, T. L. Hughes and S.-C. Zhang, Phys. Rev. B 78,

195424 (2008).

4L. Fu and C. L. Kane, Phys. Rev. Lett. 100, 096407 (2008).

5C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801

(2005).

6B. A. Bernevig, T. L. Hughes, and S. C. Zhang, Science

314, 1757 (2006).

7S. Murakami, Phys. Rev. Lett. 97, 236805 (2006).

8M. Wada, S. Murakami, F. Freimuth and G. Bihlmayer,

Phys. Rev. B 83, 121310 (2011).

9C. Liu, T. L. Hughes, X.-L. Qi, K. Wang, and S.-C. Zhang,

Phys. Rev. Lett. 100, 236601 (2008).

10D. Xiao, W. Zhu, Y. Ran, N. Nagaosa, and S. Okamoto,

Nature Commun. 2, 596 (2011).

11P. F. Zhang, Z. Liu, W. Duan, F. Liu, and J. Wu, Phys.

Rev. B 85, 201410 (2012).

12H. Weng, X. Dai, and Z. Fang, arXiv: 1309.7529 (2013); P.

Tang, B. Yan, W. Cao, S. Wu, C. Felser, W. Duan, arXiv:

1307.8054 (2013).

13M. K¨ onig, S. Wiedmann, C. Brune, A. Roth, H. Buhmann,

L. W. Molenkamp, X. L. Qi, and S. C. Zhang, Science 318,

766 (2007).

14I. Knez, R.-R. Du, and G. Sullivan, Phys. Rev. Lett. 107,

136603 (2011).

15C. Lee, X. Wei, J. W. Kysar, and J. Hone, Science 321,

385 (2008).

16C. Si, W. Duan, Z. Liu, and F. Liu, Phys. Rev. Lett. 109,

226802 (2012).

17D. S. L. Abergel, V. Apalkov, J. Berashevich, K. Ziegler,

and T. Chakraborty, Adv. Phys. 59, 261 (2010).

18C. 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, Phys. Rev. Lett. 100, 206802 (2008).

19S. Cahangirov, M. Topsakal, E. Akt¨ urk, H. Sahin and S.

Ciraci, Phys. Rev. Lett. 102, 236804 (2009).

20C.-C. Liu, W. Feng, and Y. Yao, Phys. Rev. Lett. 107,

076802 (2011).

21W.-F. Tsai, C.-Y. Huang, T.-R. Chang, H. Lin, H.-T. Jeng,

and A. Bansil, Nat. Commun. 4, 1500 (2013).

22P. 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).

23Y. Xu, B. Yan, H.-J. Zhang, J. Wang, G. Xu, P. Tang,

W. Duan, and S.-C. Zhang, Phys. Rev. Lett. 111, 136804

(2013).

24L. Song, L. Ci, H. Lu, P. B. Sorokin, C. Jin, J. Ni, A.

G. Kvashnin, 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).

25B. Radisavljevic, A. Radenovic, J. Brivio, V. Giacometti,

and A. Kis, Nat. Nanotech. 6, 147 (2011).

26S. 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, 2898 (2013).

27K. Novoselov, V. Fal, L. Colombo, P. Gellert, M. Schwab,

and K. Kim, Nature 490, 192 (2012).

28H. Min, J. E. Hill, N. A. Sinitsyn, B. R. Sahu, L. Kleinman

and A. H. MacDonald, Phys. Rev. B 74, 165310 (2006).

29E. Bianco, S. Butler, S. Jiang, O. D. Restrepo, W. Windl,

and J. E. Goldberger, ACS Nano 7, 4414 (2013).

30K. J. Koski and Y. Cui, ACS Nano, 7, 3739 (2013).

31G. Kresse and J. Furthm¨ uller, Comput. Mater. Sci. 6, 15

(1996).

32P. Broqvist, A. Alkauskas, and A. Pasquarello, Phys. Rev.

B, 78, 075203 (2008).

33J. Heyd, G. E. Scuseria, and M. Ernzerhof, J. Chem. Phys.

118, 8207 (2003); J. Heyd and G. E. Scuseria, ibid. 120,

7274 (2004).

34L. Fu and C. L. Kane, Phys. Rev. B 76, 045302 (2007).

35P. 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).

36H. Hu, M. Liu, Z. F. Wang, J. Zhu, D. Wu, H. Ding, Z.

Liu, and F. Liu, Phys. Rev. Lett. 109, 055501 (2012).

37A. H. Castro Neto and F. Guinea, Phys. Rev. Lett. 103,

026804 (2009).

38We analytically solved the eigenvalues and the correspond-

ing eigenstates at Γ for the tight binding model and ob-

tained 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 eas-

ily got ξσ = E1− E2 = (−3δ +3λ +√9δ2+ 6δλ + 9λ2)/4.

39C.-C. Liu, H. Jiang, and Y. Yao, Phys. Rev. B, 84, 195430

(2011).

40D. Huertas-Hernando, F. Guinea and A. Brataas, Phys.

Rev. B 74, 155426 (2006).

41J. Hu, J. Alicea, R. Wu, and M. Franz, Phys. Rev. Lett.

109, 266801 (2012).

42Y. Li, P. Tang, P. Chen, J. Wu, B.-L. Gu, Y. Fang, S. B.

Zhang, and W. Duan, Phys. Rev. B 87, 245127 (2013).