Strain-Induced Large Band-Gap Topological Insulator in a New Stable Silicon Allotrope: Dumbbell Silicene

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(a)a1 (b) dumbbell silicene silicene (a) (b) 16 16 a1 14 14

12 12

2 10 10 a2 a 8 8

6 6

s = 4.29 Å s = 2.58Å Frequency(THz) Frequency(THz) 4 4 Si(α) h= 0.44Å 2 2 h= 2.71Å Si(β) Si 0 0 dumbbell silicene silicene Γ M K Γ Γ M K Γ

(c) 3 (c) (d) 4.46 5.94 7.43 8.92 10.40 (d) 3 b1 -2.5 2 2 dumbbell silicene M Γ -3.0 silicene 1 1 K -3.5 0 0 -4.0 Energy(eV) HB -1 Energy(eV) -1

E nergy (eV/atom) E nergy -4.5 L B

-5.0 b2 -2 -2 2.30 3.07 3.84 4.61 5.38 -3 -3 L attice constant (angstrom) M Γ K M M Γ K M

FIG. 1: (Color online) (a) Top and side view of the dumbbell FIG. 2: (Color online) Vibrational band structure ω(~k) of (a) silicene structure. (b) Top and side view of the silicene struc- the dumbbell silicene and (b) the silicene. The slope of the red ture. Pink dashed lines mark the hexagonal lattice; ~a1 and dashed lines along the longitudinal acoustic branches near Γ ~a2 are the lattice unit vectors. (c) The first Brillouin zone. corresponds to the speed of sound and the in-plane stiffness. ~b1 and ~b2 are the reciprocal lattice unit vectors. Γ, M and K Electronic band structure of (c) the dumbbell silicene and refer to special points in the first Brillouin zone. (d) Total (d) the silicene without SOC. Dirac cone is marked by black energies of the dumbbell silicene and silicene with different rectangular box. The green dots corresponding to Si-pz. The lattice constants. Fermi level is set to zero. anisotropic strain. It implies that the hexagonal symme- ditional, we calculated the phonon vibration spectra by try is the main precondition for the present of Dirac cone using the frozen phonon method as implemented in the in the dumbbell silicene. In additional, we further show PHONOPY code37. that the boron nitride (BN) substrate is suitable to sup- Crystal structure of the dumbbell silicene is presented port the dumbbell silicen under external strain, and the in Fig. 1(a). Different from silicene [see Fig. 1(b)], in the topologically nontrivial properties of the strained dumb- dumbbell silicene there are two types of Si atoms: one is bell silicene can be retained well in the realistic growth fourfold coordinated [denoted by Si(α)], and the other on the boron nitride (BN) substrate. one is only threefold coordinated with a dangling bond The first-principle calculations based on the density [denoted by Si(β)]. It is easy to find that all the Si(α) functional theory were performed using the Vienna ab atoms keep on the same plane, like in graphene, which simulation package (VASP)30,31. We use the general- is surrounded by the Si(β) atoms to form the dumbbell ized gradient approximations (GGA) of Perdew-Burke- structure. The optimized dumbbell silicene has a hexag- 32 Ernzerhof (PBE) for electron-electron interactions onal structure with space group D6h (P 6/mmm) and ten and the projector-augmented-wave (PAW)33 pseudo- Si atoms in one unit cell, which is different from the op- potentials in the plane-wave basis with an energy cutoff timized silicene with space group D3d (P 3m1) and only of 700 eV. The Brillouin zone (BZ) was sampled using two Si atoms in one unit cell. The buckling height of the an 13×13×1 gamma-centered Monkhorst-Pack grid34. A dumbbell silicene [h in Fig. 1(a) and (b)] about 2.71 A˚ is slab model, together with a vacuum layer larger than 30 much larger than that in the silicene about 0.44 A,˚ which A, was employed to avoid spurious interactions due to may result in the larger SOC strength for the dumbbell the nonlocal nature of the correlation energy35. The lat- silicene. The obtained lattice constant and the distance tice constants are relaxed until the residual force on each between the neighboring dumbbell units (s in Fig. 1(a)) atom is less than 0.01 eV/A˚ and the total energy is less for the dumbbell silicene are 7.43 A˚ and 4.29 A,˚ respec- than 1×10−6 eV. Spin-orbit coupling (SOC)36 is included tively. These results including those for the silicene are in in the calculations after the structural relaxations. In ad- well agreement with those reported in previous published 3 work27,29. To search for the energetically most stable configu- ration, we calculate the total energies of the dumbbell silicene and the silicene with different lattice constants [see Fig. 1(c)]. The minimum energy per Si atom for the dumbbell silicene is lower than that for the silicene, implying that the stability of the dumbbell silicene is over the silicene. This may arise mainly from fourfold- coordinated Si atoms because fourfold-coordinated Si is energetically more preferable. Two distinct min- ima of the silicene in Fig. 1(c) correspond to its low- buckled (LB) and high-buckled (HB) honeycomb structures, where the HB configure is unstable due to its imag- inary phonon frequency in a large portion of the Brillouin zone38. Hence, the mentioned silicene in our whole work refers to the LB silicene. Another way to compare the stability and structural rigidity of the different silicon allotropes is by studying the phonon vibration spectrum. Our results for the phonon vibration spectra of the dumbbell silicene and the silicene are presented in Fig. 2(a) and (b), respectively. The frequencies of all modes are positive over the whole Brillouin zone for the two silicon allotropes, which shows that the two lattice structures are thermodynamically stable and their stability do not FIG. 3: (Color online) Electronic band structure of the dumb- depend on the substrates. We compare the slopes of the bell silicene when σ=−3.0% (a) without SOC and (c) with longitudinal acoustic branches near Γ, which corresponds SOC, where the inset in (c) shows its enlarged views of elec- to the speed of sound and reveals the in-plane stiffness. tronic band structure around the Fermi level in the vicinity of the Γ point. Electronic band structure of the dumbbell As seen in Fig. 2(a), the calculated speed of sound along − Γ−M silicene when σ= 5.0% (b) without SOC and (d) with SOC, the Γ−M direction in the dumbbell silicene, υs =3.0 where the right-hand panels of (b) and (d) show their en- Γ−K Km/s, is lower than the υs =3.3 Km/s value along the larged views of electronic band structures around the Fermi Γ−K direction. The lower rigidity along the Γ−M direc- level in the vicinity of the Γ point. The green and blue dots tion, corresponding to the ~a1 + ~a2 direction in Fig. 1(a), corresponding to Si-pz and Si-px,y, respectively. (even, odd) indicates that compression along this direction requires parity is denoted by (+,−) and the Fermi level is set to zero. primarily bond bending, which comes at a lower energy cost than stretching. Different from the dumbbell silicene, the in-plane elastic response of the silicene is nearly makes a contribution to pz state, which binds covalently isotropic, with nearly the same value υs=2.3 Km/s for the with each other and develops into delocalized π and π∗ speed of sound along the Γ−M and the Γ−K directions, states. Such bands are responsible for Dirac points and as shown in Fig. 2(b). We can find that the dumbbell linear band dispersion near the Fermi level. However, the silicene has the larger in-plane stiffness, while the sil- Dirac cone of the dumbbell silicene may make less con- icene has the higher mechanical flexibility. These results tribution to the transportation, because its other nodes are advantageous when accommodating lattice mismatch throughout the band structure are parabolic instead of during Chemical Vapor Deposition (CVD) growth on a Dirac-type with linear dispersion. substrate. Previous theoretical and experimental studies have Our DFT results for the electronic band structures shown that the external strain is an excellent method to without SOC of the two different silicon allotropes are tune electronic structures of 2D materials. Our results presented in Fig. 2(c) and (d). Band structure informa- also prove that the band gap of the dumbbell silicene de- tion shows that the dumbbell silicene is a semiconductor pends sensitively on the applied in-layer isotropic strain with an indirect band gap about 0.24 eV, which is much σ, as shown in Fig. 3. The in-plane isotropic strain σ larger than almost zero band gap for the silicene. In ad- is loaded synchronously in the ~a1 and ~a2 directions, and ditional, it is worth noting that the dumbbell silicene also is defined as σ = (a1/2 − a10/20)/a10/20, where a1/2 owns a Dirac cone at K point around the Fermi level, and and a10/20 are equilibrium lattice constants with and more surprisingly, like in the silicene, the Dirac cone of without strain, respectively. Without SOC, the conduc- the dumbbell silicene also originates from the pz orbital tion band (CB) and the valence band (VB) at Γ point [see the black rectangular box in Fig. 2(c) and (d)]. In tend to approach together as σ increases, and compres- fact, the especial Dirac cone is closely related to the Si(β) sion beyond 5.0% shall turn the dumbbell silicene metal- atoms. Its three of four valence electrons form bonds lic. At Γ point, the top of VB is mainly contributed with neighboring silicon atoms while the remaining one by the antibonding px±iy orbitals with fourfold degen- 4

σ (a) 0.10 (b) -5.5% w/ soc eracy, and the bottom of CB is from the bonding pz 0.3 w/ow /o SSOCOC orbital. However, by introducing SOC, the fourfold de- 0.08 generate valence bands are split, as shown in Fig. 3(c). w/w / SOCSO C 0.2 0.06 metal Most remarkably, the closed band gap is reopened when 0.1 σ=−5.0% (σ < 0 means the compression strain), and a 0.04 2D topological 0 band inversion occurs between the px±iy and pz orbitals insulator 0.02 semiconductor with a 12 meV nontrivial band gap around Γ point, ac- -0.1 Energy(eV) B and gap (eVB gap and ) companied by the exchange of their parities, as illustrated 0.00 -0.2 in Fig. 3(d). The inverted states are labeled by |pzi and -0.02 4.0 4.4 4.8 5.2 5.6 |px±iyi, and (even, odd) parity is denoted by (+,−). Such -0.3 σ Γ band-inversion character may indicates that the topolog- Compression strain ( /% ) M/2 K/2 ical phase transition from a trivial state to a nontrivial topological state is happened for the dumbbell silicene (c) as σ increases. It is not strange that the Fermi level lies outside the bulk gap for the dumbbell silicene when σ=−5.0%, because we can artiﬁcially adjust the Fermi Ef level inside the gap by applying a gate voltage. Further, we apply a rigid method of Fu and Kane39 to judge whether or not the dumbbell silicene is a 2D topological insulator by calculating its Z2 number when σ =0 and −5.0%. Such method is valid since the dumb- σ σ bell silicene has both spatial invention and time reversal -5.0% σ 1 w/o soc -5.0% + -3% 1 w/ soc (d) 2 (e) 2 symmetries (four time reversal invariant points in the 2D Brillouin zone). Inversion center in the crystal ensures εn(k) =εn(−k), where εn(k) is the electron energy for 1 1 the n-th band with spin index at k wave vector in the Brillouin zone. The time reversal symmetry makes εn(k) = εnα¯(−k), whereα ¯ is the spin opposite to α. Hence, 0 0 we ﬁnd εn(k) = εnα¯(k) when both symmetries are pre- Energy(eV) sented, i.e. electronic bands acquire Kramers’ doubly Energy(eV) degenerate. We state a time-reversal invariant periodic -1 -1 Hamiltonian H with 2N occupied bands characterized by

Bloch wave functions. A time-reversal operator matrix -2 -2 Γ Γ relates time-reversed wave functions is defined by M K M M K M FIG. 4: (Color online) (a) Dependence of the fundamental band gap for the dumbbell silicene on the in-plane isotropic A (Γ )=<µ (Γ )|Θ|µ (Γ ) >, (1) αβ i α i β −i compression strain σ. Yellow, green and pink shaded regions indicate the characters of semiconductor, topological insula- where α, β =1, 2, ..., N, |µ (Γ ) > are cell periodic α i tor and metal for the dumbbell silicene under the isotropic eigenstates of the Bloch Hamiltonian, Θ=exp(iπSy)K is compression strain, respectively. (c) Schematic diagram of the time-reversal operator (Sy is spin and K complex the electronic band evolution of the dumbbell silicene with 2 conjugation ), which Θ =−1 for spin 1/2 particles. Since SOC under the isotropic compression strain for the orbitals around the Fermi level at the Γ point. Electronic band structure of the dumbbell silicene (b) under −5.5% σ with SOC, < Θµα(Γi)|Θµβ(Γi) >=<µβ(Γi)|µα(Γi) >, (2) (d) under −3.0% σ1 without SOC, and (e) under −5.0% σ + −3.0% σ1 with SOC. The green and blue dots corresponding A(Γi) is antisymmetric at TRIM Γi. The square of its to Si-pz and Si-px,y, respectively. The Fermi level is set to Pfaffian is equal to its determinant, i.e., det[A]=Pf[A]2. zero. 1/2 Then δi=(det[A(Γi)]) /Pf[A(Γi)]=±1. Therefore, the topological invariant Z2 can be defined as 0.0), (0.5, 0.0, 0.0) and (0.5, 0.5, 0.0)] contributes to a M +1 parity when σ=0, yielding a trivial topological in- Z2 variant Z2=0. However, as the strain is increased up to (−1) = Y ξ2m(Γi), (3) m=1 σ=−5.0%, the product of parities of occupied bands is −1 at Γ point but +1 at the three other time-reversal invari- where ξ is the parities of all occupied bands at Γi, and ant momenta. Hence, like in the silicene, the dumbbell M is the number of Kramers pairs. Results show that silicene also is a 2D topological insulator when the ex- the product of parities of occupied bands at the four ternal isotropic compression strain is beyond 5.0% with time reversal invariant points [(0.0, 0.0, 0.0),(0.0, 0.5, Z2=1. The evolution of the band gap at Γ point with the 5 isotropic compression strain σ is presented in Fig. 4(a). (a) (b) σ = 0.0% w/ soc We can find that the character of the dumbbell silicene 4 shall undergo a semiconductor, topological insulator to metal process with the increase of external isotropic compression strain σ. The dumbbell silicene can maintain Si 2 to be the 2D topological insulator under the isotropic B strain σ = −4.9 ∼−5.5%. In additional, the dependence N + of the band gap for the silicene with SOC on the ap- 0 _

plied isotropic strain σ is also analyzed. The band gap Energy(eV) of the silicene is not sensitive to σ. Its Dirac cone at K -2 point is protected by the crystal symmetry, and thus can hardly be eliminated by the isotropic strain σ. Under d= 3.16Å the isotropic strain σ = 0.0 ∼ −6.0%, the band gap of -4 the silicene only changes from 1.52 ∼ 1.56 meV, which is M Γ K M smaller than the maximum topological band gap about 12 meV of the dumbbell silicene when σ=−5.0%. Hence, σ = -6.2% w/ soc it is quite promising for the achievement of the QSH ef- (c) 4 fect of the dumbbell silicene at higher temperatures than 0.08 the silicene. 0.04 We ﬁnd that the incorporating band inversion in dumb- 2 bell silicene could be created solely by external isotropic 0 _ strain, even without considering the SOC, as shown in 0 Fig. 3(b). As the isotropic strain σ is increased up to -0.04 +_

−5.0%, the top of VB (|px±iyi state) and the bottom of Energy(eV) -0.08 CB (|pzi state) is inversion, and then the exchanged bot- -2 tom of VB (|pzi state) and the previous degenerate top -0.12 of VB (|px∓iyi state) is rapidly once again inversion due to the existence of the fourfold degenerate valence bands -4 Γ -0.16 at Γ point, which turns the dumbbell silicene metallic. M K M Γ Hence, it is impossible that the dumbbell silicene can FIG. 5: (Color online) (a) Top and side view of the dumbbell turn 2D topological insulator solely by external isotropic silicene structure on the (3×3) BN sheet. Pink dashed lines strain. SOC plays a vital role in lifting the band degen- mark the hexagonal lattice. d is the distance between the eracy for the valence bands at Γ point, and thus creating dumbbell silicene and substrate. Electronic band structure a gap at the crossing points originating from the band of the dumbbell silicene on the (3×3) BN sheet with SOC inversion. However, the SOC in Si is too weak, which when (b) σ=0 and (c) σ=−6.2%. The red and blue dots leads to a tiny energy difference between the |px±iyi and stand for the contribution from the BN sheet and Si-px,y, |px∓iyi states. As seen in Fig. 4(b) and (c), the ex- respectively. The right-hand panel of (c) show its enlarged changed |pzi state and the |px∓iyi state tend to approach view of electronic band structure around the Fermi level in together as the compression strain σ is increased, then ex- the vicinity of Γ point. (even, odd) parity is denoted by (+, − change secondly their orbital compositions, similar to the ) and the Fermi level is set to zero. case without SOC. Hence, we know that the weak SOC strength and the fourfold degenerate valence bands at Γ point are the main restricted factors for the dumbbell lattice structure under σ = −5.0% with the largest non- silicene to create larger topological band gap. trivial topological band gap, as shown in Fig. 4(e). The One restricted factor for the fourfold degenerate va- optimized lattice structure of the dumbbell silicene under lence bands of the dumbbell silicene can be solved by −5.0% σ and −3.0% σ1 still has both spatial invention tuning anisotropic strain. The calculated results for the and time reversal symmetries, and hence its topological dumbbell silicene under anisotropic strain σx are pre- property still can be judged by the previous method of sented in Fig. 4(d) and (e). The anisotropic strain σ1 Fu and Kane. We find that the character of 2D topo- is only along the ~a1 direction and is defined as σ1 = logical insulator for the dumbbell silicene is retained and (a1 − a10)/a10, where a1 and a10 are equilibrium lat- the nontrivial topological band gap is increased up to tice constants along the ~a1 direction with and without 36 meV, which is sufficiently larger to realize QSH ef- strain, respectively. Results show that the fourfold de- fect at room-temperature. Unfortunately, with the fur- generate valence band at Γ point for the dumbbell silicene ther increase of the anisotropic compression strain σ1, can be lifted solely by anisotropic strain σ1, as shown in the nontrivial topological band gap for the dumbbell sil- Fig. 4(d). It may create the larger topological band gap icene under σ = −5.0% is decreased. In additional, it than that only induced by σ. Hence, we apply an ex- is worth to noting that the Dirac cone for the dumbbell tra anisotropic strain σ1 about −3.0% on the previous silicene under the anisotropic strain σ1 are destroyed as 6 a result of spatial symmetry breaking, different from the the hexagonal BN sheet is a suitable substrate to support previous case under the isotropic strain σ, as shown in the dumbbell silicene, maintaining its nontrivial topology Fig. 4(e). It is strong verified that hexagonal symmetry under high compression strain. is the main precondition for the present of Dirac cone in In summary, based on first-principles calculations, we the dumbbell silicene. have proven that the dumbbell silicene is an excellent The in-plane strain on the dumbbell silicene can be 2D topological material when the isotropic compression realized by bending its flexure substrate in experiment, strain σ loaded synchronously in the ~a and ~a direc- similar to graphene40, where the amount of the strain 1 2 tions is larger than about 5.0%. The pristine structure is proportional to 2D mode position of the dumbbell of the dumbbell silicene is found to be an indirect band silicene. The coupling between the substrate and the gap semiconductor and is more mechanical stable than topological insulators deposited on them may destroy the previous silicene. Its electronic band can be inverted the topological nontrivially or increase the SOC band between the valence band maximum of p orbitals and gap. Hence, it is important to find a proper substrate x,y the conduction band maximum of p orbital at Γ point for the dumbbell silicene on which its exotic topological z by tuning isotropic compression strain σ, accompanied by properties under high external strain can be retained for the change of Z number from 0 to 1, and thus leads to future applications. Because the BN sheet has a close 2 quantum spin Hall (QSH) state of the dumbbell silicene. lattice structure and large band gap and high dielectric The obtained maximum topological nontrivial band gap constant, here we use it as a substrate to support DB about 12 meV under isotropic compression strain σ can stanene. Fig. 5(a) shows the geometrical structure of be further improved to 35 meV by tuning anisotropic the dumbbell silicene on the (3×3) BN sheet, where the strain σ , which is useful for the application of QSH ef- lattice mismatch is less than 1.5%. The dumbbell silicene 1 fect at room temperature and the fabrication of high- almost retains the original structure with lattice constant speed spintronics devices. A Dirac cone at K point orig- about 7.50 A˚ and a buckling height h about 2.69A.˚ The inated from p orbital is found in the pristine dumbbell distance between the adjacent layers [d in Fig. 3(b)] is z silicene, similar to previous silicene, but can be easily de- about 3.16 A.˚ Cohesive energy for the heterostructure stroyed as a result of spatial symmetry breaking under of the dumbbell silicene and the hexagonal BN sheet is anisotropic strain. It implies that the hexagonal symme- defined by E =E -E -E , where E and E coh total DS BN DS BN try is the main precondition for the present of Dirac cone are the total energy of single dumbbell silicene and single in the dumbbell silicene. In additional, we confirm that hexagonal BN sheet, respectively. The obtained cohesive the boron nitride (BN) substrate is suitable to support energy is only about −0.58 eV per atom, showing a weak the dumbbell silicen under external strain, and the topo- interaction between the dumbbell silicene and the BN logically nontrivial properties of the strained dumbbell sheet. The calculated band structure of the heterostruc- silicene are retained well in the realistic growth on the ture with SOC under σ=0 is shown in Fig. 5(b). We find boron nitride (BN) substrate. We expect that the previ- that the dumbbell silicene on BN sheet remains semicon- ous silicene can be replaced by the dumbbell silicene in ducting. There is essentially no charge transfer between modern silicon-based microelectronics industry, and the the adjacent layers, and the states around the Fermi level quantum anomalous Hall effect, Chern half metallicity, are dominantly contributed by the dumbbell silicene. As and topological superconductivity can also be realized in the isotropic strain is increased up to −6.2%, the top of the dumbbell silicene, which shall make supported dumb- VB and the bottom of CB for the dumbbell silicene on bell silicene an ideal platform to study quantum states of BN sheet are inversion, and thus open a small topologi- matter and show great potential for future applications. cal band gap [see the inset of Fig. 5(c)], accompanied by the exchange of their parities. Evidently, the dumbbell The authors would like to thank the support by the silicene on the (3×3) BN substrate also is a 2D topologi- NSAF Joint Fund Jointly set up by the National Natural cal insulator whose band inversion is not affected by the Science Foundation of China and the Chinese Academy of substrate. Although the required isotropic compression Engineering Physics (Grant Nos. U1430117, U1230201). strain to realize the quantum topological phase transi- Some calculations are performed on the ScGrid of Super- tion is increased for the dumbbell silicene on BN sheet computing Center, Computer Network Information Cen- due to the inevitable lattice mismatch, we still think that ter of Chinese Academy of Sciences.

∗ Electronic address: [email protected] naga, Nat. Nanotechnol. 9, 391 (2014). † Electronic address: [email protected] 5 L. Li et al., Nat. Nanotechnol. 9, 372 (2014). 1 J. Guan, Z. Zhu, and D. Tom´anek, Phys. Rev. Lett. 113, 6 S. Banerjee, G. Periyasamy, and S. K. Pati, J. Mater. 046804 (2014). Chem. A 2, 3856 (2014). 2 F. H. L. Koppens et al., Nat. Nanotechnol. 9, 780 (2014). 7 P. Y. Huang et al., Nano Lett. 12, 1081 (2012). 3 D. Jariwala et al., ACS Nano 8, 1102 (2014). 8 S. Zhao, W. Kang, and J. Xue, Appl. Phys. Lett. 104, 4 Y. -C. Lin, D. O. Dumcenco, Y. -S. Huang, and K. Sue- 113106 (2014). 7

9 C. Ataca, H. Sahin, E. Aktürk, and S. Ciraci, J. Phys. 26 Y. Yao, F. Ye, X. Qi, S. Zhang, and Z. Fang, Phys. Rev. Chem. C 115, 3934 (2011). B 75, 041401 (2007). 10 Eduardo. V. Castro et al., Phys. Rev. Lett. 99, 216802 27 X. -L. Zhang, L. -F. Liu, and W. -M. Liu, Scientific Reports (2007). 3, 02908 (2013). 11 G. Constantinescu, A. Kuc, and T. Heine, Phys. Rev. Lett. 28 C. Liu, W. Feng, and Y. Yao, Phys. Rev. B 107, 076802 111, 036104 (2013). (2011). 12 C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 95, 226801 29 S. Cahangirov et al., Phys. Rev. B 90, 035448 (2014). (2005). 30 G. Kresse, and J. Furthmuller, Phys. Rev. B 54, 11169 13 B. A. Bernevig, T. L. Hughes, and S. -C. Zhang, Science (1996). 314, 1757 (2006). 31 G. Kresse, and J. Furthmuller, Comput. Mater. Sci. 6, 15 14 T. Zhang, J. -H. Lin, Y. -M. Yu, X. -R. Chen, and W. -M. (1996). Liu, Scientific Reports 5, 13927 (2015). 32 J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. 15 W. Luo, and H. Xiang, Nano Lett. 15, 3230 (2015). Lett. 77, 3865 (1996). 16 F. -C. Chuang et al., Nano Lett. 14, 2505 (2014). 33 P. E. Blochl, Phys. Rev. B 50, 17953 (1994). 17 X. Qian, J. Liu, L. Fu, and J. Li, Science 346, 1344 (2014). 34 H. J. Monkhorst, and J. D. Pack, Phys. Rev. B 13, 5188 18 C. Niu, G. Bihlmayer, H. Zhang, D. Wortmann, S. Blugel, (1976). and Y. Mokrousov, Phys. Rev. B 91,041303 (2015). 35 T. Olsen, J. Yan, J. J. Mortensen, and K. S. Thygesen, 19 M. Z. Hasan, and C. L. Kane, Rev. Mod. Phys. 82, 3045 Phys. Rev. Lett. 107, 156401 (2011). (2010). 36 A. Togo, F. Oba, and I. Tanaka, Phys. Rev. B 78, 134106 20 X. -L. Qi, S. -C. Zhang, and C. L. Kane, Rev. Mod. Phys. (2008). 83, 1057 (2011). 37 K. Parlinski, Z. Q. Li, and Y. Kawazoe, Phys. Rev. Lett. 21 J. Zhang, B. Zhao, Y. Yao, and Z. Yang, Scientific Reports 78, 4063 (1997). 5, 10629 (2015). 38 S. Cahangirov, M. Topsakal, E. Aktürk, H. Sahin, and S. 22 A. M. Black-Schaffer, Nature 109, 197001 (2012). Ciraci, Phys. Rev. Lett. 102, 236804 (2009). 23 R. Nandkishore, L. S. Levitov, and A. V. Chubukov, Nat. 39 L. Fu, and C. L. Kane, Phys. Rev. B 76, 045302 (2007). Phys. 8, 2208 (2012). 40 T. Yu, Z. Ni, C. Du, Y. You, Y. Wang, and Z. Shen, J. 24 Y. Wu, et al., Nat. Phys. 472, 74 (2011). Phys. Chem. Lett. 112, 12602 (2008). 25 K. F. Mak, C. Lee, J. Hone, J. Shan, and T. F. Heinz, Phys. Rev. Lett. 105, 136805 (2010).