Letter pubs.acs.org/NanoLett
Giant Topological Nontrivial Band Gaps in Chloridized Gallium Bismuthide Linyang Li, Xiaoming Zhang, Xin Chen, and Mingwen Zhao* School of Physics and State Key Laboratory of Crystal Materials, Shandong University, Jinan, Shandong 250100, China S Supporting Information *
ABSTRACT: Quantum spin Hall (QSH) effect is promising for achieving dissipationless transport devices but presently is achieved only at extremely low temperature. Searching for the large-gap QSH insulators with strong spin−orbit coupling (SOC) is the key to increase the operating temperature. We demonstrate theoretically that this can be solved in the chloridized gallium bismuthide (GaBiCl2) monolayer, which has nontrivial gaps of 0.95 eV at the Γ point, and 0.65 eV for bulk, as well as gapless edge states in the nanoribbon structures. The nontrivial gaps due to the band inversion and SOC are robust against external strain. The realization of the GaBiCl2 monolayer will be beneficial for achieving QSH effect and related applications at high temperatures. KEYWORDS: quantum spin Hall insulator, two-dimensional chloridized gallium bismuthide, gapless edge states, large band gap, band inversion, first-principles calculations
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atoms,16−20 such as bismuth (Bi), whose SOC is inherently strong, are expected to have large bulk gap. First-principles calculations indicate that the nontrivial bulk gap in the lowbuckled honeycomb structures of thallium bismuthide (TlBi) monolayer can be up to 0.56 eV.20 However, the atomic structure of TlBi resembles that of silicene or germanene9,10 and faces the same problem as these two lattices. The interactions between the undercoordinated atoms of the TlBi monolayer with substrates may destroy the topological nontrivial phase of TlBi.21 The hybridization of electronic states at the TlBi/substrate interface is also disadvantageous for the realization of QSH effect. A question naturally arises: can we simultaneously improve the stability and the nontrivial bulk gap of the group-III bismuthide monolayer via functionalization? In this contribution, we show that this can indeed be achieved in chloridized gallium bismuthide (GaBiCl2) monolayer. The chloridization removes the imaginary frequency modes from the phonon spectrum of GaBi monolayer. The bulk gap in the GaBiCl2 is greatly improved to 0.65 eV compared to 0.16 eV of pristine GaBi monolayer. This value is larger than that of TlBi monolayer. More importantly, the GaBiCl2 monolayer is free from under-coordinated atoms and thus expected to be chemically stable. The topological nontrivial property of the bulk gap is confirmed by the appearance of gapless edge states in the GaBiCl2 nanoribbon. The nontrivial bulk gap is related to
opological insulators (TIs), as new quantum states of matter, have drawn quite intensive attention recently owing to the topological nontrivial bulk gap due to spin−orbital coupling (SOC) and gapless surface or edge states protected by time-reversal symmetry.1−4 The low-energy scattering of the edge states is prohibited by the time-reversal symmetry, leading to the dissipationless transport edge channels.5,6 The edge states in two-dimensional (2D) TIs, also known as quantum spin Hall (QSH) insulators, are more robust against backscattering than the surface states in three-dimensional (3D) TIs and thus are quite promising for such applications. Experimentally, QSH effect has been observed in HgTe/CdTe7 and InAs/GaSb8 quantum wells, but the operating temperature is quite low due to their small bulk gap arising from weak SOC. Therefore, searching for new QSH insulators with large bulk gap is crucial for practical applications. Graphene was first proposed as a QSH insulator, but the bulk gap is unobservable small (∼10−3meV),4 which limits the operating regime to unrealistically low temperature. Many 2D materials, such as silicene, germanene, stanene,9,10 and metal− organic frameworks11−13 have been proposed as alternates of graphene to achieve QSH effect at high temperature. The SOC strength in the materials is greatly enhanced compared with graphene and the bulk gaps are comparable or even higher than the thermal motion energy at room temperature (∼26 meV). Another QSH insulator family is halogenated germanene and stanene,14,15 which are predicted to have a large indirect bulk gap of about 0.3 eV around the Γ point. This opens a brand approach for searching for the large-gap QSH insulators. Additionally, the 2D QSH insulators containing heavy metal © XXXX American Chemical Society
Received: November 23, 2014 Revised: January 18, 2015
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the band inversion at the Γ point. Using a tight-binding (TB) model, we demonstrate that the band inversion of the GaBiCl2 monolayer is mainly due to the built-in electric filed induced by chloridization. The topological nontrivial phase of GaBiCl2 is robust against external strain. The realization of the GaBiCl2 will be beneficial for the applications in spintronics and quantum information devices. Methods. Our first-principles calculations were performed using the plane wave basis Vienna ab initio simulation package known as VASP code,22−24 implementing the density functional theory (DFT). The electron exchange-correlation functional was treated using a generalized gradient approximation (GGA) in the form proposed by Perdew, Burke, and Ernzerhof (PBE).25 Pseudopotentials including d valence electrons are used for Ga and Bi atoms. The atomic positions and lattice vectors were fully optimized using the conjugate gradient (CG) scheme until the maximum force on each atom was less than 0.01 eV/Å. The energy cutoff of the plane waves was set to 600 eV with the energy precision of 10−5 eV. For the 2D structures, the Brillouin zone (BZ) was sampled by using a 19 × 19 × 1 Gamma-centered Monkhorst−Pack grid, whereas a 1 × 11 × 1 grid was used for the nanoribbon. The vacuum space was set to at least 20 Å in all the calculations to minimize artificial interactions between neighboring slabs. SOC was included by a second variational procedure on a fully self-consistent basis. The phonon spectra were calculated using a supercell approach within the PHONON code.26 Results and Discussions. We start from a pristine GaBi monolayer whose atomic structure resembles those of silicene and germanene9,10 as shown in Figure 1a. Ga and Bi atoms
monolayer, which are disadvantageous for the realization and practical applications. Our phonon spectrum calculations clearly show that the GaBi monolayer has imaginary frequency modes, as shown in Figure 2a, and thus is dynamically unstable. Increasing the stability of GaBi monolayer is quite necessary.
Figure 2. Phonon spectra of (a) GaBi and (b) GaBiCl2 along the highsymmetric points in the BZ.
We then try to stabilize the GaBi monolayer by halogenation. Figure 1b gives the optimized structure of GaBiCl2 monolayer. In this configuration, each Ga (or Bi) atom is bonded to a Cl atom and three Bi (or Ga) atoms and thus fully coordinated as the cases of topical III−V compounds. The lattice constant and the Ga−Bi distance are stretched to 4.77 and 2.834 Å upon chloridization, whereas the buckling height is decreased to be 0.667 Å. The Ga−Cl and Bi−Cl distances are 2.194 and 2.454 Å, respectively. The electron localization function (ELF)27,28 profile of the GaBiCl2 monolayer indicates that Cl atoms are chemically bonded to the GaBi monolayer while the covalentlike Ga−Bi bonds are preserved, as shown in Figure 1c. Obviously, electron density redistribution takes place around Ga and Bi atoms, which facilitates the protection of topological nontrivial phase of the framework. The dynamical stability of the GaBiCl2 monolayer is confirmed by the phonon spectrum, as shown in Figure 2b. It is clear that the chloridization removes the imaginary frequency modes from the phonon spectrum of GaBi monolayer. We have also calculated the phonon spectra of other halogenated GaBi monolayers, such as GaBiY2 (Y = F, Br, and I). However, only the GaBiCl2 monolayer is free from imaginary frequency modes. The strong interactions between F and GaBi tend to destroy the framework of GaBi. For the GaBiBr2 and GaBiI2, the configuration with vertical Ga−Y and Bi−Y bonds converts to a more stable configuration with titled Ga−Y and Bi−Y bonds where the C3 rotation symmetry is lifted and the halogenated GaBi monolayers are trivial insulators and thus are not considered in the following parts. The stability of GaBiCl2 monolayer is related to the moderate interactions between Cl atoms and GaBi monolayer. The plausibility of GaBiCl2 monolayer can also be hinted by the low formation energy. We calculate the formation energy of the GaBiCl2 monolayer by the difference between the total energy of GaBiCl2 and the sum of the total energies of GaBi monolayer and Cl2 molecule. The formation energy is −1.45 eV/Cl atom. The negative formation confirms the superiority of GaBiCl2 over GaBi in stability and plausibility. Even so, the realization of GaBiCl2 monolayer remains challenging because gallium prefers to form binary alloy with bismuth. 29 Fortunately, both Ga and Bi atoms can form stable compounds with Cl atoms with the chemical formulas of Ga2Cl630 and
Figure 1. Schematic representations (top and side views) of (a) GaBi and (b) GaBiCl2 monolayers. (c) ELF profiles on the plane perpendicular to the basal plane of the monolayers.
occupy two sublattices with a buckling height (h) of 0.789 Å between them along the z-direction. The lattice constant and Ga−Bi distance are a = 4.53 Å and lGa−Bi= 2.732 Å. These results are in good agreement with those reported in the previous literature.20 However, it is noteworthy that both the Ga and Bi atoms are undercoordinated compared with the stable crystal structures of III−V compounds. This may lead to dynamical instability and chemical reactivity of the GaBi B
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Figure 3. Band structures of GaBi and GaBiCl2 with and without SOC obtained from DFT calculations. (a) GaBi without SOC, (b) GaBiCl2 without SOC, (c) GaBi with SOC, and (d) GaBiCl2 with SOC. The enlarged views of the band lines in proximity to the Fermi level around Γ point are shown in the right panel. The direct band gap at the Γ point is labeled as Eg(Γ) and the global indirect band gap is labeled as Eg.
BiCl3.31 Therefore, the polymerization of Ga2Cl6 and BiCl3 may give rise to the formation of GaBiCl2 compound through the Wurtz-like reaction with sodium metal nGa 2Cl 6 + 2n BiCl3 + 8n Na → [GaBiCl 2]2n + 8n NaCl
Our DFT calculations indicate that the reaction is exothermic with an energy release of 0.75 eV/atom. A similar scheme has been successfully employed to fabricate graphitic carbon nitride materials, where cyanuric chloride (CNCl)3 molecules are polymerized via the Wurtz-like reaction with sodium metal: 2n(CNCl)3 + 6 nNa → [C6N6]n + 6 nNaCl at the temperature of 250 °C,32−34 and the energy release is 0.79 eV/atom by DFT calculations. Both Ga2Cl6 and BiCl3 become gas molecules at the temperature higher than 447 °C as cyanuric chloride, and the two reactions have very close energy release, so the experimental conditions may be very similar. The electronic band structures of pristine GaBi monolayer and GaBiCl2 monolayer obtained from DFT calculations without SOC are plotted in Figure 3. The differences between the band structures of the two structures are quite obvious. For the pristine GaBi monolayer, there is a direct band gap of about 0.11 eV at the Γ point, as shown in Figure 3a, in good agreement with the results of previous work.20 The two valence bands closest to the Fermi level that are energetically degenerated at the Γ point arise mainly from the px and py atomic orbitals of Ga and Bi atoms (denoted as pxy band), whereas the lowest conduction band is contributed by the s atomic orbital of Ga and Bi atoms (denoted as s band) as shown in Figure 4a. The states of the pxy band at the Γ point have the E symmetry, whereas that of the s band has the A1 symmetry. The GaBiCl2, however, is a gapless semiconductor with the valence and conduction bands meeting in a single point at the Γ point, as shown in Figure 3b. The A1 state (s band) is lower in energy than the E states (pxy bands), as shown in Figure 4b. Such s−pxy band inversion has also been reported for the halogenated germanene and stanene.14,15 We then turn on the SOC in the band structure calculations. A direct band gap of Eg(Γ) = 0.95 eV is opened up at the Γ point in the GaBiCl2, as shown in Figure 3d. The valence band
Figure 4. Orbital-resolved band structures with and without SOC around the Γ point obtained from DFT calculations. (a) GaBi without SOC, (b) GaBiCl2 without SOC, (c) GaBi with SOC, and (d) GaBiCl2 with SOC. The red dots represent the contributions from the s atomic orbital of Ga and Bi atoms and the green dots represent contributions from the px and py atomic orbitals of Ga and Bi atoms.
maximum moves from the Γ point, leading to an indirect band gap of Eg = 0.65 eV. The pxy band splits to P3/2 and P1/2, and the P1/2 mixes with the S1/2, as shown in Figure 4d. The direct band gap at the Γ point and the indirect band gap in the GaBiCl2 are both much larger than the values of pristine GaBi which are about 0.19 eV (direct) and 0.16 eV (indirect) as shown in C
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Figure 3c. The s−pxy band inversion takes place in the pristine GaBi owning to SOC, as shown in Figure 4c. It is noteworthy that the spin degeneracy is lifted at the zone except the Γ point. Such spin splitting has also been found in GaAs quantum wells,35 where the lifting of spin degeneracy due to SOC leads to terms linear in electron wave vector k in the effective Hamiltonian.36 The origin of the linear terms in lowdimensional systems is the structure inversion asymmetry which leads to Rashba and Dresselhaus spin−orbital terms in the Hamiltonian.37,38 The spin splitting is crucial for the field of spintronics because it allows the electric field control of spin polarization, determines the spin relaxation rate, and can be utilized for all-electric spin injection.38 The existence of gapless edge states is one of the prominent features of QSH insulators. For simplification, armchair-edge nanoribbons are always adopted to reveal the existence of gapless edge states.14,15 Therefore, we consider a GaBiCl2 nanoribbon with armchair edges, as shown in Figure 5a. The
monolayer is chloridized, two effects are introduced: (1) Ga−Bi bond stretching and (2) electron density redistribution. To study the roles of the two factors in the band inversion separately, we compress the lattice constant of the GaBiCl2 monolayer to the value of pristine GaBi monolayer, 4.53 Å. The lGa−Bi (2.731 Å) and h (0.787 Å) of the compressed GaBiCl2(cGaBiCl2) are very close to the values of pristine GaBi monolayer (lGa−Bi = 2.732 Å and h = 0.789 Å) as well. The bond stretching effect is therefore suppressed in the c-GaBiCl2. The Cl atoms are also removed from the c-GaBiCl2 to exclude the electron density redistribution effect. Our DFT calculations of the both structures indicate that s−pxy band inversion does not take place in the GaBi without chloridization, but in the cGaBiCl2, as shown in Figure 6a and b. This clearly implies that the band inversion in the GaBiCl2 is unlikely due to the bond stretching effect but is closely related to the electron density redistribution.
Figure 5. (a) Schematic representations (top and side views) of the GaBiCl2 nanoribbon with armchair edges. L represents the width of the nanoribbon. The edge atoms are passivated by H atoms represented by the small blue balls. (b) Electronic band structure of the GaBiCl2 nanoribbon with L = 8.6 nm. The gapless edge states (red lines) can be clearly seen within the bulk gap.
Figure 6. Band structures without SOC obtained from DFT and TB calculations. (a) GaBi and (b) c-GaBiCl2. The red lines represent the results of DFT calculations and the blue dotted lines represent the results of TB calculations.
To further illustrate the band inversion mechanism in the GaBiCl2 monolayer, we propose a tight-binding (TB) model involving s, px, and py atomic orbitals of Ga and Bi atoms. The effective Hamiltonian without SOC is taken as
edge atoms (Ga and Bi) are passivated by hydrogen atoms to eliminate the dangling bonds.15,39 The width of the nanoribbon (8.6 nm) is large enough to avoid interactions between the edge states of the two sides. The band structure of the nanoribbon is shown in Figure 5b. We can see explicitly that the gapless edge states (red lines) appear in bulk gap and cross linearly at the Γ point, which proves the topological nontrivial property of the bulk gap. The Fermi velocity of the edge states at the Γ point is about 5.0 × 105 m/s, comparable to that of 5.5 × 105 m/s in HgTe/CdTe quantum well,2 both of which are larger than that of 3.0 × 104 m/s in InAs/GaSb quantum well.8 Gapless edge states have also been found in GaBiCl 2 nanoribbon with zigzag edges (see Part I of the Supporting Information). The gapless edge states with high Fermi velocity are very useful for electronics and spintronics owning to their robustness against scattering. The origin of the QSH effect in the GaBiCl2 monolayer is closely related to the inversion of s and pxy bands caused by chloridization, as shown in Figure 4a and b. When GaBi
HTB =
∑ εiαciα +ciα + ∑ i,α
εαi , cα+ i ,
tijαβ(ciα +c jβ + h. c . )
i,j ,α ,β
cαi
Here, and represent the on-site energy, creation, and annihilation operators of an electron at the α-orbital of the ith atom, respectively. To compare the cases with and without chloridization, the coordinates of the Ga and Bi atoms are set to the values of c-GaBiCl2. For the GaBi monolayer without chloridization, the on-site energies of s and p orbitals are set to εsGa = −12.00 eV, εpGa = −5.67 eV, εsBi = −17.68 eV, and εpBi = −8.30 eV.40 The tαβ ij parameter is the nearest-neighbor hopping energy of an electron between α-orbital of ith atom and βorbital of jth atom (α,β ∈ (s,px, py)), which can be determined by fitting the DFT data according to TB theory (see Part II of the Supporting Information). The TB model reproduces well D
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the three bands nearest to the Fermi level, especially the order of the s and pxy bands, as shown in Figure 6a. For the cGaBiCl2, chloridization leads to electron transfer from Ga to Bi atoms, which is equivalent to a vertical build-in electric field. In the TB model, this effect can be described by reducing the differences of on-site energies between Ga and Bi atoms. It is found that when the on-site energies of Ga are set to εsGa = −12.848 eV and εpGa = −6.518 eV, whereas other parameters are kept unchanged, band inversion takes place, as shown in Figure 6b. The TB band lines are in reasonable agreement with the DFT results. This implies that the build-in electric filed induced by chloridization is the main reason for the band inversion of the GaBiCl2 monolayer. The orbital-resolved band structures of the GaBi and c-GaBiCl2 presented in Part III of the Supporting Information show that these bands are contributed mainly by the s, px, and py atomic orbitals of Ga and Bi atoms. By involving a SOC term in the TB Hamiltonian, the SOC band gap and the band profile in the region near the Fermi level can be well reproduced (see Part IV of the Supporting Information). The electronic structures of 2D frameworks can be tuned by applying external strain.5,19,20,41−43 The stability of the bulk gap of GaBiCl2 under external strain is therefore quite crucial for device applications where strain is inevitable. Figure 7 gives the
Letter
ASSOCIATED CONTENT
S Supporting Information *
There are four parts (Part I−IV): Part I shows the gapless edge states of a GaBiCl2 nanoribbon with zigzag edges. Part II describes the details of TB model without SOC. Part III describes orbital projection band structures of GaBi and cGaBiCl2. Part IV gives the TB model involving SOC. These materials are available free of charge via the Internet at http:// pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS This work is supported by the National Basic Research Program of China (No.2012CB932302), the National Natural Science Foundation of China (No.91221101 and 21433006), the 111 project (no. B13209), the Taishan Scholar Program of Shandong, and the National Super Computing Centre in Jinan.
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Figure 7. Variation of the direct band gap Eg(Γ) at the Γ point and indirect band gap Eg as a function of external strain.
variation of the nontrivial gap (direct, Eg(Γ)) at Γ point and bulk gap (indirect, Eg) in response to the external strain in the range of ±10%. The s−pxy band inversion is preserved in the strain range, indicating that the topological nontrivial phase is robust against external strain. Both the direct and indirect band gaps are insensitive to the tensile strain but decrease slightly with the increase of compress strain, especially for the direct band gap. Those properties are beneficial to the synthesis of GaBiCl2 on different substrate.44 Conclusions. In summary, from first-principles, we demonstrate theoretically that the GaBiCl2 monolayer is a QSH insulator with high stability, a giant bulk gap up to 0.65 eV, and robust topological nontrivial phase under external strain. Chloridization can improve not only the stability but also the nontrivial bulk gap of GaBi monolayer. The topological nontrivial phase of the GaBiCl2 monolayer is related to the band inversion induced by the built-in electric field. Those properties are beneficial for achieving QSH effect at high temperature. The high Fermi velocity of the edge states in the GaBiCl2 nanoribbon is quite promising for high-speed spintronics device applications. E
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