Two-Dimensional Topological Crystalline Insulator and Topological

Letter pubs.acs.org/NanoLett

Two-Dimensional Topological Crystalline Insulator and Topological Phase Transition in TlSe and TlS Monolayers Chengwang Niu,* Patrick M. Buhl, Gustav Bihlmayer, Daniel Wortmann, Stefan Blügel, and Yuriy Mokrousov Peter Grünberg Institut and Institute for Advanced Simulation, Forschungszentrum Jülich and JARA, 52425 Jülich, Germany ABSTRACT: The properties that distinguish topological crystalline insulator (TCI) and topological insulator (TI) rely on crystalline symmetry and time-reversal symmetry, respectively, which encodes different bulk and surface/edge properties. Here, we predict theoretically that electron-doped TlM (M = S and Se) (110) monolayers realize a family of two-dimensional (2D) TCIs characterized by mirror Chern number CM = −2. Remarkably, under uniaxial strain (≈ 1%), a topological phase transition between 2D TCI and 2D TI is revealed with the calculated spin Chern number CS = −1 for the 2D TI. Using spin-resolved edge states analysis, we show different edge-state behaviors, especially at the time reversal invariant points. Finally, a TlBiSe2/NaCl quantum well is proposed to realize an undoped 2D TCI with inverted gap as large as 0.37 eV, indicating the high possibility for roomtemperature observation. KEYWORDS: 2D topological crystalline insulator, 2D topological insulator, topological phase transition, edge states, mirror Chern number, spin Chern number

T

transition is helpful for both the future use and the fundamental understanding of the gapless edge states that are protected by different symmetries. In the present study, we report the identification of a phase transition between TCI and TI, based on a new family of 2D TCIs, that is, TlSe and TlS (110) monolayers, by means of the density functional theory calculations. The calculated mirror Chern number and spin Chern number for 2D TCI and 2D TI phases are respectively CM = −2 and CS = −1. The spin polarization analysis of edge states further confirms the phase transition. The density functional calculations are performed for structural relaxations and electronic structure calculations. The generalized gradient approximation (GGA) of Perdew− Burke−Ernzerhof (PBE)33 is used for the exchange correlation potential as implemented in the Vienna ab initio simulation package (VASP)34 and the FLEUR code.35 A vacuum layer of 20 Å is used to avoid interactions between nearest slabs for VASP while the film calculations are carried out with the film version of the FLEUR code. Spin−orbit coupling (SOC) is included in the calculations self-consistently. The maximally localized Wannier functions (MLWFs) are constructed using the wannier90 code.36,37 Because of the lack of a particular valence, Bi and Tl are wellknown valence skip elements with their compounds being fruitful to induce the high-Tc superconductivity.38,39 Recent ab initio calculations demonstrated that Bi-based compound BaBiO3 is also a candidate to realize the 3D TI.40 So, what

opological insulator (TI) is an unusual quantum state of matter with odd number of gapless surface or edge states inside the insulating bulk gap.1,2 The appearance of the metallic surface or edge states is ensured by the time-reversal symmetry that gives rise to the nontrivial 2 topological invariant.3 The unique properties and promising applications in spintronics and quantum computing have generated the explosion of interest in TIs,4−6 as well as other novel topologically nontrivial phases protected by other discrete symmetries.7 When the role of the time-reversal symmetry is replaced by the crystal mirror symmetry, the so-called topological crystalline insulator (TCI) can be realized.8 The TCI bears a variety of exotic topological phenomena such as the Dirac gap via crystal symmetry breaking,9,10 large Chern number quantum anomalous Hall effect,11 and straininduced superconductivity.12 While several three-dimensional (3D) TCIs have been predicted theoretically13−17 and synthesized experimentally,18−20 the material realization of two-dimensional (2D) TCIs is so far limited to theoretical prediction of IV−VI multilayers/monolayers21−24 and graphene multilayers.25 It is therefore essential to extend the domain of candidate 2D TCI materials as well as appropriate substrates that can support the growth of the 2D TCIs. The latter challenge is similar to that faced by 2D TIs where only HgTe/ CdTe and InAs/GaSb quantum wells are experimentally realized.26,27 Both the TCI and TI can be originated from a topologically trivial system through a topological phase transition by tuning the alloy composition18,19,28−30 or the crystal lattice.31,32 Therefore, a natural question arises as to whether the phase transition between TCI and TI is possible. This phase © XXXX American Chemical Society

Received: June 10, 2015 Revised: July 31, 2015

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DOI: 10.1021/acs.nanolett.5b02299 Nano Lett. XXXX, XXX, XXX−XXX

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Figure 1. (a) Crystal structure of bulk TlM (M = S and Se) with Pm3̅m space group. The shaded area depicts the (110) plane. (b) Top view of (110) TlM monolayer with bulk lattice parameters, where the unit cell is indicated by the dashed lines. The lattice changes between bulk and monolayer are marked by the arrows. (c,d) The Berry curvature distribution associated with − i and + i mirror eigenstates, respectively, of the occupied bands over the 2D Brillouin zone.

about the Tl-based compounds? Here, we focus on the naturally existing TlM (M = S and Se) alloys with the crystal structure shown in Figure 1a. TlM has the CsCl-type structure with Pm3̅m space group. Its (110) monolayer presents a rectangular lattice with two slightly different lattice parameters. Different from the (001) films of NaCl-type SnTe,21 the TlM (110) monolayer possesses only the 2-fold (C2) rotational symmetry, while not the 4-fold (C4) rotational symmetry. However, it is symmetric under the reflection z→ − z that supports to define the mirror Chern number. This (110) monolayer is the subject of our work. In (110) TlM monolayers, due to the Tl atom’s tendency to prefer the Tl1+ over the Tl2+ state, a band crosses the Fermi energy which lies below the inverted gap. It is similar to another valence skip element Bi-based TI, BaBiO3.40 Hereafter the Fermi level is tuned into the inverted gap by mimicking one electron doping for the convenience of discussion. The orbitally resolved band structures without and with SOC are plotted in Figure 2. As we can see, the band structures of TlS and TlSe are quite similar. In the absence of SOC, the lowest unoccupied bands are dominated by the Tl-pz orbital over the whole Brillouin zone, while the highest occupied bands around the X and Y points are dominated by the M-px and M-py orbitals, respectively. It is, such as TlSe, a direct-gap insulator with an energy gap of 0.15 eV (0.21 eV for TlS) at the Y point while the energy gap is 0.21 eV (0.25 eV for TlS) at the X point. When SOC is switched on, the orbital characters at both the X and Y points are inverted and a direct-gap of 0.12 eV (0.03 eV for TlS) appears at the X point. The even number of band inversions at the time reversal invariant points means that the (110) TlM monolayers can not be the 2D 2 TIs, while it could be the 2D TCIs owing to the presence of the mirror symmetry in combination with the band inversion taking place at the mirror plane z = 0. To confirm the 2D TCI phase, we calculate the Chern number of all occupied bands in the mirror plane with opposite mirror eigenvalues − i and + i41,42

Figure 2. Orbitally resolved band structures for the monolayers of TlSe (a,b) and TlS (c,d) without (a,c) and with (b,d) SOC, weighted with the S/Se-px, S/Se-py, and Tl-pz characters. The S/Se-px and S/Sepy orbitals contribute differently around the X and Y points. The Fermi level is shifted into the inverted energy gap.

C±i =

1 2π

∫BZ Ω(k)d2k

(1)

where Ω(k) is the Berry curvature of all occupied bands42 Ω(k) =

∑ ∑ 2 Im n < EF m ≠ n

⟨ψnk|υx|ψmk ⟩⟨ψmk|υy|ψnk ⟩ (εmk − εnk )2

(2)

where m, n are band indices, ψm/nk and εm/nk are the Bloch wave functions and corresponding eigenenergies of band m/n, respectively, and υx/y are the velocity operators. The Chern number and mirror Chern number are defined as C = C−i + C+i and CM = (C+i − C−i)/2, respectively. The MLWFs are constructed to calculate the Berry curvature. Figure 1c,d shows the 2D distribution of Berry curvature for all occupied bands B


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Figure 3. Variation of the energy gaps at X and Y points for TlSe monolayer versus the uniaxial strain along (a) a- and (d) b-axes. The negative values indicate the inverted energy gaps. For case with SOC, a topological phase transition occurs accompanied by a sign changes of the energy gap. Orbitally resolved band structures of the TI phase (b,e) without and (c,f) with SOC under 1.02 uniaxial strain along (b,c) a- and (e,f) b-axes are presented, respectively.

with mirror eigenvalues − i and + i, respectively. We see that, different from the SnTe multilayers,21 the regions of nonzero Berry curvature are localized slightly away from the X and Y points. The opposite sign of the Berry curvature for opposite i results in Chern numbers for each polarization C−i = 2 and C+i = −2. Thus, the Chern number is zero, but the mirror Chern number is CM = −2, indicating that the (110) TlM monolayers are 2D TCIs. As discussed above, the band inversion occurs at both the X and Y points. Thus, one might ask whether it is possible that the band inversion occurs only at the X or Y points, thus resulting in a phase transition from 2D TCI to 2D TI. Strain engineering is an effective way of modulating the electronic and topological properties in TCIs, such as the recently predicted flat band superconductivity by strain in IV−VI semiconductors.12 Here, we focus on the strain-induced phase transition in (110) TlSe monolayer. The magnitude of strain is described by a/a0 (b/b0), here a0 (b0) and a (b) denote the lattice parameters of the unstrained and strained systems, respectively. The calculated energy gaps at X and Y points without and with SOC versus the uniaxial strain are presented in Figure 3a,d. The energy gaps as well as the band topology can be effectively modified by uniaxial strain along both a- and b-axes. Under compressive strain, the band inversion takes place even without SOC, and the 2D TCI phase remains with inhanced gap when SOC is taken into account, indicating that the 2D TCI phase is robust against compressive strain. The normal energy gap increases with the tensile strain increasing at both the X and Y points for the case without SOC. Considering SOC, a band gap closing and reopening occurs individually at the X and Y points under different critical strains (vertical solid lines in Figure 3a,d). When the lattice parameter lies between the two critical values, the band inversion occurs only at the X points for a-axis and Y points for b-axis. The corresponding orbitally resolved band structures without and with SOC, as shown in Figures 3, reveal this band inversion clearly. To determine the band topology of these strain-induced 2D insulators, we calculate the spin Chern number,43 which generalizes the 2D 2 topological invariant even without the time-reversal symmetries, using CS = (C+ − C−)/2, where C+ and C− are the Chern numbers for the spin-up and spin-down manifolds of the occupied states. For both spins, the Chern numbers are respectively C± = ∓1, leading to a spin Chern

number CS = −1. Thus, a phase transition from 2D TCI to 2D TI can be effectively tuned by uniaxial strain. With further increasing strain, the band inversion disappears at both the X and Y points, and the system becomes a trivial insulator. Moreover, our results suggest that the relaxed monolayer is located very close to the boundary between the 2D TCI and the 2D TI (≈ 1.01), indicating an excellent opportunity to investigate the topological phase transition experimentally. To check the topological phase transition, we investigate the edge states of a 1D nanoribbon, since the existence of gapless edge states is an important signature of both the 2D TCIs and 2D TIs. The values of CM = −2 and CS = −1 indicate that there are two pairs and one pair of gapless edge states in bulk energy gap for 2D TCI and 2D TI, respectively. To illustrate this, we performed calculations of the edge-state band structures using MLWFs, which can reproduce the band dispersion of TlSe without and with 2% uniaxial strain (along the b-axis) quite precisely. Figure 4 displays the edge states of a 80-atom wide TlSe nanoribbon edged with Tl atoms. One can clearly see that two pairs of nontrivial edge states cross at Γ̅ and X̅ for the TCI

Figure 4. Band structures of 1D nanoribbon edged with Tl atoms for TlSe in (a) TCI phase and (b) TI phase. The corresponding projected bulk band structures are represented by the shaded areas. Edge states are colored with the expectation value of σz for indicating the spin polarization on one particular side of 1D nanoribbon. Insets in (a) and (b) show the 2D and projected 1D Brillouin zones, and the corresponding zoom-in at the Γ̅ point. C


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Nano Letters and one pair of that cross at Γ̅ for the TI phase. This is in direct agreement with the values of mirror Chern number and spin Chern numbers and is a natural consequence of the bulk band inversion at X and Y points for TCI (Figure 2a,b) and only at Y point for TI (Figure 3e,f), which are projected onto X̅ and Γ̅ points as shown in the inset of Figure 4a. Note that the different dispersion of Dirac crossings in TCI phase from that in IV−VI monolayers21−24 is due to the edge termination. In contrast to 2D TI, whose edge state is determined by time-reversal symmetry,1,2 the key property of the 2D TCI relies on crystal mirror symmetry,8 resulting in the gapless edge state are not spin degenerate.44 In order to indicate the spin texture, the matrix element of the Pauli matrices σα (α = x, y, z) are computed in the basis of the MLWFs and the edge states are colored with the expectation value of the σz. The edge states, as shown in Figure 4a, are spin-polarized and change directions of spin when crossing the time reversal invariant points for the 2D TCI. At the time reversal invariant points, the expectation value of the σz is zero. This is obviously different from the situation of the 2D TI as shown in Figure 4b, where the opposite spins are degenerate at the time reversal invariant points. Finally, one critical point has to be addressed to make sure that the insulating 2D TCI phase can be obtained on a latticematching substrate, which is important for the further experimental investigations and device applications. To test out this possibility, we propose a quantum well structure with one TlBiSe2 monolayer (Bi-doped 2 × 1 TlSe) sandwiched between NaCl layers (lattice mismatch is about 2.5%) that retains the mirror symmetry.24 As shown in the inset of Figure 5a, both the topology and the electronic properties are sensitive to the thickness of NaCl layers. Above four NaCl layers, a wide region of insulating 2D TCI phase is realized, and the global energy gap can reach as much as 0.37 eV for 16 NaCl layers, indicating the feasibility of experimentally observation of 2D TCI at room temperature. To get a better insight, Figure 5a shows the orbitally resolved band structures with SOC for eight NaCl layers. As we can see, the Fermi level lies above the top of the valence band and is pinned in the inverted gap, which proves the insulating states are indeed obtained. Similar to pure TISe monolayer, the band inversion occurs at an even number of time reversal invariant points (Γ and Y), which is the hallmark of TCI. For the edge states, a 96-atom wide nanoribbon terminated with Se atoms was considered. As shown in Figure 5b, the appearance of two pairs of nontrivial edge states confirms conclusively the 2D TCI phase of the TlBiSe2/NaCl quantum well. We also exam the feasibility of the Bi substitution for Tl by calculating the formation energy, defined as Ef = Etotal − Epure − μBi + μTl, where Etotal and Epure are the total energies of the supercell with and without the Bi dopants. μBi and μTl are the chemical potentials of Bi and Tl atoms, respectively. The calculated formation energies of TlBiSe2 monolayer and TlBiSe2/NaCl quantum well are respectively −0.33 and −0.20 eV, indicating that the Bi substitution for Tl atom is realistic. In addition, it is interesting to note that the TlBiSe2 monolayer can also be obtained from the well know TI TlBiSe2 along the (112) direction. In summary, using first-principles calculations, we identify that TlSe and TlS monolayers are promising 2D TCIs with direct band gap of 0.12 and 0.03 eV, respectively. The inverted gaps of TlM are tunable via uniaxial strain engineering, and a topological phase transition from 2D TCI to 2D TI can be obtained by tensile strain. Details of mirror symmetry and time

Figure 5. (a) Orbitally resolved band structures for Bi-doped 2 × 1 TlSe sandwiched between eight NaCl layers with SOC, weighted with the Se-px, Se-py, and Tl/Bi-pz characters. The Fermi level lies in the energy gap, indicating the insulating character. The inset is the global energy gap as a function of NaCl thickness. (b) Band structures of 1D nanoribbon edged with Se atoms. The inset shows the 2D and projected 1D Brillouin zones.

reversal symmetry protected edge states for 2D TCI and 2D TI are identified by spin-polarized edge states analysis. This phase transition can serve as a seed for realizing the quantum anomalous Hall effect with tunable Chern numbers. At last, we reveal the possibility of the insulating Tl-based 2D TCI with a band gap large enough for room-temperature applications.

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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 was supported by the Priority Program 1666 of the German Research Foundation (DFG), the Virtual Institute for Topological Insulators (VITI), and the Project No. VH-NG513 of the Helmholtz Association (HGF). We acknowledge computing time on the supercomputers JUQUEEN and JUROPA at Jülich Supercomputing Centre and JARA-HPC of RWTH Aachen University.

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Two-Dimensional Topological Crystalline Insulator and Topological

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