Prediction of Two-Dimensional Topological Crystalline Insulator in

Letter pubs.acs.org/NanoLett

Prediction of Two-Dimensional Topological Crystalline Insulator in PbSe Monolayer Ernesto O. Wrasse and Tome M. Schmidt* Instituto de Física, Universidade Federal de Uberlândia, Caixa Postal 593, CEP 38400-902, Uberlândia, Minas Gerais, Brazil

ABSTRACT: Two-dimensional (2D) topological crystalline insulator, a new class where states are protected by lattice symmetry instead of by time-reversal symmetry, is predicted in PbSe monolayer based on first-principles electronic structure calculations. A combination of strong spin−orbit interaction with quantum confinement effects in PbSe monolayer lead to a topological phase transition with an even number of band inversion momentum space points. We demonstrate that the PbSe nanostructure presents pairs of spin-polarized Dirac cones coming from the monolayer edges, where each Dirac pair presents a unique spin alignment, leading to a quantum spin Hall system. More importantly, due to the quantum confinement this 2D nanostructure presents larger band gap as compared to its parent narrow band gap trivial insulator bulk PbSe, favoring a roomtemperature 2D band gap with crystalline-protected Dirac states at the edges, turning this system interesting to combine nontrivial topological states with nanoelectronic and spintronic applications. KEYWORDS: 2D topological crystalline insulator, 2D lead chalcogenide, Dirac states, first-principles calculations

T

realized TI was a quantum well TI, giving rise to a quantum spin Hall (QSH) system.12 Epitaxially TCI materials have been recently synthesized,13,14 showing that 2D TCI materials are feasible. The 2D nanostructures have the potential to present larger band gap, as well to be desirable to combine TI with nanoelectronic devices. In this work, we show that while bulk PbSe is a trivial insulator, monolayers of PbSe are 2D TCI, presenting protected Dirac edge states with spin locked to the momentum, forming a QSH system. This 2D PbSe monolayer is different from proposals where topological states in 3D chalcogenides are tuned to throw pressure 15 or alloy compounds.16 In this system, it does not need any external mechanical deformation or alloy composition to induce a band inversion. The inverted bands are a consequence of two merged effects, the high SO interaction of PbSe, coupled with quantum confinement effects, which act different in the bands around the Fermi level. This 2D TCI is more interesting than the alloy -based TCI because the crystal symmetry is better preserved.

opological insulators (TIs) form a class of materials in which spin−orbit (SO) interactions invert the bands around the Fermi level, leading to a topological quantum phase where protected metallic states lie on the edges of the material. These edge states are massless Dirac Fermions and present a unique helical spin texture that turn these materials into promising ones for many applications, such as spintronics and quantum devices. These Dirac-like states are protected by timereversal symmetry (TRS)1 or by crystal symmetry.2−4 The latter one, called topological crystalline insulator (TCI), presents an even number of band inversions driven by SO interactions, which is in contrast to the TRS-protected TI that have an odd number of band inversions.1 The topological class is determined by the space group symmetry of the material lattice.5 Few materials have been shown to be in the TCI class, the bulk SnTe,6 and for a range of the composition x in Pb1−xSnxTe,7 and Pb1−xSnxSe8−10 alloys. All of them are three-dimensional (3D) TCI, and very recently theoretical calculations showed that thin films of them are predicted to be 2D TCI.11 No other material have been observed or even predicted to be in this new class of TI, and pure 2D TCI nanostructure (not coming from a reduction of 3D TCI) has not been predicted at all, although the first experimentally © 2014 American Chemical Society

Received: July 2, 2014 Revised: August 28, 2014 Published: September 8, 2014 5717

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inverted band point, between 50 and 75% of the total SO (see Figure 2), and then it open ups again. This band inversion occurs at X point, and we have an even number of X points (four) on the 2D first BZ. This system cannot present protected edge states by TRS within the well-known Z2 TI;1 rather, it may lead to a 2D TCI2,3 because the band inversions are at the momentum space mirror planes and the C4 rotational symmetry is present. This topological phase transition due to SO interactions turns a single PbSe monolayer feasible to present edge states protected by lattice symmetry and not by TRS, giving rise to a QSH system. Protected Edge States. In order to check the topological phase transition, we look for the protected Dirac states by introducing edges on the PbSe sheet, forming a nanoribbon. We construct nanoribbons with retangular shape, being infinite in one direction (x-direction from Figure 1) and keeping only the X−Γ−X line in the first BZ, using Bloch conditions. By restricting the Bloch conditions in the y-direction, we introduce edges on the PbSe sheet perpendicular to the y-direction of Figure 1. The first consequence of the 2D finiteness is an enlargement of the band gap due to quantum confinement effects. As we can see in Figure 3, we obtain a larger band gap

First-principles calculations have been performed within the density functional theory using the generalized gradient approximation to the exchange-correlation functional.17 The wave functions are expanded in plane wave basis set with cutoff of 300 eV. Fully relativistic PAW pseudopotentials including spin−orbit interactions have been used with VASP code.18 Such an approach has already been used to study electronic structure of 3D TI.19,20 The parity of the wave functions at the band edges have been obtained by projecting them onto spherical harmonics. PbSe Monolayer. The crystal structure of a PbSe monolayer presents a square Bravais lattice, symmetry point group D4, with a pair of Pb and Se atoms in the unit cell as shown in Figure 1. The first Brillouin zone (BZ) of a PbSe

Figure 1. (a) Top view of PbSe monolayer, showing the unit cell in the x−y square. (b) First BZ of the PbSe monolayer.

monolayer is also square with four X points on the side centers and four M points on the corners, as shown in Figure 1. This BZ can be obtained from a projection of the 3D bulk BZ onto the (001) surface, where the bulk L and X points of the facecentered cubic lattice are projected onto the X and M points of the 2D first BZ, respectively. The time-reversal invariant momentum (TRIM) points are Γ, M, and two X points (labeled X and Y in Figure 1). The direct band gap is at the momentum space X point as shown in Figure 2. Without the

Figure 3. Band structure of PbSe nanoribbons. The values on top are the nanoribbon widths, and ∞ means PbSe sheet. The red lines are the protected edge states.

(shadow areas) but we verify the presence of two bands inside the band gap, even for a nanoribbon width of just 2.4 nm. For narrow nanoribbons, the edge states do not form a metallic system but they already present some linear energy dispersion, typically of topologically protected states. The band gap of the edge states becomes zero for nanoribbons larger than 17 nm, forming a Dirac cone centered along the Γ−X direction of the first BZ, similar to the topological states observed on the surface of the 3D TCI SnTe.3 It is interesting to note that when the Bloch conditions are restored along both directions (labeled ∞ in Figure 3), the bands protected by symmetry merge with respective bulk bands, opening up a band gap. The edges of a PbSe nanoribbon can be constructed in two different ways according to symmetry constraints. We can cut the cell perpendicular to the y-direction, as shown in Figure 1, keeping the mirror plane along the x-direction, making the edges symmetric. Or we can destroy the x-mirror plane by making the edges nonsymmetric. The number of Dirac pairs depends on the nanoribbon symmetry. The PbSe sheet presents four band inversion momentum space points in the first BZ, so four Dirac pairs are expected. For symmetric edge nanoribbon, there are two Dirac crossings located along the Γ− X−Γ direction, one before and other after the band inversion momentum space X-point. While one Dirac crossing presents all spin up, the other Dirac crossing presents all spin down (see

Figure 2. Band structure around the Fermi level for a single atomic layer of PbSe. Blue color is a projection onto Pb 6p orbitals, while red color is onto Se 4p orbitals.

inclusion of SO interactions the valence band maximum (VBM) of a PbSe monolayer is derived mostly from Se 4p orbitals, while the conduction band minimum (CBM) is predominantly Pb 6p orbitals. As a function of the four base atoms Pb1, Pb2, Se1, and Se2, the wave function of the VBM can be written as Φ+VBM = |Pz⟩Se1 − |Pz⟩Se2 (red in Figure 2), while the CBM can be written as Φ−CBM = |Px ± iPy⟩Pb1 − |Px ± iPy⟩Pb2 (blue in Figure 2), presenting opposite parities. This scenario for the band edges of PbSe monolayer without SO interactions is quite similar to that of the bulk PbSe.21 By turn, on SO interactions we observe that the band gap decreases to an 5718

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Figure 4). This texture is only permitted if the spins are aligned perpendicular to the nanoribbon sheet, that is, spin up (down)

On the other hand for the nonsymmetric edge nanoribbon only two Dirac crossings are observed. As one edge is not symmetric with respect to the other edge, the pair of Dirac cones at the band inversion momentum space X point become massives (see Figure 4-b). The asymmetric nanoribbon originates a small band gap at the X point due to the mass acquired, and this effect increases the particle-hole asymmetry in the band structure of the edge states. Recently, mass acquisition has been observed in the 3D TCI Pb1−xSnxSe alloy due to symmetry-breaking interactions.9 The spin texture of this nonsymmetric nanoribbon presents one Dirac crossing with spin up and the other Dirac with spin down, forming a QSH system. The spin is aligned perpendicular to the nanoribbon plane. Discussion and Conclusions. To characterize the Dirac cone we plot in Figure 5 the charge density of the states around the Fermi level. We observe that the protected Dirac cones come mostly from regions close to the edges of the nanoribbon but with some spreading toward the center of the ribbon. Also, we can see that the charge density for hole and electron states are quite similar, indicating small particle-hole asymmetry around the Fermi level. This asymmetry is more visible far from the Dirac crossing. The charge density presented in Figure 5 is for a nanoribbon width of 17.4 nm. For narrower nanoribbons, we do not obtain the correct spin texture, and the Dirac states come from all parts of the nanoribbon surface. That means that there is a minimum ribbon width to get the protected crossing Dirac states; this occurs because for narrow nanoribbon the Dirac cones interact to each other, opening a band gap (as shown in Figure 2). It is interesting to compare the electronic and spin texture of this 2D TCI, PbSe monolayer, with those of 3D TCIs as well as 2D TI. The 3D TCIs present four Dirac cones on the surface located out of the TRIM points,3,6−10 where each state pair coming from each Dirac cone presents opposite spin polarization.3,7 On the other hand, 2D TIs present a Dirac pair also with opposite spin polarization but located at a TRIM point.12,23 While bulk SnTe is a 3D TCI, ultrathin layers of SnTe are 2D TCI with a pair of identical mirror eigenvalues, crossing at non-TRIM points.11 Here, for a PbSe monolayer the Dirac crossings are also in pairs, located at non-TRIM points, but by increasing the number of PbSe layers the band gap is so large that the SO coupling is not enough to induce a band inversion, becoming a trivial insulator. This is different from the SnTe, where up to five layers it is a trivial insulator and after that it becomes a 2D TCI.11 Also, this PbSe monolayer is different from the Bi(111) and Sb(111) stacking bilayers.

Figure 4. Band structure of PbSe nanoribbons. Panel a shows for a symmetric edges nanoribbon, while panel b is for a nonsymmetric edges nanoribbon. The arrows mean the spin orientation with respect to the nanoribbon plane.

is aligned upward (downward) from the ribbon plane. This spin texture is different from those of 3D TCI22 but in both cases the two-fold rotation symmetry is preserved. Here, each edge has two pairs of crystalline-protected states and each Dirac has one defined spin, as shown in Figure 4. This momentum-spin locking forms a QSH system. For this symmetric edge nanoribbon, we observe two additional nondispersing crossings centering exactly at the BZ edge X-point, one close to the CBM and other close to the VBM. These two massless Dirac crossings, as shown in Figure 4a, present a spin texture similar to edge states protected by TRS with each branch of the Dirac cone has opposite spin. This similarity with TRS-protected TI is because the crossings occur at one band inversion momentum space point. Also it is interesting to note that the band velocities of the Dirac crossings located along the ΓX direction are higher than those of the Dirac cones at the Xpoint. Different Dirac velocities have been observed on the surface of the 3D TCI Pb1−xSnxSe,9 as well on the [111] surface of SnTe.6

Figure 5. Electron and hole charge densities around the Fermi level. The charge density of the Dirac states indicated by arrows on the left (right) band structure comes mostly from the left (right) edge of the nanoribbon. The Fermi level is at zero energy. 5719

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(8) Dziawa, P.; Kowalski, B. J.; Dybko, K.; Buczko, R.; Szczerbakow, A.; Szot, M.; Lusakowska, E.; Balasubramanian, T.; Wojek, B. M.; Berntsen, M. H.; Tjernberg, O.; Story, T. Nat. Mater. 2012, 11, 1023. (9) Okada, Y.; Serbyn, M.; Lin, H.; Walkup, D.; Zhou, W.; Dhital, C.; Neupane, M.; Xu, S.; Wang, Y. J.; Sankar, R.; Chou, F.; Bansil, A.; Hasan, M. Z.; Wilson, S. D.; Fu, L.; Madhavan, V. Science 2013, 341, 1496. (10) Wojek, B. M.; Buczko, R.; Safaei, S.; Dziawa, P.; Kowalski, B. J.; Berntsen, M. H.; Balasubramanian, T.; Leandersson, M.; Szczerbakow, A.; Kacman, P.; Story, T.; Tjernber, O. Phys. Rev. B 2013, 87, 115106. (11) Liu, J.; Hsieh, T. H.; Wei, P.; Duan, W.; Moodera, J.; Fu, L. Nat. Mater. 2014, 13, 178. (12) Konig, M.; Wiedmann, S.; Brune, C.; Roth, A.; Buhmann, H.; Molenkamp, L. W.; Qi, X.-L.; Zhang, S.-C. Science 2007, 318, 766. (13) Taskin, A. A.; Yang, F.; Sasaki, S.; Segawa, K.; Ando, Y. Phys. Rev. B 2014, 89, 121302(R). (14) Guo, H.; Yan, C.-H.; Liu, J.-W.; Wang, Z.-Y.; Wu, R.; Zhang, Z.D.; Wang, L.-L.; He, K.; Ma, X.-C.; Ji, S.-H.; Duan, W.-H.; Chen, X.; Xue, Q.-K. APL Mater. 2014, 2, 056106. (15) Barone, P.; Rauch, T.; Di Sante, D.; Henk, J.; Mertig, I.; Picozzi, S. Phys. Rev. B 2013, 88, 045207. (16) Yang, K.; Setyawan, W.; Wang, S.; Nardelli, M. B.; Curtarolo, S. Nat. Mater. 2012, 11, 614. (17) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. Rev. Lett. 1996, 77, 3865. (18) Kresse, G.; Furthmüller, J. Phys. Rev. B 1996, 54, 11169. (19) Schmidt, T. M.; Miwa, R. H.; Fazzio, A. Phys. Rev. B 2011, 84, 245418. (20) Abdalla, L. B.; Seixas, L.; Schmidt, T. M.; Miwa, R. H.; Fazzio, A. Phys. Rev. B 2013, 88, 045312. (21) Wrasse, E. O.; Baierle, R. J.; Fazzio, A.; Schmidt, T. M. Phys. Rev. B 2013, 87, 085428. (22) Wang, Y. J.; Tsal, W.-F.; Lin, H.; Xu, S.-Y.; Neupane, M.; Hasan, M. Z.; Bansil, A. Phys. Rev. B 2013, 87, 235317. (23) Kane, C. L.; Mele, E. J. Phys. Rev. Lett. 2005, 95, 146802. (24) Liu, Z.; Liu, C.-X.; Wu, Y.-S.; Duan, W.-H.; Liu, F.; Wu, J. Phys. Rev. Lett. 2011, 107, 136805. (25) Zhang, P.; Liu, Z.; Duan, W.; Liu, F.; Wu, J. Phys. Rev. B 2012, 85, 201410(R).

Ultrathin Bi(111) bilayers are TI protected by TRS from 1 to 8 bilayers,24 while ultrathin Sb(111) film is trivial up to 3 bilayers, then it is a 2D TI up to 8 bilayers, then it is a 3D TI up to 22 bilayers.25 Bulk chalcogenides are narrow band gap materials. So it is important to mention that the calculated PbSe monolayer band gap is 0.23 eV, which is larger than the calculated bulk PbSe of 0.02 eV.21 This enlargement of the band gap is a consequence of the quantum confinement. As this result has been obtained within the DFT approach, which underestimates the band gap, it is expected experimentally a larger band gap for the PbSe sheet as compared to the bulk PbSe, turning this system even more interesting for applications, especially to combine TI materials with nanoelectronic devices. Also, this TCI nanostructure is more interesting as compared to the alloybased TCI, because the crystal symmetry is better preserved because there are no compounds. In conclusion, we show that a single atomic layer of PbSe is a 2D TCI, presenting protected states at the nanoribbon edges forming a QSH system. The topological quantum phase takes place due to the strong SO interactions observed in this system together to confinement effects. As the quantum confinement acts different in the bands around the Fermi level, the SO interactions invert the levels at the edges of valence and conduction bands. The 2D TCI presents pairs of Dirac cones along the Γ−X direction, one before and other after the band inversion momentum space X-point. Different from the TRS protected TI, the spin of each Dirac crossing of the 2D TCI presents all spin aligned in one direction, and the Dirac pair has opposite spins. The PbSe monolayer presents larger band gap as compared to its parent trivial insulator bulk PbSe, as well it is larger than most narrow band gap bulk chalcogenides. Considering also that the crystal symmetry is bettter preserved in this 2D TCI as compared to the alloy-based TCIs, it becomes more interesting to combine nontrivial topological states with nanoelectronic and spintronic devices.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: tschmidt@infis.ufu.br. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the Brazilian agencies FAPEMIG, CAPES, CNPq, and Cenapad-Unicamp computational facilities.

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REFERENCES

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