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Bulk and surface electronic structure of SnBi4Te7 topological insulator
Applied Surface Science 267 (2013) 146–149
Contents lists available at SciVerse ScienceDirect
Applied Surface Science journal homepage: www.elsevier.com/locate/apsusc
Bulk and surface electronic structure of SnBi4 Te7 topological insulator M.G. Vergniory a,∗ , T.V. Menshchikova a,b , S.V. Eremeev a,b,c , E.V. Chulkov a,d a
Donostia International Physics Center, 20018 San Sebastian/Donostia, Basque Country, Spain Tomsk State University, pr. Lenina 36, 634050, Tomsk, Russia c Institute of Strength Physics and Materials Science, pr. Academicheskii 2/4, 634021, Tomsk, Russia d Departamento de Física de Materiales, Centro de Física de Materiales CFM-MPC and Centro Mixto CSIC-UPV/EHU, Facultad de Ciencias Químicas, UPV/EHU, Apdo. 1072, 20080 San Sebastián/Donostia, Basque Country, Spain b
a r t i c l e
i n f o
Article history: Available online 28 August 2012 Keywords: Topological insulators Electronic structure Surface states
a b s t r a c t Using density functional theory with the spin–orbit coupling included we analyze the bulk and surface electronic structure of SnBi4 Te7 ternary compound. It was revealed that this material is a strong topological insulator with a bulk band gap of about 100 meV and a robust surface state around the point. We find that the topological nature of the surface state remains robust with different terminations of the surface. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Three-dimensional topological insulators (TIs) have attracted great attention due to their topologically protected surface states (SS) [1,2]. These materials are narrow gap semiconductors characterized by an inverted energy gap owing to strong spin–orbit coupling (SOC). In three dimensions (3D) TIs differ from band insulators in that a metallic SS arises in the bulk energy gap. Unlike SSs in ordinary materials, this SS shows linear dispersion, forming a Dirac cone with a crossing (Dirac) point at/close to the Fermi level (EF ) [1]. As distinct from the Dirac cone in graphene, this topological SS carries only one electron per momentum with a spin that changes its direction consistently with a change of momentum. The topological origin of the SSs protects them from surface perturbation. The unique electronic properties of the surface of the topological insulators make these materials important for many potential applications, particularly in spintronics and quantum computing. Several families of TI have been proposed theoretically and confirmed experimentally. The first discovery was a two-dimensional TI in an HgTe super lattices [3]. Subsequently first-principle calculations [2,4–6] showed that bulk Bi2 Te3 , Bi2 Se3 and Sb2 Te3 are 3D strong TIs with the topological SS mostly located in the first five layer block adjacent to the vacuum side. The existence of the Dirac cone was confirmed experimentally by angle-resolved photoemission spectra (ARPES) [7–9]. Motivated by their application potential, the search for new topological insulators has extended to ternary compounds [10–14]. Recently a type of ternary thallium
∗ Corresponding author. Tel.: +34 943 01 54 22; fax: +34 943 01 56 00. E-mail address: maia [email protected] (M.G. Vergniory). 0169-4332/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.apsusc.2012.08.073
based II–V–VI2 chalcogenides (TlBiTe2 and TlBiSe2 ) has also been proposed theoretically [15,16] and confirmed experimentally [9,17]. One family more of ternary compounds is based on binary TIs (Bi2 Te3 , Bi2 Se3 and Sb2 Te3 ). Among them PbBi4 Te7 and PbBi2 Te4 were predicted to be TI [11,12] and confirmed experimentally [12,13]. In this work we have presented detailed ab initio calculation results for Sn-based TI, SnBi4 Te7 .
2. Calculation methods The structural optimization and electronic structure calculation were performed within the density functional formalism implemented in VASP [18,19]. In this plane-wave code, the interaction between the ion cores and valence electrons was described within the projector augmented method [20]. For the description of the exchange-correlation energy, we have used the generalized gradient approximation [21]. The Hamiltonian contained the scalar relativistic corrections and the SOC was taken into account by the second variation method [22]. We have used a 6 × 6 × 6 and 11 × 11 × 1 k-point grid for bulk and slab self-consistent calculation respectively, the kinetic energy cut-off for the plane-wave basis set was 220 eV and in the surface calculation the vacuum space ˚ between slabs was 12 A. We have used symmetric surface termination setups, thus the upper and lower surfaces were identical. In these cases, deviations from the ideal stoichiometry occur, so the position of the Fermi level depends on the termination and film thickness. Therefore, we have adjusted the bands of the film calculations to the projected bulk band structure and have obtained the Fermi level from the latter.
M.G. Vergniory et al. / Applied Surface Science 267 (2013) 146–149
147
(Fig. 2(b)) while the bottom of the conduction band is produced by Bi and Sn orbitals. The SOC inverts the gap edges in the vicinity of the point (Fig. 2(c)). Such an inversion unambiguously indicates a change of the parity of the occupied states at . Together with the parity in other time-reversal invariant momenta (three equivalent M points) it gives a reliable evidence of the fact that this compound is 3D topological insulator.
3.2. Surface properties
Fig. 1. Atomic structure of SnBi4 Te7 (a) and the bulk Brillouin zone and the surface Brillouin zone of hexagonal cell (b).
3. Results and discussion 3.1. Bulk properties The SnBi4 Te7 compound has a hexagonal unit cell with alternating 5-layers (5L) and 7-layers (7L) blocks in the following way: Te–Bi–Te–Bi–Te and Te–Bi–Te–Sn–Te–Bi–Te respectively [23] (Fig. 1(a)). We have used experimental lattice parameters a = 4.3921 A˚ and c = 23.99 A˚ with optimized atomic positions in the unit cell. The calculated charge density distribution (see Fig. 2(a)) indicates that the binding within the 5L and 7L blocks is ionic-covalent whereas the binding between the blocks is determined by the van der Waals (vdW) forces, like in the case of Bi2 Te3 . The equilibrium vdW interlayer spacing is 2.516 A˚ slightly less than the similar space ˚ in Bi2 Te3 (2.801 A). In Fig. 2 the bulk electronic band structure calculated without SOC (w/o SOC) Fig. 2(b) and with SOC Fig. 2(c) included along the high symmetry lines of the Brillouin zone (see Fig. 1(b)) is presented. The effect of the SOC leads to an elevation of the upper edge of the valence band and the lowering of the lower edge of the conduction band, particularly it induces more pronounced changes in the spectrum near the point. In the case of w/o SOC included the top of the valence band is composed predominantly of Te states
Owing to the topological nature of this material its surface should hold gapless surface state with Dirac dispersion. Because the crystal structure of SnBi4 Te7 is constituted by alternating 5L and 7L building blocks, the surfaces formed after cleavage of this crystal have two possible terminations. The 5L-terminated and 7L-terminated surfaces were simulated by 41 atomic layer (5L–7L–5L–7L–5L–7L–5L) and 43 atomic layer (7L–5L–7L–5L–7L–5L–7L) slabs, respectively. Fig. 3 shows the band structure of the surface terminated by 5L (a) and 7L (b) blocks and the spin polarization of the Dirac cone for different energy contours for both terminations (c and d). The dispersion of the Dirac cone in the case of the 5L-terminated surface is wider than in the 7Lterminated surface. The position of the Dirac point with respect to the Fermi energy is different too, while in the case of 5L-terminated surface it is located at 85 meV below, for the case of 7L-terminated surface we find it at 12 meV below the Fermi energy. In both cases it is near the projection of the bulk states of the valence band. For both cases, once they get closer to the conduction band the surface contour becomes hexagonal and even has a snowflake shape for 7Lterminated surface just below the conduction band bottom. Such anisotropy of the dispersion of the surface state can lead to intraband scattering. [24] At the same time both surfaces demonstrate typical for TI clockwise spin helicity in the Dirac cone (Fig. 3(c and d)). One should note that at the 5L-terminated surface the out-ofplane spin component Sz is not negligible even at lower energies, close to the Dirac point, that is due to hybridization with the bulk valence states. The charge density distribution of the states at the surface with 5L and 7L termination differs considerably. The charge density of the SS on the 7L-terminated surface is almost completely localized within the 7L block (Fig. 3b on the right side), while it is significantly different in the case of 5L-terminated surface, which is mostly localized in two blocks. This result is very interesting if we compare it
Fig. 2. The charge density distribution in the (1120) plane (a); bulk band spectra without (w/o SOC) (b) and with (SOC) (c) spin–orbit coupling included. The size of the color circles is proportional to partial weights of the states.
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M.G. Vergniory et al. / Applied Surface Science 267 (2013) 146–149
Fig. 3. Band spectrum of the SnBi4 Te7 (0 0 0 1) surface terminating by (a) 5L and (b) 7L blocks. The green shade indicates the projection of the bulk states. On the right of each panel the probability density of the Dirac SS at the point is shown. Calculated spin structure of the Dirac state as represented by projections of S on cartesian axes at energies of 0.075, 0.125 and 0.175 eV for the 5L-terminated surface (c) and of 0.05, 0.075 and 0.125 eV for the 7L-terminated surface (d). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)
with PbBi4 Te7 [11], where the charge density distribution of the SS for the case of the 5L-bock termination was completely localized in the 7L-block. This difference is due to the fact that both, Te-7L and Te-5L states, contribute to the bottom of the bulk conduction band (see Fig. 2(c)) while in the case of PbBi4 Te7 they are separated in energy and only the Te-7L state located at the gap edge. 4. Summary and conclusions To summarize, it has been established that SnBi4 Te7 is a threedimensional topological insulator. Owing to the more complicated crystal structure than Bi2 Te3 -type TI this material can have different surface terminations. We find some differences between the dispersion of the Dirac topological surface state for 5L- and 7Lterminated surfaces. It has been shown that this state can deeply penetrate beneath the surface in the 5L-terminated case due to peculiarities of the bulk band structure. The revealed differences between 5L- and 7L-terminated surfaces can be responsible for peculiarities in spin-polarized current in SnBi4 Te7 depending on the surface termination.
(grant no. FIS 2004-06490-C03-01). The calculations where perfomed on Arina supercomputer of the Basque Country University.
References [1] [2] [3] [4] [5] [6] [7] [8]
[9]
[10] [11] [12]
Acknowledgments [13]
The work was supported by the University of the Basque Country, Departamento de Educación del Gobierno Vasco, and MCyT
M.Z. Hasan, C.L. Kane, Reviews of Modern Physics 82 (2010) 3045. H. Zhang, C.X. Liu, X.-L. Qi, Z. Fang, S.C. Zhang, Nature Physics 5 (2009) 438. B.A. Bernevig, T.L. Hughes, S.C. Zhang, Science 314 (2006) 1757. S.V. Eremeev, Yu.M. Koroteev, E.V. Chulkov, JETP Letters 91 (2010) 387. J.H. Song, H. Jin, A. Freeman, Physical Review Letters 105 (2010) 096403. W. Zhang, R. Yu, J.H. Zhang, X. Dai, Z. Fang, New Journal of Physics 12 (2010) 065013. Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D. Grauer, Y.S. Hor, R.J. Cava, M.Z. Hasan, Nature Physics 5 (2009) 398. Y.L. Chen, J.G. Analytis, J.H. Chu, Z.K. Liu, S.K. Mo, X.L. Qi, J.H. Zhang, D.H. Lu, X. Dai, Z. Fang, S.C. Zhang, I.R. Fisher, Z. Hussain, Z.X. Shen, Science 325 (2009) 178. K. Kuroda, M. Ye, A. Kimura, S.V. Eremeev, E.E. Krasovskii, E.V. Chulkov, Y. Ueda, K. Miyamoto, T. Okuda, K. Shimada, H. Namatame, M. Taniguchi, Physical Review Letters 105 (2010) 146801. S. Chadov, X. Qi, J. Kübler, G.H. Fecher, S.C. Zhang, Nature Materials 9 (2010) 541. S.V. Eremeev, Yu.M. Koroteev, E.V. Chulkov, JETP Letters 92 (2010) 161. S.V. Eremeev, G. Landolt, T.V. Menshchikova, B. Slomski, Y.M. Koroteev, Z.S. Aliev, M.B. Babanly, J. Henk, A. Ernst, L. Patthey, A. Eich, A.A. Khajetoorians, J. Hagemeister, O. Pietzsch, J. Wiebe, R. Wiesendanger, P.M. Echenique, S.S. Tsirkin, I.R. Amiraslanov, J.H. Dil, E.V. Chulkov, Nature Communications 3 (2012) 635. K. Kuroda, H. Miyahara, M. Ye, S.V. Eremeev, Yu.M. Koroteev, E.E. Krasovskii, E.V. Chulkov, S. Hiramoto, C. Moriyoshi, Y. Kuroiwa, K. Miyamoto, T. Okuda, M. Arita, K. Shimada, H. Namatame, M. Taniguchi, Y. Ueda, A. Kimura, Physical Review Letters 108 (2012) 206803.
M.G. Vergniory et al. / Applied Surface Science 267 (2013) 146–149 [14] T.V. Menshchikova, S.V. Eremeev, Yu.M. Koroteev, V.M. Kuznetsov, E.V. Chulkov, JETP Letters 93 (2011) 15. [15] S.V. Eremeev, Yu.M. Koroteev, E.V. Chulkov, JETP Letters 91 (2010) 594. [16] S.V. Eremeev, G. Bihlmayer, M. Vergniory, Yu.M. Koroteev, T.V. Menshchikova, J. Henk, A. Ernst, E.V. Chulkov, Physical Review B 83 (2011) 205129. [17] T. Sato, K. Segawa, H. Guo, K. Sugawara, S. Souma, T. Takahashi, Y. Ando, Physical Review Letters 105 (2010) 136802. [18] G. Kresse, J. Hafner, Physical Review B 48 (1993) 13115.
149
[19] G. Kresse, J. Furthmüller, Computation Materials Science 6 (1996) 15. [20] G. Kresse, D. Joubert, Physical Review B 59 (1998) 1758. [21] J.P. Perdew, K. Burke, M. Ernzerhof, Physical Review Letters 77 (1996) 3865. [22] D.D. Koelling, B.N. Harmon, Journal of Physics C 10 (1977) 3107. [23] O.G. Karpinskii, L.E. Shelimova, M.A. Kretova, E.S. Avilova, V.S. Zemskov, Inorganic Materials 39 (2003) 305. [24] L. Fu, Physical Review Letters 103 (2009) 266801.
Contents lists available at SciVerse ScienceDirect
Applied Surface Science journal homepage: www.elsevier.com/locate/apsusc
Bulk and surface electronic structure of SnBi4 Te7 topological insulator M.G. Vergniory a,∗ , T.V. Menshchikova a,b , S.V. Eremeev a,b,c , E.V. Chulkov a,d a
Donostia International Physics Center, 20018 San Sebastian/Donostia, Basque Country, Spain Tomsk State University, pr. Lenina 36, 634050, Tomsk, Russia c Institute of Strength Physics and Materials Science, pr. Academicheskii 2/4, 634021, Tomsk, Russia d Departamento de Física de Materiales, Centro de Física de Materiales CFM-MPC and Centro Mixto CSIC-UPV/EHU, Facultad de Ciencias Químicas, UPV/EHU, Apdo. 1072, 20080 San Sebastián/Donostia, Basque Country, Spain b
a r t i c l e
i n f o
Article history: Available online 28 August 2012 Keywords: Topological insulators Electronic structure Surface states
a b s t r a c t Using density functional theory with the spin–orbit coupling included we analyze the bulk and surface electronic structure of SnBi4 Te7 ternary compound. It was revealed that this material is a strong topological insulator with a bulk band gap of about 100 meV and a robust surface state around the point. We find that the topological nature of the surface state remains robust with different terminations of the surface. © 2012 Elsevier B.V. All rights reserved.
1. Introduction Three-dimensional topological insulators (TIs) have attracted great attention due to their topologically protected surface states (SS) [1,2]. These materials are narrow gap semiconductors characterized by an inverted energy gap owing to strong spin–orbit coupling (SOC). In three dimensions (3D) TIs differ from band insulators in that a metallic SS arises in the bulk energy gap. Unlike SSs in ordinary materials, this SS shows linear dispersion, forming a Dirac cone with a crossing (Dirac) point at/close to the Fermi level (EF ) [1]. As distinct from the Dirac cone in graphene, this topological SS carries only one electron per momentum with a spin that changes its direction consistently with a change of momentum. The topological origin of the SSs protects them from surface perturbation. The unique electronic properties of the surface of the topological insulators make these materials important for many potential applications, particularly in spintronics and quantum computing. Several families of TI have been proposed theoretically and confirmed experimentally. The first discovery was a two-dimensional TI in an HgTe super lattices [3]. Subsequently first-principle calculations [2,4–6] showed that bulk Bi2 Te3 , Bi2 Se3 and Sb2 Te3 are 3D strong TIs with the topological SS mostly located in the first five layer block adjacent to the vacuum side. The existence of the Dirac cone was confirmed experimentally by angle-resolved photoemission spectra (ARPES) [7–9]. Motivated by their application potential, the search for new topological insulators has extended to ternary compounds [10–14]. Recently a type of ternary thallium
∗ Corresponding author. Tel.: +34 943 01 54 22; fax: +34 943 01 56 00. E-mail address: maia [email protected] (M.G. Vergniory). 0169-4332/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.apsusc.2012.08.073
based II–V–VI2 chalcogenides (TlBiTe2 and TlBiSe2 ) has also been proposed theoretically [15,16] and confirmed experimentally [9,17]. One family more of ternary compounds is based on binary TIs (Bi2 Te3 , Bi2 Se3 and Sb2 Te3 ). Among them PbBi4 Te7 and PbBi2 Te4 were predicted to be TI [11,12] and confirmed experimentally [12,13]. In this work we have presented detailed ab initio calculation results for Sn-based TI, SnBi4 Te7 .
2. Calculation methods The structural optimization and electronic structure calculation were performed within the density functional formalism implemented in VASP [18,19]. In this plane-wave code, the interaction between the ion cores and valence electrons was described within the projector augmented method [20]. For the description of the exchange-correlation energy, we have used the generalized gradient approximation [21]. The Hamiltonian contained the scalar relativistic corrections and the SOC was taken into account by the second variation method [22]. We have used a 6 × 6 × 6 and 11 × 11 × 1 k-point grid for bulk and slab self-consistent calculation respectively, the kinetic energy cut-off for the plane-wave basis set was 220 eV and in the surface calculation the vacuum space ˚ between slabs was 12 A. We have used symmetric surface termination setups, thus the upper and lower surfaces were identical. In these cases, deviations from the ideal stoichiometry occur, so the position of the Fermi level depends on the termination and film thickness. Therefore, we have adjusted the bands of the film calculations to the projected bulk band structure and have obtained the Fermi level from the latter.
M.G. Vergniory et al. / Applied Surface Science 267 (2013) 146–149
147
(Fig. 2(b)) while the bottom of the conduction band is produced by Bi and Sn orbitals. The SOC inverts the gap edges in the vicinity of the point (Fig. 2(c)). Such an inversion unambiguously indicates a change of the parity of the occupied states at . Together with the parity in other time-reversal invariant momenta (three equivalent M points) it gives a reliable evidence of the fact that this compound is 3D topological insulator.
3.2. Surface properties
Fig. 1. Atomic structure of SnBi4 Te7 (a) and the bulk Brillouin zone and the surface Brillouin zone of hexagonal cell (b).
3. Results and discussion 3.1. Bulk properties The SnBi4 Te7 compound has a hexagonal unit cell with alternating 5-layers (5L) and 7-layers (7L) blocks in the following way: Te–Bi–Te–Bi–Te and Te–Bi–Te–Sn–Te–Bi–Te respectively [23] (Fig. 1(a)). We have used experimental lattice parameters a = 4.3921 A˚ and c = 23.99 A˚ with optimized atomic positions in the unit cell. The calculated charge density distribution (see Fig. 2(a)) indicates that the binding within the 5L and 7L blocks is ionic-covalent whereas the binding between the blocks is determined by the van der Waals (vdW) forces, like in the case of Bi2 Te3 . The equilibrium vdW interlayer spacing is 2.516 A˚ slightly less than the similar space ˚ in Bi2 Te3 (2.801 A). In Fig. 2 the bulk electronic band structure calculated without SOC (w/o SOC) Fig. 2(b) and with SOC Fig. 2(c) included along the high symmetry lines of the Brillouin zone (see Fig. 1(b)) is presented. The effect of the SOC leads to an elevation of the upper edge of the valence band and the lowering of the lower edge of the conduction band, particularly it induces more pronounced changes in the spectrum near the point. In the case of w/o SOC included the top of the valence band is composed predominantly of Te states
Owing to the topological nature of this material its surface should hold gapless surface state with Dirac dispersion. Because the crystal structure of SnBi4 Te7 is constituted by alternating 5L and 7L building blocks, the surfaces formed after cleavage of this crystal have two possible terminations. The 5L-terminated and 7L-terminated surfaces were simulated by 41 atomic layer (5L–7L–5L–7L–5L–7L–5L) and 43 atomic layer (7L–5L–7L–5L–7L–5L–7L) slabs, respectively. Fig. 3 shows the band structure of the surface terminated by 5L (a) and 7L (b) blocks and the spin polarization of the Dirac cone for different energy contours for both terminations (c and d). The dispersion of the Dirac cone in the case of the 5L-terminated surface is wider than in the 7Lterminated surface. The position of the Dirac point with respect to the Fermi energy is different too, while in the case of 5L-terminated surface it is located at 85 meV below, for the case of 7L-terminated surface we find it at 12 meV below the Fermi energy. In both cases it is near the projection of the bulk states of the valence band. For both cases, once they get closer to the conduction band the surface contour becomes hexagonal and even has a snowflake shape for 7Lterminated surface just below the conduction band bottom. Such anisotropy of the dispersion of the surface state can lead to intraband scattering. [24] At the same time both surfaces demonstrate typical for TI clockwise spin helicity in the Dirac cone (Fig. 3(c and d)). One should note that at the 5L-terminated surface the out-ofplane spin component Sz is not negligible even at lower energies, close to the Dirac point, that is due to hybridization with the bulk valence states. The charge density distribution of the states at the surface with 5L and 7L termination differs considerably. The charge density of the SS on the 7L-terminated surface is almost completely localized within the 7L block (Fig. 3b on the right side), while it is significantly different in the case of 5L-terminated surface, which is mostly localized in two blocks. This result is very interesting if we compare it
Fig. 2. The charge density distribution in the (1120) plane (a); bulk band spectra without (w/o SOC) (b) and with (SOC) (c) spin–orbit coupling included. The size of the color circles is proportional to partial weights of the states.
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Fig. 3. Band spectrum of the SnBi4 Te7 (0 0 0 1) surface terminating by (a) 5L and (b) 7L blocks. The green shade indicates the projection of the bulk states. On the right of each panel the probability density of the Dirac SS at the point is shown. Calculated spin structure of the Dirac state as represented by projections of S on cartesian axes at energies of 0.075, 0.125 and 0.175 eV for the 5L-terminated surface (c) and of 0.05, 0.075 and 0.125 eV for the 7L-terminated surface (d). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of the article.)
with PbBi4 Te7 [11], where the charge density distribution of the SS for the case of the 5L-bock termination was completely localized in the 7L-block. This difference is due to the fact that both, Te-7L and Te-5L states, contribute to the bottom of the bulk conduction band (see Fig. 2(c)) while in the case of PbBi4 Te7 they are separated in energy and only the Te-7L state located at the gap edge. 4. Summary and conclusions To summarize, it has been established that SnBi4 Te7 is a threedimensional topological insulator. Owing to the more complicated crystal structure than Bi2 Te3 -type TI this material can have different surface terminations. We find some differences between the dispersion of the Dirac topological surface state for 5L- and 7Lterminated surfaces. It has been shown that this state can deeply penetrate beneath the surface in the 5L-terminated case due to peculiarities of the bulk band structure. The revealed differences between 5L- and 7L-terminated surfaces can be responsible for peculiarities in spin-polarized current in SnBi4 Te7 depending on the surface termination.
(grant no. FIS 2004-06490-C03-01). The calculations where perfomed on Arina supercomputer of the Basque Country University.
References [1] [2] [3] [4] [5] [6] [7] [8]
[9]
[10] [11] [12]
Acknowledgments [13]
The work was supported by the University of the Basque Country, Departamento de Educación del Gobierno Vasco, and MCyT
M.Z. Hasan, C.L. Kane, Reviews of Modern Physics 82 (2010) 3045. H. Zhang, C.X. Liu, X.-L. Qi, Z. Fang, S.C. Zhang, Nature Physics 5 (2009) 438. B.A. Bernevig, T.L. Hughes, S.C. Zhang, Science 314 (2006) 1757. S.V. Eremeev, Yu.M. Koroteev, E.V. Chulkov, JETP Letters 91 (2010) 387. J.H. Song, H. Jin, A. Freeman, Physical Review Letters 105 (2010) 096403. W. Zhang, R. Yu, J.H. Zhang, X. Dai, Z. Fang, New Journal of Physics 12 (2010) 065013. Y. Xia, D. Qian, D. Hsieh, L. Wray, A. Pal, H. Lin, A. Bansil, D. Grauer, Y.S. Hor, R.J. Cava, M.Z. Hasan, Nature Physics 5 (2009) 398. Y.L. Chen, J.G. Analytis, J.H. Chu, Z.K. Liu, S.K. Mo, X.L. Qi, J.H. Zhang, D.H. Lu, X. Dai, Z. Fang, S.C. Zhang, I.R. Fisher, Z. Hussain, Z.X. Shen, Science 325 (2009) 178. K. Kuroda, M. Ye, A. Kimura, S.V. Eremeev, E.E. Krasovskii, E.V. Chulkov, Y. Ueda, K. Miyamoto, T. Okuda, K. Shimada, H. Namatame, M. Taniguchi, Physical Review Letters 105 (2010) 146801. S. Chadov, X. Qi, J. Kübler, G.H. Fecher, S.C. Zhang, Nature Materials 9 (2010) 541. S.V. Eremeev, Yu.M. Koroteev, E.V. Chulkov, JETP Letters 92 (2010) 161. S.V. Eremeev, G. Landolt, T.V. Menshchikova, B. Slomski, Y.M. Koroteev, Z.S. Aliev, M.B. Babanly, J. Henk, A. Ernst, L. Patthey, A. Eich, A.A. Khajetoorians, J. Hagemeister, O. Pietzsch, J. Wiebe, R. Wiesendanger, P.M. Echenique, S.S. Tsirkin, I.R. Amiraslanov, J.H. Dil, E.V. Chulkov, Nature Communications 3 (2012) 635. K. Kuroda, H. Miyahara, M. Ye, S.V. Eremeev, Yu.M. Koroteev, E.E. Krasovskii, E.V. Chulkov, S. Hiramoto, C. Moriyoshi, Y. Kuroiwa, K. Miyamoto, T. Okuda, M. Arita, K. Shimada, H. Namatame, M. Taniguchi, Y. Ueda, A. Kimura, Physical Review Letters 108 (2012) 206803.
M.G. Vergniory et al. / Applied Surface Science 267 (2013) 146–149 [14] T.V. Menshchikova, S.V. Eremeev, Yu.M. Koroteev, V.M. Kuznetsov, E.V. Chulkov, JETP Letters 93 (2011) 15. [15] S.V. Eremeev, Yu.M. Koroteev, E.V. Chulkov, JETP Letters 91 (2010) 594. [16] S.V. Eremeev, G. Bihlmayer, M. Vergniory, Yu.M. Koroteev, T.V. Menshchikova, J. Henk, A. Ernst, E.V. Chulkov, Physical Review B 83 (2011) 205129. [17] T. Sato, K. Segawa, H. Guo, K. Sugawara, S. Souma, T. Takahashi, Y. Ando, Physical Review Letters 105 (2010) 136802. [18] G. Kresse, J. Hafner, Physical Review B 48 (1993) 13115.
149
[19] G. Kresse, J. Furthmüller, Computation Materials Science 6 (1996) 15. [20] G. Kresse, D. Joubert, Physical Review B 59 (1998) 1758. [21] J.P. Perdew, K. Burke, M. Ernzerhof, Physical Review Letters 77 (1996) 3865. [22] D.D. Koelling, B.N. Harmon, Journal of Physics C 10 (1977) 3107. [23] O.G. Karpinskii, L.E. Shelimova, M.A. Kretova, E.S. Avilova, V.S. Zemskov, Inorganic Materials 39 (2003) 305. [24] L. Fu, Physical Review Letters 103 (2009) 266801.