Electronic states of SnTe and PbTe (001) monolayers with supports

Surface Science 639 (2015) 54–65

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Electronic states of SnTe and PbTe (001) monolayers with supports Katsuyoshi Kobayashi Department of Physics, Faculty of Science, Ochanomizu University, 2-1-1 Otsuka, Bunkyo-ku, Tokyo 112-8610, Japan

a r t i c l e

i n f o

Article history: Received 24 October 2014 Accepted 7 April 2015 Available online 14 April 2015 Keywords: Topological crystalline insulator Electronic structure Density functional calculation

a b s t r a c t Electronic states of SnTe and PbTe (001) monolayers are theoretically studied by density-functional calculations. The systems investigated are freestanding monolayers, monolayers on NaBr substrates, and monolayers sandwiched between two NaBr surfaces. Though isolated PbTe monolayers assumed in a planar structure have the edge states that suggest topological crystalline insulating states, the planar structure of freestanding monolayers is unstable and the mirror symmetry is lost by buckling. Calculations of monolayers on NaBr substrates show that though the attractive interaction between monolayers and substrates reduces buckling amplitude in monolayers, substrates produce a new asymmetry of electrostatic potential in monolayers. Theoretical calculations show that the SnTe monolayers sandwiched between two NaBr surfaces have the edge states suggesting that they are two-dimensional topological crystalline insulators. © 2015 Elsevier B.V. All rights reserved.

1. Introduction The topological crystalline insulator (TCI) is a new concept recently introduced by Fu [1]. In TCIs point group symmetries such as mirror symmetry are quantities classifying electronic states of insulators. SnTe is the first realistic material theoretically proposed for the TCI [2]. The topological surface states were experimentally observed in SnTe [3], Pb1 − xSnxSe [4] and Pb1 − xSnxTe [5]. These TCIs are threedimensional materials. Recently two-dimensional (2D) TCIs were theoretically proposed [6], where SnTe thin films are studied as an example of 2D TCIs. Theoretical studies on 2D TCIs were subsequently published by other groups [7,8]. These previous works studied freestanding 2D TCIs. In this paper we present a numerical study on electronic states of SnTe and PbTe (001) monolayers with supports, and show that SnTe (001) monolayers sandwiched between two NaBr (001) surfaces have the edge states suggesting 2D TCI states. In contrast to the time-reversal symmetry that characterizes topological insulators [9], crystal symmetries are easily broken under usual experimental conditions of thin films. The proposed 2D TCIs use the mirror symmetry with respect to the center planes of thin films. When TCI thin films are grown on substrates, the mirror symmetry with respect to the mirror plane of thin films is broken. In a strict sense gaps always open when TCI films are grown on substrates. However, the size of gaps depends on the strength of interaction with substrates. If substrates are properly chosen, gaps may be small enough to be negligible in ordinary experimental conductions. Another important point in thin films is the stability of planar structures. It is known that the carbon monolayer called graphene has a

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http://dx.doi.org/10.1016/j.susc.2015.04.009 0039-6028/© 2015 Elsevier B.V. All rights reserved.

planar structure, but the silicon monolayer called silicene has a buckled structure [10]. Both bulk SnTe and PbTe have the rock-salt structure of alkali halides. It is known that the (001) surfaces of alkali halides have rumpling structures. The (001) surface of PbTe also has a buckled structure. The heights of the topmost Pb and Te layers differ by 0.22 Å which is 7% of the bulk interlayer distance [11]. This buckling amplitude is in similar order as that of other alkali halides such as NaCl [12]. Both structural changes in thin films and the interaction with substrates may lead to symmetry breaking in 2D TCIs. In this paper we study these effects on electronic states of SnTe and PbTe (001) monolayers. We calculate the electronic states of the monolayers on substrates. A factor choosing substrates is lattice matching. The experimental values of bulk lattice constants for several crystals in the rock-salt structure are listed in Table 1. Band gap values are also shown in the table. We select the crystals that have lattice constants relatively close to those of SnTe and PbTe. Only the lattice constants smaller than those of SnTe and PbTe are shown. This is due to the tendency of the electronic states of SnTe and PbTe to change from normal insulators to TCIs with decreasing lattice constant [2,13]. In this paper we choose the NaBr (001) surface for substrates. The reason is that the bulk lattice constant of NaBr is smaller than those of SnTe and PbTe, but it is not too small such as that of NaCl. The two conditions of lattice matching and band inversion are expected to be satisfied. It may seem that the difference in bulk lattice constants between NaBr and SnTe or PbTe is large. However, there are reports that bilayer Bi films are grown on Bi2Te3 [16], Bi2Te2Se [17], and Bi2Se3 [18] substrates in a pseudomorphic structure. The lattice mismatch of bulk lattices in the last case is about 10%. Therefore, it may be possible for monolayers to grow on substrates even with a relatively large mismatch. The calculations in the present paper show that the in-plane

K. Kobayashi / Surface Science 639 (2015) 54–65 Table 1 Experimental lattice constants and band gap values of several bulk crystals in the rock-salt structure. The data of tin and lead chalcogenides are obtained from Ref. [14]. Those of alkali halides are from Ref. [15]. The band gap values differ depending on temperature and experiments. Crystal

Lattice constant (Å)

Band gap value (eV)

SnTe PbS PbSe PbTe LiI NaCl NaBr KCl RbF

6.3268 5.936 6.117 6.462 6.0228 5.6402 5.9772 6.2931 5.6516

0.19 0.286–0.41 0.145–0.278 0.187–0.310 6.1 8.6–9.0 7.1–7.7 8.48–8.7 10.3–10.4

lattice constants of SnTe and PbTe monolayers are smaller than those of their bulk ones. This property improves the lattice matching of the monolayers with the substrate. We calculate the electronic states of freestanding monolayers, monolayers on NaBr substrates, and monolayers sandwiched between two NaBr surfaces. We find that SnTe monolayers sandwiched between two NaBr (001) surfaces have the edge states suggesting 2D TCI states. 2. Method of calculations Electronic states are calculated by a density-functional method. The WIEN2k package [19] is used. This package uses the pull-potential linearized augmented plane wave method. The spin–orbital interaction (SOI) is included in the calculations. Since the force calculation with SOI is not yet implemented in the WIEN2k package, atom positions of systems are determined not by force but by total energy. We use the revised Perdew–Burke–Ernzerhof generalized gradient approximation (GGA), the so called PBEsol [20] for the exchange-correlation potential in total-energy calculations. This potential reproduces experimental values of lattice constants within a 1% accuracy. However, this potential produces smaller values of band gaps than experimental ones. This is crucial in the case of small-gap semiconductors such as SnTe and PbTe. Therefore, we use the modified Becke–Johnson (mBJ) potential [21] in band–structure calculations. Calculations of thin films are performed in supercell geometries. The widths of vacuum regions are larger than 10 Å. The numbers of wave vectors used in self-consistent calculations are 10 × 10 × 10 in 3-dimensional Brillouin zones for bulk calculations and 10 × 10 in 2D Brillouin zones for thin film calculations. The density-functional calculations are performed only for bulks and thin films. We also calculate the electronic states of wires with finite widths in order to investigate explicitly the existence of edge states which are signatures of TCIs. The electronic states of the wires are calculated by using the maximally localized Wannier function (MLWF) method [22–24]. We use the Wannier90 package [25] in order to obtain effective Hamiltonian expressed in terms of the MLWF basis. We use also the Wien2Wannier package [26] for interface calculations between the results of WIEN2k calculations and MLWF calculations. Three MLWFs per atom and spin are used for expressing the p bands of the SnTe and PbTe monolayers. Three MLWFs per atom and spin are also used for expressing the p states of Br in NaBr. No MLWF is used for the Na atoms. The outer and inner energy windows are used for constructing MLWFs from entangled bands [23]. The windows are chosen depending on the system. For an isolated SnTe monolayer they are the ranges between − 4.9 and 9.9 eV for the outer window and between −4.9 and 3.5 eV for the inner one. For a SnTe monolayer sandwiched between two NaBr surfaces they are between −6.3 and 8.1 eV for the outer window and between − 6.3 and 1.2 eV for the inner one. Here the origin of energy is the Fermi energy. The outer window is chosen as narrow as possible on the condition that it covers the p bands of SnTe monolayers. The p bands of Br are narrow and included in the energy range of the valence p bands of SnTe

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monolayers. The inner window is the frozen window. It is chosen as wide as possible on the condition that it excludes the conduction bands having the characters of Na atoms. Similar ranges of windows are chosen for PbTe monolayers. 8 × 8 wave vectors in 2D Brillouin zones are used to construct MLWFs. The calculated bulk lattice constants are 6.285 Å for SnTe and 6.441 Å for PbTe. Their deviations from the experimental values are 0.7% and 0.3%, respectively. The band gap values calculated by the mBJ potential are 0.189 eV for SnTe and 0.183 eV for PbTe. These values are calculated at the theoretically optimized lattice constants. The experimental value for SnTe is 0.19 eV at 300 K [14]. Those for PbTe are 0.187 eV at 4 K and 0.310 eV at 300 K [14]. The calculated band gap value for PbTe is similar to that obtained by quasiparticle calculations and other methods [27]. The calculated values using the PBEsol potential are 0.211 eV for SnTe and 0.008 eV for PbTe. It seems that the GGA potential is inadequate for reproducing the band gap of PbTe. The reason for reduction in the band gap of SnTe by the mBJ calculation is that the valence and conduction bands are inverted in SnTe. For reference we optimize atomic positions using forces calculated without SOI. This calculation is performed mainly for investigating whether planar structures of monolayers are stable or not. The PBEsol potential is used in the calculation. The geometry optimization is performed until absolute values of forces are less than 1 mRy/Bohr radius. The atomic positions obtained by the calculations without SOI are not much different from those with SOI. For example, the bulk lattice constants calculated without SOI are 6.285 Å for SnTe and 6.444 Å for PbTe. These values are almost the same as those calculated with SOI shown above. We checked that the geometrical parameters of the freestanding SnTe monolayer such as the in-plane lattice constant and the interlayer distance of buckling are also almost the same between the results obtained with and without SOI. We performed also calculations of heavier element Bi, and checked, for example, that the difference in the interlayer distance of a one-bilayer Bi film is less than 1% between calculations with and without SOI. It may not need to emphasize that electronic states are different. These results suggest that the results of geometry optimization calculated without SOI may be used for reference. The structural parameters obtained in this study are summarized in Appendix A.

3. Calculated results 3.1. Freestanding SnTe and PbTe (001) monolayers In this subsection we show the electronic states of freestanding SnTe and PbTe (001) monolayers. First we show results obtained by assuming planar structures. We determine the equilibrium in-plane lattice constants of SnTe and PbTe (001) monolayers by the total-energy calculations. The determined values are 4.292 Å for SnTe and 4.413 Å for PbTe. These values correspond to 6.070 and 6.241 Å for the lattice constant in the bulk cubic structure. Therefore, the inter-atomic distances of the SnTe and PbTe monolayers are contracted by 4.1 and 3.4% in comparison with the bulk structures, respectively. Fig. 1 shows band structures of the freestanding monolayers calculated at the determined lattice constants. Γ, X, and M are the points in the 2D Brillouin zone given by (0, 0), (π/a2D, 0), and (π/a2D, π/a2D), respectively. Here, a2D is the lattice constant of the 2D square lattice. The origin of energy is the Fermi energy. The SnTe monolayer is a semimetal, though the overlap of the valence and conduction bands is negligibly small. The PbTe monolayer is an insulator with a band gap of 0.16 eV. Fig. 2 shows band structures of wires calculated by the MLWF method. The unit cell of the wires consists of 100 atoms. The widths of the wires are 149 Å for SnTe and 153 Å for PbTe. The edge lines of the wires are parallel to [010] of the three-dimensional bulk structure. Since the SnTe monolayer is semimetal, the valence bands are connected with the conduction bands and there is no energy gap. The 2D bands

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K. Kobayashi / Surface Science 639 (2015) 54–65

Energy (eV)

(b)

Energy (eV)

(a)

Γ

X

M

Γ

Γ

X

M

Γ

Fig. 1. Band structures of SnTe (a) and PbTe (b) monolayers in planar structures.

of the PbTe monolayer have an energy gap between 0 and 0.16 eV. The states in this gap are edge states. The edge bands link the valence and conduction bands. Two edge bands cross linearly. There are two crossing points in the Brillouin zone. These characters are in common with the surface states of bulk SnTe that is a TCI [2]. Therefore we identify the edge states in the present system as topological edge states of a 2D TCI. We calculate the electronic states of the SnTe and PbTe monolayers at the bulk lattice constants. We found that edge states having similar characters exist in the SnTe monolayer, but they are absent in the PbTe monolayer. This result is parallel to the fact that bulk SnTe is a three-dimensional TCI, but PbTe is not. The result of the present calculations is different from those of other groups [6,8]. The calculations of the latter show that SnTe monolayers are topologically trivial. With increasing thickness of the SnTe films the band gap decreases and the transition to topologically nontrivial states occurs. Two factors are considered as the origins of this discrepancy. One is the method of calculations. Other groups used a tight-binding (TB) method [28]. The TB parameters are determined to reproduce the bulk bands of SnTe. They do not necessarily reproduce accurately the electronic states of SnTe thin films. In particular monolayers may be exceptional because the interlayer interaction is absent. Second is the difference in the lattice constant. Since the parameters for bulk bands are used, the TB calculation corresponds to the monolayer in the bulk lattice constant. The in-plane lattice constant of the monolayer determined by the density-functional calculation is smaller than the bulk lattice constant. The band inversion is enhanced

Energy (eV)

(b)

Energy (eV)

(a)

with decreasing the lattice constant [2,13]. Therefore, the result of the density-functional calculation tends to show topologically non-trivial electronic states. Next we show the results for non-planar structures. We assume that Sn, Pb, and Te atoms maintain the square sublattices, and Te atoms are displaced on the vertical center lines of the squares formed by the Sn and Pb sublattices. Only the distance between Sn or Pb and Te layers and the in-plane lattice constants varies. The determined interlayer distances are 1.421 Å for SnTe and 1.416 Å for PbTe. The in-plane lattice constants are 3.816 Å for SnTe and 3.930 Å for PbTe. The nearest-neighbor bond lengths are 3.050 Å for SnTe and 3.119 Å for PbTe. Those for planar structures are 3.035 Å for SnTe and 3.120 Å for PbTe. Therefore the bond lengths are little changed by buckling. The differences in total energy between the planar and buckled structures are 0.338 eV per unit cell for SnTe and 0.211 eV per unit cell for PbTe. The buckled structure is stabler than the planar one in both cases. Fig. 3 shows the band structures of the SnTe and PbTe monolayers in the buckled structures. The two band structures are similar, and both band structures are considerably different from those of the planar structures. The SnTe monolayer is an insulator with a small band gap of 0.03 eV. The PbTe monolayer is a semimetal. The Rashba-like spin splitting appears around the time-reversal symmetric points in the Brillouin zone. The buckling induces an electrical field between the charged Sn or Pb and Te layers. Therefore, spin splitting appears. The splitting in the PbTe monolayer is larger than that in the SnTe monolayer. This result is consistent with the difference in strength of SOI. Since the mirror symmetry is broken, both systems are topologically trivial.

0

π/a Wave Number

2π/a

0

π/a

2π/a

Wave Number

Fig. 2. Band structures of SnTe (a) and PbTe (b) monolayer wires in planar structures. a is the lattice constant of the one-dimensional lattice.

K. Kobayashi / Surface Science 639 (2015) 54–65

Energy (eV)

(b)

Energy (eV)

(a)

57

X

M

X

M

Fig. 3. Band structures of SnTe (a) and PbTe (b) monolayers in buckled structures.

Recently a theoretical study proposes that PbSe monolayers are TCIs [29]. However, this study investigates freestanding PbSe monolayers and it seems that the planar structure is assumed for the monolayers. Therefore, it is expected that the freestanding PbSe monolayers are not TCIs when structure optimization is taken into account. The results in this subsection show that the freestanding SnTe and PbTe (001) monolayers undergo buckling and they are not TCIs. The monolayers need supports in order to maintain the planar structures. In the next subsection the electronic states of monolayers on NaBr substrates are shown. 3.2. Monolayers on NaBr (001) substrates 3.2.1. SnTe monolayers on NaBr substrates We show electronic states of SnTe monolayers on NaBr (001) substrates. First we show results obtained by assuming planar structures for SnTe monolayers. We use thin films of a few layers for the NaBr substrates. The atomic positions of the NaBr films are fixed in the bulk structure. We determine the distance between a SnTe monolayer and the topmost NaBr layer of the substrates by total-energy calculations. Calculations are performed in the configuration that Sn and Te atoms are placed just above the Br and Na atoms, respectively. The optimized distance is 3.053, 3.078, and 3.076 Å for the 1, 2, and 3layer NaBr substrates, respectively. The difference is less than 1%. We show results calculated for the 3-layer NaBr substrate below. We also calculated the total energy in the configuration that Sn and Te atoms are placed just above the Na and Br atoms, respectively. However, we

(a)

did not find a minimum in total energy and the total energy is always larger than that calculated at the same distance in the former configuration. This result suggests that Sn and Te are positively and negatively charged, respectively. Fig. 4(a) shows a band structure of a SnTe monolayer on a NaBr substrate. The band structure is similar to that of the freestanding SnTe monolayer in the planar structure shown in Fig. 1. However, there are several differences. A large difference is the splitting of the degenerated states. The splitting is large near the Fermi energy and the X point. The origin of the splitting is discussed in Section 3.2.3. Another difference is the degree of the band inversion. The energy difference between the lowest conduction band and the highest valence band at X is 0.639 eV for the freestanding monolayer. It is 0.452 eV for the monolayer on the substrate. The electrostatic potential of the substrate is considered the main origin of this reduction in the band inversion as explained below. Though it is not shown in the figure, the p bands of Br atoms in the substrate appear in the energy range between − 6 and − 4 eV below the Fermi energy. Their band widths are narrow, and the hybridization with them is negligible for the states of the SnTe monolayer near the Fermi energy. Unoccupied states of the substrate Na atoms appear in the energy range more than 2 eV above the Fermi energy. Their hybridization is also negligible for the states near the Fermi energy. It can be thought that the main effect of the substrate on the SnTe monolayer is an addition of an electrostatic potential of the ionic crystal substrate. The states of SnTe near the Fermi energy consist mainly of p states of Sn and Te atoms. The energy level of the p orbital of Sn is higher

Energy (eV)

Energy (eV)

(b)

X

M

0 Wave Number

Fig. 4. Band structures of the SnTe monolayer (a) and wire (b) on a NaBr substrate. a is the lattice constant of the one-dimensional lattice. The SnTe monolayer has a planar structure.

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K. Kobayashi / Surface Science 639 (2015) 54–65

than that of Te [28]. In the inverted band structures the top of the Te bands is higher than the bottom of the Sn bands. The Sn and Te atoms in the monolayer are located on top of the Br and Na atoms of the substrate, respectively. The energy level of the p orbital of Sn tends to shift upward by the electrostatic potential of the negatively charged Br atoms located just below the Sn atoms. On the other hand, the energy level of Te tends to be lower by the Na atoms. Therefore the electrostatic potential of the ionic crystal substrates tends to decrease the band inversion. The in-plane lattice constant of the monolayer on the substrate is slightly reduced by 1.5% in comparison to the freestanding case, which tends to increase the band inversion. However, since the effect of the electrostatic potential of the substrate is stronger than that of the in-plane contraction, the band inversion is reduced. In order to confirm the explanation above more quantitatively we calculate the partial density of states (PDOS). PDOS is the density of states weighted with atomic orbitals centered at each atom. Fig. 5 shows PDOSs of an isolated SnTe monolayer, a SnTe monolayer on NaBr substrate, and a SnTe monolayer sandwiched between two NaBr surfaces. The last sandwiched system will be discussed later in Section 3.3. The PDOSs show that p-state characters of Sn and Te atoms are strong in the conduction and valence bands, respectively. The energy difference between the PDOS spectra of Sn atoms in the conduction bands and Te atoms in the valence bands near the Fermi energy increases constantly with additions of the NaBr surfaces. The electrostatic potential in the sandwiched structure is expected to be roughly twice that of the monolayer on the substrate because the electrostatic potential from the upper side of the monolayer is added to that from the lower side. Therefore, the energy difference between the p states

Partial Density of States (states/eV)

(b)

Partial Density of States (states/eV)

(a)

of Sn and Te atoms in the sandwiched structure is also larger than that of the monolayer on the substrate. We calculate diagonal elements of the Hamiltonian matrix represented in terms of the MLWFs. The difference between the averages of diagonal elements of the MLWFs centered at the Sn and Te atoms are 1.54 eV, 1.69 eV, and 1.92 eV for isolated SnTe monolayer, SnTe monolayer on NaBr surface, and SnTe monolayers sandwiched between NaBr surfaces, respectively. The monotonic change in energy difference coincides with the change in PDOS spectra. We estimate the Coulomb potential by the NaBr surfaces using a point charge model. Point charges with positive and negative elementary charges are placed at lattice points of a semi-infinite NaCl lattice. Potential energy is calculated at a point just above a lattice point. When the distance of the point from the surface of the semi-infinite lattice is the same as the nearest neighbor lattice distance, potential energy is 0.306 eV. Here we use the lattice constant of the bulk NaBr. Therefore, the difference in the potential energy between the Sn and Te atom sites is about 0.6 eV. This value is larger than the change of about 0.2 eV in the diagonal matrix elements of the MLWFs with the addition of a NaBr surface. The reason for this reduction may be the spatial distribution of wave functions. The Coulomb potential at the interstitial region between Sn and Te atoms is weaker than those at the centers of the atoms. Since the p states are distributed in regions far from the atom centers, the average of the Coulomb potential is smaller. These calculations show that the main reason for the systematic change in the band energy is the electrostatic potential of the substrate. In contrast to the freestanding SnTe monolayer in the planar structure, the SnTe monolayer on the NaBr substrate is a semiconductor with a band gap of 0.103 eV. Fig. 4(b) shows a band structure of a

Energy (eV)

Energy (eV)

Partial Density of States (states/eV)

(c)

Energy (eV) Fig. 5. Partial density of states of an isolated SnTe monolayer (a), a SnTe monolayer on NaBr substrate (b), and a SnTe monolayer sandwiched between two NaBr surfaces (c). Solid and broken lines show PDOSs of the p states of the Sn and Te atoms, respectively.

K. Kobayashi / Surface Science 639 (2015) 54–65

SnTe wire with a finite width on a NaBr substrate. The MLWF method is used in the calculation. The unit cell of the SnTe wire is 100 cells of the 2D unit cell of the SnTe monolayer. The width of the SnTe wire is 299 Å. The unit cell of the NaBr substrate is 150 cells of the 2D unit cell. A periodic boundary condition is imposed on the NaBr substrate. This models the wires with finite widths on substrates with infinite areas. However, the lateral size of substrates is not so important because the states near the Fermi energy are well localized within the SnTe monolayer. The calculated result shows no edge state in the band gap. The reason for absence of edge states is spatial asymmetry in the electronic states of the SnTe monolayer as discussed in Section 3.2.3. Next we consider the buckling in the SnTe monolayer. We calculate the total energy by independently varying the heights of the Sn and Te layers on the substrate. The optimized heights shift from those in the planar structure toward the substrate side by 0.030 Å for the Sn layer and toward the vacuum side by 0.345 Å for the Te layer. The interlayer distance between the Sn and Te layers by buckling is 0.375 Å. The bond length between Sn and Te atoms is 3.012 Å which is close to that of the freestanding SnTe monolayer. The buckling in the SnTe monolayer on the substrate is much reduced from that of the freestanding SnTe monolayer. The energy difference between the planar and buckled structures of the SnTe monolayer on the NaBr substrate is 0.009 eV which is negligibly small in comparison with that of the freestanding SnTe monolayer. The interaction with the substrate tends to stabilize the planar structure of monolayers. We optimize the atomic structure of the SnTe monolayer on the 3-layer NaBr substrate by forces acting on atoms calculated without SOI. In this case we optimize the heights of all sublattice layers including the NaBr substrate except for the bottom layer of the NaBr substrate fixed on a plane. The optimized atomic positions are listed in Table 6. The obtained distance between Sn and Te layers is 0.465 Å which is slightly larger than that shown above. The optimized distance between Na and Br layers in the topmost NaBr layer is 0.189 Å. The Na layer shifts toward the vacuum side. The buckling in the topmost NaBr layer may be the reason for the enhancement of buckling in the SnTe monolayer. We optimize also the atomic structure of a bare NaBr substrate without the SnTe monolayer. The optimized atomic positions are listed in Table 6. The distances between the Na and Br layers are 0.150 Å for the topmost layer and 0.045 Å for the second layer. These values are similar to those of the NaCl (001) surface [12]. In the bare NaBr surface the Br layer in the topmost layer and the Na layer in the second layer shift toward the vacuum side as usually seen in the rumpling structure of alkali halide surfaces. In the case with the SnTe monolayer also the Na layer in the topmost layer of the substrate, which is the second layer of the system, shifts to the vacuum side. However, since the buckling of the SnTe monolayer is larger than that of the topmost layer of the bare NaBr surface, that in the topmost layer of the substrate is also large. We optimize the atomic positions of the SnTe monolayer on the NaBr substrate with fixing all the substrate Na and Br atoms in the bulk positions. In this case the distance between Sn and Te layers by buckling is 0.390 Å which is close to the value obtained by the calculation with SOI. A theoretical calculation of the bare NaCl (001) surface using a slab having a 12 layer thickness shows that buckling reconstruction is restricted to the topmost and second layers [30]. The reconstruction in the layers below them can be neglected. An experiment of NaCl and KCl surfaces shows similar results [12]. Furthermore, NaBr is an insulator with a large band gap. Wavefunctions of occupied states are well localized at atoms and the transfer between atoms is very small. These results suggest justification of the use of the three-layer slab for NaBr substrates. The small difference in the buckling amplitude of the topmost NaBr layer between the bare NaBr surface and that covered with SnTe monolayer also suggests the justification of the use of the threelayer slab.

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3.2.2. PbTe monolayers on NaBr substrates First we show results obtained by assuming the planar structure for the PbTe monolayer. We determine the distance between the PbTe monolayer and the topmost layer of the NaBr substrates by the totalenergy calculation. The distances determined are 3.124, 3.137, 3.146, 3.152, and 3.152 Å for the 1, 2, 3, 4, and 5-layer NaBr substrates, respectively. The difference is less than 1%. We use 3.146 Å and show results calculated for the 3-layer NaBr substrate below. The obtained distance is larger than that between the SnTe monolayer and the NaBr substrates. This result is consistent with the difference in the interlayer distances between bulk SnTe and PbTe. Fig. 6(a) shows a calculated band structure of a PbTe monolayer on a 3-layer NaBr substrate. The band structure is similar to that of the isolated PbTe monolayer. The reason for the similarity is the same as that of the SnTe monolayer on NaBr substrates. It is the compensation of the reduction of the band inversion by the electrostatic potential of the substrates and the enhancement of the band inversion by the in-plane contraction of the PbTe monolayer. The degree of the in-plane contraction of the PbTe monolayer is larger than that of the SnTe monolayer. The PbTe monolayer on NaBr substrates is a semiconductor with a band gap. We calculate the band gap value with changing the number of layers of the NaBr substrate. The band gap values obtained are 0.0661, 0.0921, 0.0905, 0.0910 eV for the 1, 2, 3, and 4-layer NaBr substrates, respectively. These values are calculated with fixing the distance between PbTe monolayer and NaBr substrates at the optimized value for the 3-layer NaBr substrate. The band gap value is small for the monolayer NaBr substrate. This is due to the fact that the effect of the electrostatic potential of the substrate is strongest in the monolayer NaBr case. When a NaBr layer is added below the NaBr monolayer, the second NaBr layer weakens the electrostatic potential of the first layer. Therefore the band gap value increases in the 2-layer NaBr substrate. In the present system it can be concluded that the band gap value is regarded as converged at 3 or 4 NaBr layers. The result that the band gap increases with a weakening of the electrostatic potential of substrates implies that the band is inverted in the present system. Fig. 6(b) shows a band structure of a PbTe wire with a finite width on a NaBr substrate calculated by the MLWF method. The unit cells of the PbTe wire and the NaBr substrate are 100 and 150 cells of the 2D unit cell, respectively. The periodic boundary condition is imposed on the substrate. The spacing between the edges of the adjacent PbTe wires is 50 cells. We verified that both the width of a PbTe wire and the spacing of the wires are large enough to spatially separate the wave functions of edge states localized at each edge. The 2D bands of this system have an energy gap between 0 and 0.9 eV. Edge states appear in this gap. Though a gap of 0.019 eV opens in the edge-state bands, these edge states may be regarded as states originated from the topological edge states of TCIs. The reason for opening of the gap is breaking of mirror symmetry with respect to the PbTe plane by the substrate. However the degree of the symmetry breaking is not strong in the case of PbTe. Next we consider the buckling. First we determine the heights of the Pb and Te layers by the total-energy calculation with SOI. The optimized heights shift to the vacuum side by 0.056 Å for the Pb layer and 0.869 Å for the Te layer measured from the heights in the planar structure. The distance between the Pb and Te layers is 0.813 Å. The bond length between the Pb and Te atoms is 3.097 Å. The buckling is smaller than that of the freestanding monolayer, but it is larger than that of the SnTe monolayer on the NaBr substrate. A reason for the latter may be the larger lattice mismatch between the monolayer and the substrate than the SnTe monolayer. We also optimize the atomic positions without SOI. The result is listed in Table 6. The optimized distance between the Pb and Te layers is 0.799 Å. That between the Na and Br layer in the topmost NaBr layer is 0.169 Å. When the atoms of the NaBr substrate are fixed in the bulk positions, the distance between the Pb and Te layers is 0.820 Å. These results do not differ much from those of the calculation with SOI.

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K. Kobayashi / Surface Science 639 (2015) 54–65

(b) Energy (eV)

Energy (eV)

(a)

X

M

0 Wave Number

Fig. 6. Band structure of PbTe monolayer (a) and wire (b) on a NaBr substrate. a is the lattice constant of the one-dimensional lattice. The PbTe monolayer has a planar structure.

A buckling distance of about 0.8 Å is large enough for the edge state of the PbTe monolayer on the substrate to disappear. We calculate the band structure of the PbTe monolayer on the substrate in the buckled structure. The obtained band structure is similar to that of the isolated PbTe monolayer in the buckled structure shown in Fig. 3. The electronic state is a semiconductor with an indirect gap of 0.414 eV. The Rashbalike spin splitting appears. The bottom of the conduction band and the top of the valence band are located at intermediate points between X and M and between Γ and M, respectively. Therefore we need structures to suppress the buckling in the monolayers in order to preserve the TCI state. In Section 3.3 we present the electronic states of monolayers sandwiched between two surfaces. 3.2.3. Origin of the splitting in energy bands of SnTe and PbTe monolayers on substrates We discuss the origin of the splitting in energy bands of the SnTe and PbTe monolayers on NaBr substrates in the planar structure. Fig. 7 shows band structures of isolated SnTe and PbTe monolayers calculated by the MLWF method. The MLWFs used in the calculations are those obtained from the calculations for the systems of the monolayers on NaBr substrates. The present calculations are performed in order to extract the pure changes in the electronic states of monolayers influenced by substrates. The effect of direct orbital hybridization between monolayers and substrates is separated. The calculated band structures show two differences between SnTe and PbTe monolayers. One is that the PbTe monolayer is a semiconductor with a gap, but the SnTe monolayer has no gap. Second is that the splitting of degenerated states is

Energy (eV)

(b)

Energy (eV)

(a)

small in PbTe but it is large in SnTe. These two points suggest that the effect of the substrate is large in SnTe, and the band gap in the SnTe monolayer on the substrate opens due to the hybridization with the substrate. In order to clarify the origin of splitting of the degenerated states we perform a TB calculation for the isolated SnTe monolayer in the planar structure. We use a usual TB method with one s and three p orbitals for each atom. The overlap integrals are neglected. The second nearest neighbor transfers are taken into account in order to qualitatively reproduce the lowest conduction band. The TB parameters are determined to reproduce the energy band obtained by the density-functional calculation. The density-functional calculation is performed for the isolated SnTe monolayer in the planar structure at the 2D lattice constant of the NaBr (001) surface. Table 2 shows the determined TB parameters. εs and εp are on-site energies of the s and p orbitals. Δ is splitting energy of the p orbitals of an atom by SOI. One third of Δ is the matrix element of SOI. The matrix elements between Sn and Te atoms are the nearest neighbor transfers. Those between Sn atoms and between Te atoms are the second nearest neighbor ones. Fig. 8 shows band structures calculated by the TB method. Fig. 8(b) shows a band structure calculated by the TB method, where the intra-atomic matrix element of 0.5 eV is taken into account between the s and pz orbitals of Sn atoms. Here the z axis is perpendicular to the monolayer plane. The calculated band structure reproduces the features of the band structure shown in Fig. 7(a). The origin of splitting in the energy bands is considered to be the σ–π hybridization in the monolayers. We checked that the inclusion of the

X

M

X

M

Fig. 7. Band structures of isolated SnTe (a) and PbTe (b) monolayers in planar structures calculated by the MLWF method. The MLWFs of monolayers on NaBr substrates are used.

K. Kobayashi / Surface Science 639 (2015) 54–65

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3.3. Monolayers sandwiched between two NaBr (001) surfaces

Table 2 Tight-binding parameters. On-site energy (eV)

Sn

Te

εs εp Δ

−7.1978 0.0272 0.5245

−9.2736 −1.1991 0.7649

Transfer energy (eV)

Sn–Te

Sn–Sn

Te–Te

Vss Vsp Vps Vppσ Vppπ

−0.9414 0.9641 −1.6066 1.8773 −0.6534

−0.2363 0.0130 −0.0130 0.4142 −0.1479

0.1893 0.1378 −0.1378 −0.2468 0.1576

intra-s–pz matrix element for Te atoms produces only negligible influences on the results. The interaction with substrates may be considered to divide mainly into two effects. One is the electrostatic potential. The second is the orbital hybridization. When thin films with thickness more than one layer are placed on substrates, the electrostatic potential of the substrate induces asymmetry in the on-site energy of atomic orbitals between the upper and lower layers. The mirror symmetry in thin films breaks. The asymmetry in the on-site energy between the upper and lower layers is absent in the case of monolayers on substrates. However, the electric field generated by substrates induces on-site coupling between the s and pz orbitals. This produces mirror asymmetry in monolayers and the splitting of degenerated states. The orbital hybridization between monolayers and substrates also indirectly couples the σ and π orbitals in monolayers. For example, both the s and pz orbitals in the monolayers hybridize with the pz orbitals of Br atoms in the topmost layer. Therefore, the s and pz orbitals in the monolayers couple in the second order mediated by the pz orbitals of Br atoms. A comparison between Figs. 4, 6, and 7 shows the following results. The coupling between the σ and π orbitals due to the electrostatic potential of the substrate is strong in the case of the SnTe monolayer. The effect of the orbital hybridization with the substrate Br atoms appears mainly as a constant shift by about 0.5 eV of the lowest conduction band. Since the lowest conduction band is composed mainly of the pz orbitals of the Sn and Te atoms, it is reasonable that their hybridization with the pz orbitals of Br atoms is larger. On the other hand, the coupling between the σ and π orbitals due to the electrostatic potential of the substrate is weak in the case of the PbTe monolayer. The effect of the orbital hybridization with the substrate Br atoms appears as splitting of energy bands as well as a shift of the lowest conduction.

Energy (eV)

(b)

Energy (eV)

(a)

3.3.1. SnTe monolayer between two NaBr surfaces We consider a system of a SnTe monolayer sandwiched between two NaBr (001) thin films. The thickness of the NaBr thin films is 3 layers. In supercell calculations the NaBr thin films are separated by a vacuum region with a width more than 10 Å. First we assume planar structures for the SnTe monolayer and NaBr thin films. The distance between the SnTe monolayer and the topmost layers of the NaBr films is determined by the total-energy calculation. The total energy is calculated in the configuration that Na and Br atoms are positioned just above and below Te and Sn atoms, respectively. The determined distance is 3.137 Å which is larger by 2% than that of the SnTe monolayer on NaBr substrates. The total energy is calculated in the configuration with interchanging the positions of Na and Br atoms in the upper NaBr surface. In this case no minimum in total energy is found and the total energy is always larger than that in the former configuration. This result is similar to that of the SnTe monolayer on a NaBr surface. The mirror symmetry is naturally preserved in this sandwiched system. The total energy calculated in the configuration with the interchange of atoms in both sides of NaBr surfaces is always larger than those in the two configurations above. Fig. 9(a) shows the band structure of a SnTe monolayer sandwiched between two 3-layer NaBr thin films. The system is a semiconductor with a band gap of 0.072 eV. Fig. 9(b) shows a band structure of a SnTe wire sandwiched between two NaBr thin films. The unit cell of the wire consists of 150 2D unit cells. The width of the wire is 448 Å. The edge line of the wire is parallel to [010] of the three-dimensional bulk structure. Edge states exist in the band gap of the thin film. The edge states have characters in common with those of the isolated PbTe monolayers. Therefore, the SnTe monolayer sandwiched between the two NaBr surfaces can be identified as a TCI. Fig. 10 shows a wave function of the edge states. The wave function is expressed by squared absolute values of the coefficients of MLWFs. The figure shows the state with wave number k = 0.977π/a and energy E = 0.016 eV. Here, a is the lattice constant of the one-dimensional lattice. In the calculation of the wave function, we use a system composed of the upper NaBr film and SnTe monolayer with a width of 150 2D unit cells and the lower NaBr film with a width of 200 2D unit cells. The periodic boundary condition is imposed on the lower NaBr film. This structure models a wire on a NaBr substrate. The unit cell of the SnTe monolayer wire consists of two inequivalent atomic rows. One row has a Sn atom at an edge and the other has a Te atom at the edge of the same side. Fig. 10(a) shows the wave function on a plane perpendicular to the edge line and containing the atomic row that has a Sn atom at the edge. Fig. 10(b) shows the wave function on both atomic rows.

X

M

X

M

Fig. 8. Band structures of a SnTe monolayer calculated by a tight-binding method. The intra-atomic matrix element between the s and pz orbitals of the Sn atoms is 0 (a) and 0.5 eV (b).

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K. Kobayashi / Surface Science 639 (2015) 54–65

(a)

Energy (eV)

Energy (eV)

(b)

X

(1/2)π/a

M

π/a

(3/2)π/a

Wave Number Fig. 9. Band structures of SnTe monolayer (a) and wire (b) sandwiched between two NaBr films. a is the lattice constant of the one-dimensional lattice.

Fig. 10(a) shows that most of the wave function is localized within the SnTe layer. The leakage in the NaBr films is only 3%. Therefore, the separation of 50 2D unit cells in the NaBr substrate is large enough for the wire on the substrate to regard as an isolated one. The wave function is also calculated for a SnTe monolayer wire between two NaBr wires with the same finite width of 150 2D unit cells. The obtained results show only negligible difference from those of the system with imposing the periodic boundary condition on the NaBr substrate. The Sn and Te atoms in this wire structure are alternately placed along the rows perpendicular to the edge line. The amplitude of the wave function in the SnTe monolayer also alternately changes along the rows. Fig. 10(a) shows that the wave function of this state has a larger amplitude at the Sn sites. The two sites with larger and smaller amplitudes form respective envelope lines in Fig. 10(b). The Sn and Te sites have larger and smaller amplitudes in both rows, respectively. Fig. 10(b) shows that the decay of the wave function is slow. The wave function of the edge state distributes in the region about 100 Å from the edge. The reason for the slow decay may be the small band gap in the present system. This result suggests that preparations of atomically straight edges are not necessarily required in order to observe experimentally the edge states, for example, by scanning tunneling spectroscopy. Small roughness in edge lines may be acceptable. Fig. 11 shows band structures of a SnTe wire having edges with a different orientation. The edge lines are parallel to the [110] direction of

(a)

the bulk NaCl structure. In this case two types of edges are considered. One is terminated by the Sn atoms and the other is by the Te atoms. The figure shows results for both edges. The unit cell of the SnTe monolayer wire is 150 2D unit cells. The two NaBr surfaces are also wires with the same width. The edge states of the [110] edge appear near the center as well as the edge of the Brillouin zone. The band dispersions of the edge states are different between the Sn and Te terminated edges. They seem similar to those of the conduction and valence bands, respectively. This result is reasonable because the states in the conduction and valence bands are composed mainly of the p states of the Sn and Te atoms, respectively. The properties of the edge states of the [110] edges are similar to those of the surface states of the (111) surfaces of bulk SnTe [31]. The surface states appear near two points in the Brillouin zone. The band dispersions are different between the Sn and Te terminated surfaces, and they are similar to those of the conduction and valence bands. The band gap of the SnTe monolayer between two NaBr surfaces is smaller than that of the isolated SnTe monolayer. This is considered to be due to the electrostatic potentials of the two NaBr surfaces on both sides of the monolayer. The reduction of the band inversion by the electrostatic potential is roughly twice that of the SnTe monolayer on the NaBr substrates. However, since the mirror symmetry is recovered, the edge states appear. We calculate the difference in total energy between the present combined system and the system separated into a SnTe monolayer

Squared Amplitude of Wave Function

(b)

z( )

x(

)

x(

)

Fig. 10. Wave function of an edge state of a SnTe monolayer wire between two NaBr films. (a) Cross sectional view of distribution on a plane perpendicular to the edge line. Open circles show the positions of atoms. Closed circles express the distribution of the wave function. The size of the closed circles is proportional to the squared amplitude of the wave function. (b) Squared amplitude of the wave function in the SnTe monolayer on two inequivalent atomic rows perpendicular to the edge line. Closed and open circles show the amplitude on the rows having Sn and Te atoms at the edge, respectively. x = 0 is the position of the edge atoms in the wire.

K. Kobayashi / Surface Science 639 (2015) 54–65

Energy (eV)

(b)

Energy (eV)

(a)

63

π/a

0

π/a

0

Wave Number

Wave Number

Fig. 11. Band structures of SnTe monolayer wires sandwiched between two NaBr films. Edges of the wires are terminated by Sn (a) and Te (b) atoms. a is the lattice constant of the onedimensional lattice.

and two 3-layer NaBr films. The energy of the combined system is lower than that of the separated system by 0.488 eV per unit cell. In the totalenergy calculation we take the buckling structure into account for the SnTe monolayer. This result suggests that the present combined system is a stable structure. We optimize also all the atomic positions without SOI. The result is listed in Table 6. The distance between the Sn and Te layers by buckling is less than 0.01 Å. The small buckling in the monolayer may be induced by the buckling in the surface NaBr layer. We confirm that the buckling in the monolayer is reduced by increasing the thickness of the NaBr films to 4 layers. We calculate the electronic states in the optimized positions. The energy splitting of the edge states by this small buckling is less than 0.01 eV, and the edge states remain in the wire. Therefore the property of the SnTe monolayer between two NaBr surfaces as a TCI is robust. Recently 2D structures of tin and lead chalcogenides are theoretically studied [32]. In this study the low-energy structure proposed for the freestanding 2D SnTe and PbTe is a distorted NaCl structure. The distorted NaCl structure is a double-layer structure consisting of two layers of a 2D NaCl lattice. The cohesive energy between two layers of the 2D NaCl lattice contributes to the stability of the distorted NaCl structure. Therefore, low-energy 2D structures placed on substrates may differ from the freestanding structure because the cohesive energy

(a)

between a 2D layer and a substrate also contributes to the stability. In order to estimate roughly the cohesive energy we perform totalenergy calculations. We assume planar structures for the SnTe and NaBr layers and optimize the interlayer distances. When a single SnTe layer is placed on a single NaBr layer, the cohesive energy is 0.469 eV per unit cell. When a single SnTe layer is further placed on the SnTe layer with the NaBr layer, the cohesive energy is 1.243 eV per unit cell. This result suggests that SnTe monolayers exist on NaBr substrates only as metastable structures. It may be difficult to experimentally fabricate SnTe monolayers on NaBr surfaces. However, if we choose substrates with a larger cohesive energy, SnTe monolayers may grow on substrates. In this case electronic states similar to those shown in this paper are expected to obtain. In the same paper a litharge structure is investigated as a candidate for low-energy 2D structures. pffiffiffi pThe ffiffiffi litharge structure is a single-layer buckled structure with a 2  2 unit cell. The buckling direction alternates in the monolayers. Our theoretical calculations show that the litharge structure is slightly stabler than the buckled one. The energy difference between them is 0.027 eV per 1 × 1 unit cell. This is smaller than the cohesive energy of the SnTe monolayer and two NaBr surfaces. Therefore, even when the litharge structure is taken into account, the planar structure is expected for the low-energy structure of the SnTe monolayer between two NaBr surfaces. We optimized the atomic

Energy (eV)

Energy (eV)

(b)

X

M

(1/2)π/a

π/a

(3/2)π/a

Wave Number Fig. 12. Band structures of PbTe monolayer (a) and wire (b) sandwiched between two NaBr films. a is the lattice constant of the one-dimensional lattice.

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K. Kobayashi / Surface Science 639 (2015) 54–65

structure of a SnTe monolayer sandwiched between two NaBr surfaces with an initial condition of the SnTe monolayer in the litharge structure. The obtained structure of the SnTe monolayer is the planar one. Therefore the planar structure is stable when monolayers are sandwiched between two surfaces.

3.3.2. PbTe monolayer between two NaBr surfaces First we show results obtained by assuming the planar structure for the PbTe monolayer and NaBr thin films. We determine the distance between the monolayer and the topmost layer of the NaBr films by the total-energy calculation. The determined distance is 3.182 Å. Fig. 12(a) shows a band structure of a PbTe monolayer sandwiched between two 3-layer NaBr films. The system is a semiconductor with a band gap of 0.048 eV. Fig. 12(b) shows a band structure of a SnTe wire sandwiched between two NaBr films. No edge state exists in the band gap of the thin film. The PbTe monolayer sandwiched between two NaBr surfaces is a normal semiconductor. The band gap value at X is 0.688 eV for the isolated PbTe monolayer and 0.332 eV for the PbTe monolayer on the NaBr substrate. In both cases the band gap is calculated for the planar structure. The electrostatic potential of the substrate reduces the inverted band gap by 0.356 eV. Therefore it is reasonable that the band gap of the PbTe monolayer sandwiched between two NaBr surfaces is 0.048 eV and the inverted band changes to the normal one. We optimize the atomic positions without SOI. The result is listed in Table 6. The distance between Pb and Te layers in the PbTe monolayer is 0.084 Å. Therefore the almost planar structure is stabilized by the supports on both sides. However, in spite of the small buckling the PbTe monolayer sandwiched between two NaBr surfaces is a normal insulator.

4. Conclusion In this paper we presented calculations of electronic states of SnTe and PbTe monolayers without and with supports. The focus is the possibility of realization of 2D TCIs. The isolated PbTe monolayers assumed in the planar structure have edge states with characters in common with TCIs. Therefore, they are identified as 2D TCIs. However, the freestanding monolayers undergo buckling, and they are not TCIs. When these monolayers are placed on NaBr (001) substrates, the amplitude of buckling is reduced by the attractive interaction with the flat surfaces. When the planar structure is assumed for the monolayers, edge states appear in the band gap of 2D states of the PbTe monolayer. However, when the constraint of the planar structure is removed, the buckling amplitude is still too large and the edge states disappear. In order to maintain the monolayers in the planar structure we consider a structure of the monolayers sandwiched between two NaBr surfaces. The theoretical calculations show that the SnTe monolayer between two NaBr surfaces has the edges states and, theretofore, is regarded as a candidate for 2D TCIs. In the case of PbTe monolayers the inverted band of the isolated PbTe monolayer changes to the normal one by the electrostatic potential on both sides of the monolayer. In the present paper we judge the topological property of the systems by the existence of edge states that have characters in common with TCIs. In order to discuss the topological properties more rigorously it is desirable to calculate the mirror Chern numbers defined for systems having the mirror symmetry [33]. A merit of using alkali halides for supports is that they are insulators with large band gaps. The electronic states of alkali halides are separated in energy from those near the Fermi energy of the narrow gap semiconductors. The effect of orbital hybridization between these states can be neglected. The main influence of the supports on the monolayers is the electrostatic potentials by the ionic atoms.

The electrostatic potential of the alkali halide surfaces tends to reduce the band inversion of the monolayers due to the staggered sign of the electric field applied on sublattices. However, the reduction of the band inversion is not so much in the case of the SnTe monolayer. The band structure remains an inverted one. In this paper we presented results for the SnTe and PbTe monolayers. The case of multilayers may be interesting. The materials studied in the original proposal of the 2D TCI are SnTe multilayer films [6]. This is because the TB calculation of SnTe thin films in the structures with the bulk lattice constant shows a transition to 2D TCI at a certain thickness [6,8]. If NaBr surfaces are used for the supports of multilayers, inhomogeneous electric fields are applied on the multilayers. This may give different properties from the isolated multilayers. A merit of multilayers may be the stability of planar structures. It is expected that the influence of buckling is small and the mirror symmetry is preserved. In this paper we chose NaBr surfaces as an example of the supports of monolayers. Studies on other materials for supports may be important. Alkali halide surfaces with different lattice constants may control the topological property of the monolayers. Other lead chalcogenides such as PbS and PbSe are in the same rock salt structure, but their band gaps are narrow [14]. In this case, in addition to the control of lattice constants, orbital hybridization between the monolayers and supports gives a considerable effect on the electronic states near the Fermi energy. A recent theoretical study shows that the superlattices composed of PbTe and SnTe layers are weak topological insulators [34]. When PbSe and PbS are used for supports, complex and various results are expected to be obtained. As discussed in Section 3.3.1 the experimental realization of the systems studied in this paper may not be easy. An important point is fabrication of stable monolayers on substrates. The desirable properties for substrates are that the cohesive energy with the monolayers is larger, but the electronic states of the monolayers are not much affected by substrates. The candidates for materials satisfying these demands may be alkaline earth chalcogenides. They are ionic crystals having the same crystal structure as alkali halides, but they are composed of divalent ions which are expected to enhance the cohesive energy with monolayers. Studies on these materials are interesting.

Acknowledgments Numerical calculations were performed by using supercomputers at the Institute of Solid State Physics and Information Technology Center, The University of Tokyo. This work was supported by JSPS KAKENHI Grant Number 26390063. After submitting this manuscript, a theoretical work on isolated IVVI monolayers was published [35] and two theoretical manuscripts on IV-VI monolayer superlattices were uploaded on a preprint server [36, 37].

Appendix A. Structural parameters obtained in this study This appendix summarizes the structural parameters obtained in this study.

Table 3 Interlayer distance between the Sn or Pb and Te layers of monolayers for freestanding monolayers, monolayers on a NaBr surface, and monolayers sandwiched between two NaBr surfaces. The distance is shown in Å.

Freestanding monolayers Monolayers on a surface Sandwiched monolayers

SnTe

PbTe

1.412 0.465 0.009

1.404 0.799 0.084

K. Kobayashi / Surface Science 639 (2015) 54–65 Table 4 In-plane lattice constant of isolated monolayers shown in Å.

Planar structure Buckled structure

SnTe

PbTe

4.292 3.816

4.413 3.930

Table 5 Interlayer distance between a monolayer and the nearest neighbor NaBr layer for monolayers on a NaBr surface and monolayers sandwiched between two NaBr surfaces. The planar structure is assumed. The distance is shown in Å.

Monolayers on a surface Sandwiched monolayers

SnTe

PbTe

3.076 3.137

3.146 3.182

Table 6 Height of layers obtained by optimization shown in Å. The heights of atoms in the bottom NaBr layer are fixed at zero.

Na Br Na Br Sn or Pb Te Na Br Na Br Na Br

Monolayers on a substrate

Sandwiched structure

NaBr surface

SnTe

PbTe

SnTe

PbTe

2.924 3.025 6.098 5.908 8.943 9.408

2.918 3.023 6.051 5.882 9.021 9.820

2.916 2.938 5.888 5.890 9.023 9.032 12.168 12.173 15.176 15.129 18.025 18.165

2.937 2.940 5.880 5.917 9.099 9.015 12.185 12.264 15.253 15.172 18.067 18.222

3.000 2.955 5.843 5.994

Table 3 shows buckling amplitude of monolayers. Structural optimization is performed without SOI. The buckling amplitude is also calculated with SOI for freestanding monolayers. The interlayer distance is 1.421 and 1.415 Å for the SnTe and PbTe monolayers, respectively. The values with and without SOI do not differ much. Table 4 shows the in-plane lattice constant of isolated monolayers. The lattice constant is determined by total energy calculations with SOI. The in-plane lattice constant is also calculated without SOI for the buckled structures. The lattice constant is 3.814 and 3.930 Å for the SnTe and PbTe monolayers, respectively. These values are almost the same as those with SOI. Table 5 shows the interlayer distance between a monolayer and the nearest neighbor NaBr layer when the planar structure is assumed. The distance is determined by total energy calculations with SOI. The inplane lattice constant is 4.227 Å which is the lattice constant of a NaBr surface.

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Table 6 shows heights of atomic layers determined by structure optimization. The optimization is performed by force calculations without SOI. The substrate is a three-layer NaBr film. The height of the bottom NaBr layer is fixed at zero in the optimization. The in-plane lattice constant is 4.227 Å. Na and Br atoms are positioned just above and below the Sn or Pb and Te atoms, respectively. Stacking of layers in the NaBr films is that of the usual NaCl structure. References [1] L. Fu, Phys. Rev. Lett. 106 (2011) 106802. [2] T.H. Hsieh, H.L. Lin, J. Liu, W. Duan, A. Bansil, L. Fu, Nat. Commun. 3 (2012) 982. [3] Y. Tanaka, Z. Ren, T. Sato, K. Nakayama, S. Souma, T. Takahashi, K. Segawa, Y. Ando, Nat. Phys. 8 (2012) 800. [4] P. Dziawa, B.J. Kowalski, K. Dybko, R. Buczko, A. Szczerbakow, M. Szot, E. Łusakowska, T. Balasubramanian, B.M. Wojek, M.H. Berntsen, et al., Nat. Mater. 11 (2012) 1023. [5] S.-Y. Xu, C. Liu, N. Alidoust, M. Neupane, D. Qian, I. Belopolski, J. Denlinger, Y. Wang, H. Lin, L. Wray, et al., Nat. Commun. 3 (2012) 1192. [6] J. Liu, T.H. Hsieh, P. Wei, W. Duan, J. Moodera, L. Fu, Nat. Mater. 13 (2014) 178. [7] M. Ezawa, New J. Phys. 16 (2014) 065015. [8] H. Ozawa, A. Yamakage, M. Sato, Y. Tanaka, Phys. Rev. B 90 (2014) 045309. [9] C.L. Kane, E.J. Mele, Phys. Rev. Lett. 95 (2005) 146802. [10] K. Takeda, K. Shiraishi, Phys. Rev. B 50 (1994) 14916. [11] A.A. Lazarides, C.B. Duke, A. Paton, A. Kahn, Phys. Rev. B 52 (1995) 14895. [12] J. Vogt, H. Weiss, Surf. Sci. 491 (2001) 155. [13] P. Barone, D. Di Sante, S. Picozzi, Phys. Status Solidi (RRL) 7 (2013) 1102. [14] O. Madelung (Ed.), Semiconductors: Other Than Group IV Elements and III–V Compounds, Springer, Berlin, 1992. [15] D.B. Sirdeshmukh, L. Sirdeshmukh, K.G. Subhadra, Alkali Halides: A Handbook of Physical Properties, Springer, Berlin, 2010. 6. [16] T. Hirahara, G. Bihlmayer, Y. Sakamoto, M. Yamada, H. Miyazaki, S.-I. Kimura, S. Blügel, S. Hasegawa, Phys. Rev. Lett. 107 (2011) 166801. [17] S.H. Kim, K.-H. Jin, J. Park, J.S. Kim, S.-H. Jhi, T.-H. Kim, H.W. Yeom, Phys. Rev. B 89 (2014) 155436. [18] A. Eich, M. Michiardi, G. Bihlmayer, X.-G. Zhu, J.-L. Mi, B.B. Iversen, R. Wiesendanger, P. Hofmann, A.A. Khajetoorians, J. Wiebe, ArXiv E-prints, 2014. [19] P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka, J. Luitz, WIEN2k, An Augmented Plane Wave + Local Orbitals Program for Calculating Crystal Properties, Karlheinz Schwarz, Techn. Universität, Wien, Austria, 2001, ISBN 3-9501031-1-2. [20] J.P. Perdew, A. Ruzsinszky, G.I. Csonka, O.A. Vydrov, G.E. Scuseria, L.A. Constantin, X. Zhou, K. Burke, Phys. Rev. Lett. 100 (2008) 136406. [21] F. Tran, P. Blaha, Phys. Rev. Lett. 102 (2009) 226401. [22] N. Marzari, D. Vanderbilt, Phys. Rev. B 56 (1997) 12847. [23] I. Souza, N. Marzari, D. Vanderbilt, Phys. Rev. B 65 (2001) 035109. [24] N. Marzari, A.A. Mostofi, J.R. Yates, I. Souza, D. Vanderbilt, Rev. Mod. Phys. 84 (2012) 1419. [25] A.A. Mostofi, J.R. Yates, Y.-S. Lee, I. Souza, D. Vanderbilt, N. Marzari, Comput. Phys. Commun. 178 (2008) 685. [26] J. Kuneš, R. Arita, P. Wissgott, A. Toschi, H. Ikeda, K. Held, Comput. Phys. Commun. 181 (2010) 1888. [27] A. Svane, N.E. Christensen, M. Cardona, A.N. Chantis, M. van Schilfgaarde, T. Kotani, Phys. Rev. B 81 (2012) 245120. [28] C.S. Lent, M.A. Bowen, J.D. Dow, R.S. Allgaier, O.F. Sankey, E.S. Ho, Superlattices Microstruct. 2 (1986) 491. [29] E.O. Wrasse, T.M. Schmidt, Nano Lett. 14 (2014) 5717. [30] B. Li, A. Michaelides, M. Scheffler, Phys. Rev. B 76 (2007) 075401. [31] J. Liu, W. Duan, L. Fu, Phys. Rev. B 88 (2013) 241303. [32] A.K. Singh, R.G. Hennig, Appl. Phys. Lett. 105 (2014) 042103. [33] J.C.Y. Teo, L. Fu, C.L. Kane, Phys. Rev. B 78 (2008) 045426. [34] G. Yang, J. Liu, L. Fu, W. Duan, C. Liu, Phys. Rev. B 89 (2004) 085312. [35] J. Liu, X. Qian, L. Fu, Nano Lett. 15 (2015) 2657. [36] C. Niu, P.M. Buhl, G. Bihlmayer, D. Wortmann, S. Blugel, Y. Mokrousov, arXiv: 1503.01939 (2015). [37] Y. Kim, C.L. Kane, E.J. Mele, A.M. Rappe, arXiv:1503.05966 (2015).