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Topological phase transitions of (Bix Sb1−x )2Se3 alloys by density functional theory
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 J. Phys.: Condens. Matter 27 255501 (http://iopscience.iop.org/0953-8984/27/25/255501) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 155.69.4.4 This content was downloaded on 10/06/2015 at 17:36
Please note that terms and conditions apply.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 27 (2015) 255501 (6pp)
doi:10.1088/0953-8984/27/25/255501
Topological phase transitions of (BixSb1−x)2Se3 alloys by density functional theory L B Abdalla1 , Jose´ E Padilha3 , T M Schmidt2 , R H Miwa 2 and A Fazzio1 1 2 3
Instituto de F´ısica, Universidade de S˜ao Paulo, CP 66318, 05315-970, S˜ao Paulo, SP, Brazil Universidade Federal de Uberlˆandia, Uberlˆandia, MG, 38408-100, Brazil Universidade Federal do Paran´a, Campus Avanc¸ado de Jandaia do Sul, PR, Brazil
E-mail: [email protected] and [email protected] Received 16 January 2015, revised 2 April 2015 Accepted for publication 9 April 2015 Published 5 June 2015 Abstract
We have performed an ab initio total energy investigation of the topological phase transition, and the electronic properties of topologically protected surface states of (Bix Sb1−x )2 Se3 alloys. In order to provide an accurate alloy concentration for the phase transition, we have considered the special quasirandom structures to describe the alloy system. The trivial→topological transition concentration was obtained by (i) the calculation of the band gap closing as a function of Bi concentration (x), and (ii) the calculation of the Z2 topological invariant number. We show that there is a topological phase transition, for x around 0.4, verified for both procedures (i) and (ii). We also show that in the concentration range 0.4 < x < 0.7, the alloy does not present any other band at the Fermi level besides the Dirac cone, where the Dirac point is far from the bulk states. This indicates that a possible suppression of the scattering process due to bulk states will occur. Keywords: topological insulators, topological phase transition, (Bix Sb1−x )2 Se3 (Some figures may appear in colour only in the online journal)
direct or indirect [5, 6], its relatively large value, together with its spin polarized surface Dirac cone, makes these materials good candidates for future applications in spintronics devices [7–12]. However, one of the main challenges in these materials is to avoid the backscattering from bulk states, as the Dirac point of the surface states at the point is close to the bulk ones [13]. Several proposals have been made to overcome this issue, such as the use of alloys of TI materials [14–20], as well as the use of extended defects, such as stacking faults [21]. In this work we studied the topological phase transition and the energy position of the Dirac point with respect to the bulk states in (Bix Sb1−x )2 Se3 . We considered alloy concentrations (x(Bi)) from x = 0 Sb2 Se3 (trivial insulator) to x = 1 Bi2 Se3 (TI). In order to provide a suitable description of the (Bix Sb1−x )2 Se3 alloy system, the atomic configurations were generated using the special quasirandom structure (SQS) approach. The topological phase transition was verified by: (i) the calculation of the band gap closing as a function of Bi concentration (x), which is based on the adiabatic
1. Introduction
Recently, topological insulators (TIs) have ranked as one of the most promising materials in condensed matter physics, due to their remarkable electronic properties. They present a band inversion in their bulk structure, driven by a strong spin– orbit coupling (SOC), and topologically protected metallic states in a Dirac cone shape arise at the interface/surface with a trivial insulator [1, 2]. These metallic states, in a threedimensional (3D) TI, present a spin polarization locked with the momentum, which is protected by time reversal symmetry. Such surface states are due to the bulk topological phase, which is characterized by a Z2 invariant number [3, 4]. Nowadays, the most studied of the 3D TIs, are the binary compounds Bi2 Se3 and Bi2 Te3 , which are strong TIs. In the bulk structure, these materials present a bandgap of 0.3 eV and 0.13–0.17 eV. In contact with a trivial insulator a single Dirac cone appears in their surface at the point. Although in the literature there is a discussion about the bulk band gap being 0953-8984/15/255501+06$33.00
1
© 2015 IOP Publishing Ltd Printed in the UK
J. Phys.: Condens. Matter 27 (2015) 255501
L B Abdalla et al
up to 16 different concentrations, going from 0 to 1 in hops of 0.042. For the lattice parameters a and c of the hexagonal cell, we used the experimental data for Bi2 Se3 (a = 4.138 Å and c = 28.640 Å) [30], and for the Sb2 Se3 structure we used the parameters considered in [31] (a = 4.076 Å and c = 30.90 Å). The lattice vectors of the alloys were taken linearly between both pristine systems, following the Vegard’s law, keeping the hexagonal symmetry of the system [18, 32]. For each concentration of the bulk phase all atoms were relaxed until the forces were smaller than 10 meV Å−1 . The SQS structure obtained in the bulk with minimal correlation function, were duplicated in the zˆ direction ([0 0 0 1] surface ), so that the slab will have six QLs, with a vacuum of 12 Å to avoid spurious interactions between their images. Without stacking faults (a problem we want to avoid), there are three possible ways to construct the surface (A–B (Surface 1), A– C (Surface 2) or B–C (Surface 3) endings), as depicted in figure 1(b). For each concentration we choose the surface with lowest energy, and only this one was considered in our study. A TI is referred to as a material with topological invariant Z2 = 1 in 2 dimensions, and ν0 = 1 in 3 dimensions. It is customary to refer to Z2 as ν0 in many papers, because both represent a strong topological invariant. We can write down the topological invariants ν0 as the product of two Z2 invariants for each plane containing 4 TRIMs points. And we can write the Z2 number as the change in polarization between two pairs of TRIMs points. We know from [3] that:
Figure 1. (a) Unit cell of the bulk Sb2 Se3 geometry, with hexagonal lattice parameter. Here we can see three different QLs, each one rotated by 120◦ degrees from the next one. (b) In order to avoid stacking faults we decide to create all possible configurations of surfaces. Knowing that we have three inequivalent QLs, it is only possible to make a surface with A–B, A–C and B–C ending, as seen by the choice of a different color for each quintuple layer. (c) Brillouin zone of the bulk hexagonal cell with high symmetry points. (d) Projection of the 3D Brillouin zone, in its surface, making a surface Brillouin zone, and the projected high symmetry points.
continuity argument, and (ii) explicit calculation of the Z2 topological invariant number. We show that by increasing the Bi concentration, there is a topological phase transition, trivial→TI, for x around 0.4. We also verified that in the range of 0.4 < x < 0.7 the Dirac point lies inside the bulk band gap. In this composition range the electronic transport measurements are expected to be free of bulk scattering, as the Dirac point is far from the bulk states.
Z2 = P (k, t = 0) − P (k, t = T /2).
(1)
Mapping this system on a 2D brillouin, i.e. (k, t) → (kx , ky ) zone we π have: Z2 = P (Gx = 0, Gy = 0) − P ,0 π π π 2 = P , 0 − P , . (2) 2 2 2 And P can be written as:
2. Method
The calculations were performed with DFT [22, 23], by using the pseudopotentials method, with projected augmented wave potentials (PAW) [24, 25] to describe the electron– ion interaction. For the exchange-correlation functional we used the Perdew–Burke–Ernzenhof generalized gradient approximation (PBE-GGA) [26] as implemented in the Vienna Ab Initio Simulation Package code (VASP) [27]. The Kohn– Sham orbitals are expanded in a plane-wave basis set with an energy cutoff of 400 eV in all systems. The Brillouin zone was sampled by a centered scheme, using a number of k points up to 4 × 4 × 4 and 4 × 4 × 1 for the bulk and surface calculations. In our calculations no GW corrections were considered, however, no significant changes will occur in the band gap of the materials as the gap at point for Bi2 Se3 with GW is 0.30 eV [28] that is close to the GGA band gap ≈0.27 eV. The structures used in the calculations were constructed through the SQS procedure [29]. For the bulk we used a 2 × 2 × 1 supercell with 3 quintuple layers (QLs), as depicted in figure 1(a). The interchange was made between the Bi and Sb atoms, keeping the Se atoms as spectators. We considered
P = P I − P II .
(3)
Where I and II stand for Kramer pairs, we know that for a time reversal invariant Hamiltonian we have 2N occupied states in N pairs of states. The calculation of polarization as proposed by Kane is made only on TRIM points, and can be written as follows: π 1 P = A(k)dk, (4) 2π π where A(k) is the Berry connection. To determine the Z2 topological invariant number, we implemented in the VASP code the calculation procedure proposed by Fu and Kane [3]. This implementation can be applied in all plane wave basis set code. The Bloch wave functions, with spin–orbit interactions, can be written as 1 σ σ ψn, (r ) = r |Um,i = √ Cn, ,K ei(k+Kj )·r , (5) i i j j σ where is the volume of the unit cell, Cn, are the wave i ,K j function coefficients; |Um,i are the Bloch vectors at the time
2
J. Phys.: Condens. Matter 27 (2015) 255501
L B Abdalla et al
Figure 2. (a) Bulk band structure for the alloy (Bix Sb1−x )2 Se3 , between 0 and 1. (b) Gap variation of (Bix Sb1−x )2 Se3 . In green we have a straight line fitting all decreasing gap sizes, and in blue we have another line fitting values of increasing gap sizes.
reversal invariant momenta points (TRIMs) i ; σ and n are the spin and band index. The topological invariant Z2 can be obtained by the equation: (−1)Z2 =
TRIM δi ;
Pf(ω(i )) δi = √ , det ω(i )
As the wavefunction is the same after this cycle, we have = P − 2mπ , with m an integer. This could rise as a problem to P 2π evaluate normal polarization as the Chern number, because the polarization could change by an integer m, depending on the gauge. But for time reversal symmetry (TRS) the evaluation of the invariant Z2 is done by the subtraction of P I − P II both calculated in the same TRIM point. So even with a gauge I − P II = P I − P II fixing the gauge change we would have P in these points.
(6)
where the Pfaffian Pf(ω(i )) is obtained by calculating the unitary matrix ωnm (i ): ωnm (i ) = Un,−i ||Um,i = dr Un,−i |r r |Um,i .
(7)
3. Results and discussions
where is the time reversal symmetry operator. For plane wave basis set, the ω (from equation (4)) reduces to ↑∗ ↓∗ −Cn, ,−2 −K Cm, ,K ωnm = i
K j
+
K j
i
j
↓∗
i
j
↑∗
Cn, ,−2 −K Cm, ,K . i
i
j
i
In figure 2 we present the evolution of the band structure (figure 2(a)), and the band gap at point (figure 2(b)) as a function of the concentration x of Bi, going from x = 0 to x = 1. As we can see, by increasing the Bi concentration the band gap of the system decreases until 0.4 and starts to increase again. This V shape behaviour of the band gap is characteristic of a topological phase transition. This behaviour can be understood using the adiabatic continuity argument, which states that if a Hamiltonian of a system is adiabatically transformed into another without closing the band band, the system must share the same topological class. On the other hand, during this transformation, if the band gap closes, there must be a topological phase transition. Based on this argument, for x = 1 we have the Bi2 Se3 that is a TI, whereas for x = 0 (Sb2 Se3 ) we have a trivial insulator. Thus, we can conclude that around a Bi concentration of 0.4 the system undergoes a trivial→TI phase transition [33]. The adiabatic continuity argument is a strong signature of a topological phase transition. Another indication is the band inversion due to the spin–orbit interaction [34]. For Bi2 Se3 there is an inversion of pz orbitals, while for Sb2 Se3 there is not, as shown in figures 3(a) and (d). Also, we present in figure 3(b) and (c) the projection of the pz orbitals for Bi
(8)
j
In 3D systems there are 8 TRIMs and 4 Z2 invariants ν0 ; (ν1 , ν2 , ν3 ). A strong TI characterised by ν0 = 1, can be obtained by: 8 (−1)ν0 = δi . (9) i=1
Ab initio programs can give us a generic gauge [35] on the m = eiζ (k) |ψm . In that sense the Berry states as follows, |ψ connection could change as: = A(k) − ∂ ζ (k) A(k) (10) ∂k For cyclic changes, such as in the Brillouin Zone (as defined for the name Berry phase) we have that: = P − (ζ (kf ) − ζ (ki )) P 2π
(11) 3
J. Phys.: Condens. Matter 27 (2015) 255501
L B Abdalla et al
Figure 3. Band structure around the Fermi level of (Bix Sb1−x )2 Se3 for: (a) x = 0, (b) x = 0.375, (c) x = 0.459 and (d) x = 1. The projection in the pz orbitals of Bi/Sb are indicated by the blue dots, while the Se pz orbitals are indicated by the red dots.
Figure 4. Band structures of slabs of (Bix Sb1−x )2 Se3 : (a) x = 0 (Sb2 Se3 ), (b) x = 0.33, (c) x = 0.67, and (d) x = 1 (Bi2 Se3 ). The black lines corresponds to the projection in the bulk of the material and the red lines are the projection in its surface.
concentrations of 0.375 and 0.459. We find that for the former there is no such orbital inversion, whereas for Bi concentration of 45.9%, the energy positions of the Se pz orbitals at the band gap are inverted near the point. In order to provide further support to the trivial→TI transition, we calculate the Z2 number (equation (6)) for (Bix Sb1−x )2 Se3 as a function of the alloy concentration. We find Z2 = 0 for Bi concentration up to 0.4, while for concentrations larger than 40% we obtained Z2 = 1. Thus, based on the adiabatic argument, and the evaluation of the Z2 number we mapped with great precision the trivial→TI transition in (Bix Sb1−x )2 Se3 as a function of the alloy concentration. One of the main features in TIs, is the presence of protected metallic states on the surface. We calculated the slab band structure of (Bix Sb1−x )2 Se3 for several concentrations x between 0 and 1. The formation of those protected states can
be tuned by the alloy concentration. As shown in figure 4(a) there are no metallic states lying in the bandgap of Sb2 Se3 . By increasing the Bi concentration to x = 0.375, figure 4(b), the band gap of the system is smaller, but it is still absent of metallic surface states. Further increase in the Bi concentration (x > 0.4) topologically protected metallic arises, as depicted in figure 4(c) (x = 0.667). Here we have considered nine QLs to calculate the electronic band structure of the slab. These metallic states are preserved up to x = 1, figure 4(d). It is worth noting that, in agreement with previous reported results, the Dirac Point (DP) locates close in energy to the bulk valence band, figure 4(d). However, we find that for Bi concentrations between 0.4 < x < 0.7, the DP lies above the bulk valence band (figure 4(c)), that could suppress the electronic scattering process due to the bulk states. The upward displacement of the with respect to the bulk states, upon the presence of Sb 4
J. Phys.: Condens. Matter 27 (2015) 255501
L B Abdalla et al
Figure 5. Spin texture for the upper (upper panel) (lower (lower panel)) part of the Dirac cone in (a) Bi2 Se3 and (b) (Bi0.667 Sb0.333 )2 Se3 . The energy slice to obtain the spin texture was 0.1 eV for the unoccupied part and −0.01 for the occupied part. The color scale represents the amount of Sz .
atom in (Bix Sb1−x )2 Se3 alloys, can be attributed to the weaker SOC of Sb compared with Bi atoms [18]. Also, we observe a break of denegeracy between the surface states coming from the bottom and top surface, depicted in figure 4(c). This is due to a break of symmetry between the top and bottom surface. In addition we verify that not only can the energy dispersion of the electronic band structure be controlled by the alloy concentration, but also we find different pictures of the spin-texture near the DP of (Bix Sb1−x )2 Se3 as a function of the alloy concentration. In figure 5 we present the helical spin texture of (Bix Sb1−x )2 Se3 for x = 1 (left) and x = 0.667 (right). Here we took a slice of the surface band structure near the Fermi level. We note that in the pristine Bi2 Se3 case the Sx and Sy spin contributions are locked in the xˆ · yˆ plane, with a Sz contribution that oscillates around the Dirac Cone. This latter contribution has positive and negative values and goes through some nodes. The consequence is a S spin vector tilted from the xˆ · y, ˆ at some points of the Brillouin zone, by a certain angle going from ±1.15◦ for the conduction band and ±1.55◦ for the valence band. As the system is doped with Sb the Sz contribution gets more significant, but the oscillations have lesser nodes. For Bi concentration of 66% the energy slice plotted in figure 5(b) we see an S vector with angles from the plane varying from ±1.60◦ for the conduction band and ±5.38◦ for the valence band. From the figure 5 we also observe that the TRS is always preserved in the bulk Bi2 Se3 , and in the alloy.
4. Summary
In this paper we have performed a detailed and systematic investigation of the topological phases and the surface electronic properties of (Bix Sb1−x )2 Se3 . In order to provide an accurate description of the trivial→TI phase transition, the alloy system was described by using the SQS approach. The topological phases were determined through: (i) the bandgap closing, based on the adiabatic arguments, and (ii) explicitly calculation of the Z2 topological invariant number for all alloy concentrations considered in our study. In both procedures, (i) and (ii), we find that the trivial→TI phase transition occurs for Bi concentration x = 0.4. Different pictures of the energy dispersion of topologically protected surface states depending on the Bi/Sb ratio were observed. For the alloy concentration range of 0.4 < x < 0.7, the Dirac crossing point moves upward lying in the bandgap of the (Bix Sb1−x )2 Se3 alloy, being far from the bulk states. This result suggests a suppression of the scattering process ruled by bulk valence states.
Acknowledgments
We would like to acknowledge the Brazilian funding agencies: CAPES, CNPq and FAPESP. 5
J. Phys.: Condens. Matter 27 (2015) 255501
L B Abdalla et al
[17] Tang J et al 2014 Nano Lett. 14 5423 [18] Niu C, Dai Y, Zhu Y, Ma Y, Yu L, Han S and Huang B 2012 Sci. Rep. 2 976 [19] Ren Z, Taskin A A, Sasaki S, Segawa K and Ando Y 2010 Phys. Rev. B 82 241306 [20] Xiong J, Petersen A, Qu D, Hor Y, Cava R and Ong N 2012 Physica E 44 917 [21] Seixas L, Abdalla L B, Schmidt T M, Miwa R H and Fazzio A 2013 J. Appl. Phys. 113 023705 [22] Hohenberg P and Kohn W 1964 Phys. Rev. 136 B864 [23] Kohn W and Sham L 1965 Phys. Rev. 140 A1133 [24] Blochl P 1994 Phys. Rev. B 50 17953 [25] Kresse G and Joubert D 1999 Phys. Rev. B 59 1758 [26] Perdew J P, Burke K and Ernzenhof M 1996 Phys. Rev. Lett. 77 3865 [27] Kresse G and Furthm¨uller J 1996 Phys. Rev. B 54 11169 [28] Nechaev I A et al 2013 Phys. Rev. B 87 121111 [29] Zunger A, Wei S H, Ferreira L G and Bernard J E 1990 Phys. Rev. Lett. 65 353 [30] Zhang W, Yu R, Zhang H J, Dai X and Fang Z 2010 New J. Phys. 12 065013 [31] Li W, Wei X Y, Zhu J X, Ting C S and Chen Y 2014 Phys. Rev. B 89 035101 [32] Anderson T and Krause H 1974 Acta Crystallogr. 30 1307 [33] Liu J and Vanderbilt D 2013 Phys. Rev. B 88 224202 [34] Zhang H, Liu C X, Qi X L, Dai X, Fang Z and Zhang S C 2009 Nat. Phys. 5 438 [35] Yu R, Qi X L, Bernevig A, Fang Z and Dai X Phys. Rev. B 84 075119
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
Hasan M Z and Kane C 2010 Rev. Mod. Phys. 82 3045 Qi X L and Zhang S C 2011 Rev. Mod. Phys. 83 1057 Fu L and Kane C L 2006 Phys. Rev. B 74 195312 Kane C L and Mele E J 2005 Phys. Rev. Lett. 95 146802 Agullera I, Friedrich C, Bihlmayer G and Bl¨ugel S 2013 Phys. Rev. B 88 045206 Nechaev I A et al 2013 Phys. Rev. B 87 121111 Checkelsky J, Hor Y, Liu M H, Qu D X, Cava R and Ong N 2009 Phys. Rev. Lett. 103 246601 Analytis J G, Chu J-H, Chen Y, Corredor F, McDonald R D, Shen Z X and Fisher I R 2010 Phys. Rev. B 81 205407 Eto K, Ren Z, Taskin A A, Segawa K and Ando Y 2010 Phys. Rev. B 81 195309 Butch N P, Kirshenbaum K, Syers P, Sushkov A B, Jenkins G S, Drew H D and Paglione J 2010 Phys. Rev. B 81 241301 Abdalla L B, Seixas L, Schmidt T M, Miwa R H and Fazzio A 2013 Phys. Rev. B 88 045312 Schmidt T M, Miwa R H and Fazzio A 2011 Phys. Rev. B 84 245418 Kim S et al 2011 Phys. Rev. Lett. 107 056803 Smith M J, Knight R J and Spencer C W 1962 J. Appl. Phys. 33 2186 Ren Z, Taskin A A, Sasaki S, Segawa K and Ando Y 2011 Phys. Rev. B 84 165311 Taskin A A, Ren Z, Sasaki S, Segawa K and Ando Y 2011 Phys. Rev. Lett. 107 016801
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Topological phase transitions of (Bix Sb1−x )2Se3 alloys by density functional theory
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2015 J. Phys.: Condens. Matter 27 255501 (http://iopscience.iop.org/0953-8984/27/25/255501) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 155.69.4.4 This content was downloaded on 10/06/2015 at 17:36
Please note that terms and conditions apply.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 27 (2015) 255501 (6pp)
doi:10.1088/0953-8984/27/25/255501
Topological phase transitions of (BixSb1−x)2Se3 alloys by density functional theory L B Abdalla1 , Jose´ E Padilha3 , T M Schmidt2 , R H Miwa 2 and A Fazzio1 1 2 3
Instituto de F´ısica, Universidade de S˜ao Paulo, CP 66318, 05315-970, S˜ao Paulo, SP, Brazil Universidade Federal de Uberlˆandia, Uberlˆandia, MG, 38408-100, Brazil Universidade Federal do Paran´a, Campus Avanc¸ado de Jandaia do Sul, PR, Brazil
E-mail: [email protected] and [email protected] Received 16 January 2015, revised 2 April 2015 Accepted for publication 9 April 2015 Published 5 June 2015 Abstract
We have performed an ab initio total energy investigation of the topological phase transition, and the electronic properties of topologically protected surface states of (Bix Sb1−x )2 Se3 alloys. In order to provide an accurate alloy concentration for the phase transition, we have considered the special quasirandom structures to describe the alloy system. The trivial→topological transition concentration was obtained by (i) the calculation of the band gap closing as a function of Bi concentration (x), and (ii) the calculation of the Z2 topological invariant number. We show that there is a topological phase transition, for x around 0.4, verified for both procedures (i) and (ii). We also show that in the concentration range 0.4 < x < 0.7, the alloy does not present any other band at the Fermi level besides the Dirac cone, where the Dirac point is far from the bulk states. This indicates that a possible suppression of the scattering process due to bulk states will occur. Keywords: topological insulators, topological phase transition, (Bix Sb1−x )2 Se3 (Some figures may appear in colour only in the online journal)
direct or indirect [5, 6], its relatively large value, together with its spin polarized surface Dirac cone, makes these materials good candidates for future applications in spintronics devices [7–12]. However, one of the main challenges in these materials is to avoid the backscattering from bulk states, as the Dirac point of the surface states at the point is close to the bulk ones [13]. Several proposals have been made to overcome this issue, such as the use of alloys of TI materials [14–20], as well as the use of extended defects, such as stacking faults [21]. In this work we studied the topological phase transition and the energy position of the Dirac point with respect to the bulk states in (Bix Sb1−x )2 Se3 . We considered alloy concentrations (x(Bi)) from x = 0 Sb2 Se3 (trivial insulator) to x = 1 Bi2 Se3 (TI). In order to provide a suitable description of the (Bix Sb1−x )2 Se3 alloy system, the atomic configurations were generated using the special quasirandom structure (SQS) approach. The topological phase transition was verified by: (i) the calculation of the band gap closing as a function of Bi concentration (x), which is based on the adiabatic
1. Introduction
Recently, topological insulators (TIs) have ranked as one of the most promising materials in condensed matter physics, due to their remarkable electronic properties. They present a band inversion in their bulk structure, driven by a strong spin– orbit coupling (SOC), and topologically protected metallic states in a Dirac cone shape arise at the interface/surface with a trivial insulator [1, 2]. These metallic states, in a threedimensional (3D) TI, present a spin polarization locked with the momentum, which is protected by time reversal symmetry. Such surface states are due to the bulk topological phase, which is characterized by a Z2 invariant number [3, 4]. Nowadays, the most studied of the 3D TIs, are the binary compounds Bi2 Se3 and Bi2 Te3 , which are strong TIs. In the bulk structure, these materials present a bandgap of 0.3 eV and 0.13–0.17 eV. In contact with a trivial insulator a single Dirac cone appears in their surface at the point. Although in the literature there is a discussion about the bulk band gap being 0953-8984/15/255501+06$33.00
1
© 2015 IOP Publishing Ltd Printed in the UK
J. Phys.: Condens. Matter 27 (2015) 255501
L B Abdalla et al
up to 16 different concentrations, going from 0 to 1 in hops of 0.042. For the lattice parameters a and c of the hexagonal cell, we used the experimental data for Bi2 Se3 (a = 4.138 Å and c = 28.640 Å) [30], and for the Sb2 Se3 structure we used the parameters considered in [31] (a = 4.076 Å and c = 30.90 Å). The lattice vectors of the alloys were taken linearly between both pristine systems, following the Vegard’s law, keeping the hexagonal symmetry of the system [18, 32]. For each concentration of the bulk phase all atoms were relaxed until the forces were smaller than 10 meV Å−1 . The SQS structure obtained in the bulk with minimal correlation function, were duplicated in the zˆ direction ([0 0 0 1] surface ), so that the slab will have six QLs, with a vacuum of 12 Å to avoid spurious interactions between their images. Without stacking faults (a problem we want to avoid), there are three possible ways to construct the surface (A–B (Surface 1), A– C (Surface 2) or B–C (Surface 3) endings), as depicted in figure 1(b). For each concentration we choose the surface with lowest energy, and only this one was considered in our study. A TI is referred to as a material with topological invariant Z2 = 1 in 2 dimensions, and ν0 = 1 in 3 dimensions. It is customary to refer to Z2 as ν0 in many papers, because both represent a strong topological invariant. We can write down the topological invariants ν0 as the product of two Z2 invariants for each plane containing 4 TRIMs points. And we can write the Z2 number as the change in polarization between two pairs of TRIMs points. We know from [3] that:
Figure 1. (a) Unit cell of the bulk Sb2 Se3 geometry, with hexagonal lattice parameter. Here we can see three different QLs, each one rotated by 120◦ degrees from the next one. (b) In order to avoid stacking faults we decide to create all possible configurations of surfaces. Knowing that we have three inequivalent QLs, it is only possible to make a surface with A–B, A–C and B–C ending, as seen by the choice of a different color for each quintuple layer. (c) Brillouin zone of the bulk hexagonal cell with high symmetry points. (d) Projection of the 3D Brillouin zone, in its surface, making a surface Brillouin zone, and the projected high symmetry points.
continuity argument, and (ii) explicit calculation of the Z2 topological invariant number. We show that by increasing the Bi concentration, there is a topological phase transition, trivial→TI, for x around 0.4. We also verified that in the range of 0.4 < x < 0.7 the Dirac point lies inside the bulk band gap. In this composition range the electronic transport measurements are expected to be free of bulk scattering, as the Dirac point is far from the bulk states.
Z2 = P (k, t = 0) − P (k, t = T /2).
(1)
Mapping this system on a 2D brillouin, i.e. (k, t) → (kx , ky ) zone we π have: Z2 = P (Gx = 0, Gy = 0) − P ,0 π π π 2 = P , 0 − P , . (2) 2 2 2 And P can be written as:
2. Method
The calculations were performed with DFT [22, 23], by using the pseudopotentials method, with projected augmented wave potentials (PAW) [24, 25] to describe the electron– ion interaction. For the exchange-correlation functional we used the Perdew–Burke–Ernzenhof generalized gradient approximation (PBE-GGA) [26] as implemented in the Vienna Ab Initio Simulation Package code (VASP) [27]. The Kohn– Sham orbitals are expanded in a plane-wave basis set with an energy cutoff of 400 eV in all systems. The Brillouin zone was sampled by a centered scheme, using a number of k points up to 4 × 4 × 4 and 4 × 4 × 1 for the bulk and surface calculations. In our calculations no GW corrections were considered, however, no significant changes will occur in the band gap of the materials as the gap at point for Bi2 Se3 with GW is 0.30 eV [28] that is close to the GGA band gap ≈0.27 eV. The structures used in the calculations were constructed through the SQS procedure [29]. For the bulk we used a 2 × 2 × 1 supercell with 3 quintuple layers (QLs), as depicted in figure 1(a). The interchange was made between the Bi and Sb atoms, keeping the Se atoms as spectators. We considered
P = P I − P II .
(3)
Where I and II stand for Kramer pairs, we know that for a time reversal invariant Hamiltonian we have 2N occupied states in N pairs of states. The calculation of polarization as proposed by Kane is made only on TRIM points, and can be written as follows: π 1 P = A(k)dk, (4) 2π π where A(k) is the Berry connection. To determine the Z2 topological invariant number, we implemented in the VASP code the calculation procedure proposed by Fu and Kane [3]. This implementation can be applied in all plane wave basis set code. The Bloch wave functions, with spin–orbit interactions, can be written as 1 σ σ ψn, (r ) = r |Um,i = √ Cn, ,K ei(k+Kj )·r , (5) i i j j σ where is the volume of the unit cell, Cn, are the wave i ,K j function coefficients; |Um,i are the Bloch vectors at the time
2
J. Phys.: Condens. Matter 27 (2015) 255501
L B Abdalla et al
Figure 2. (a) Bulk band structure for the alloy (Bix Sb1−x )2 Se3 , between 0 and 1. (b) Gap variation of (Bix Sb1−x )2 Se3 . In green we have a straight line fitting all decreasing gap sizes, and in blue we have another line fitting values of increasing gap sizes.
reversal invariant momenta points (TRIMs) i ; σ and n are the spin and band index. The topological invariant Z2 can be obtained by the equation: (−1)Z2 =
TRIM δi ;
Pf(ω(i )) δi = √ , det ω(i )
As the wavefunction is the same after this cycle, we have = P − 2mπ , with m an integer. This could rise as a problem to P 2π evaluate normal polarization as the Chern number, because the polarization could change by an integer m, depending on the gauge. But for time reversal symmetry (TRS) the evaluation of the invariant Z2 is done by the subtraction of P I − P II both calculated in the same TRIM point. So even with a gauge I − P II = P I − P II fixing the gauge change we would have P in these points.
(6)
where the Pfaffian Pf(ω(i )) is obtained by calculating the unitary matrix ωnm (i ): ωnm (i ) = Un,−i ||Um,i = dr Un,−i |r r |Um,i .
(7)
3. Results and discussions
where is the time reversal symmetry operator. For plane wave basis set, the ω (from equation (4)) reduces to ↑∗ ↓∗ −Cn, ,−2 −K Cm, ,K ωnm = i
K j
+
K j
i
j
↓∗
i
j
↑∗
Cn, ,−2 −K Cm, ,K . i
i
j
i
In figure 2 we present the evolution of the band structure (figure 2(a)), and the band gap at point (figure 2(b)) as a function of the concentration x of Bi, going from x = 0 to x = 1. As we can see, by increasing the Bi concentration the band gap of the system decreases until 0.4 and starts to increase again. This V shape behaviour of the band gap is characteristic of a topological phase transition. This behaviour can be understood using the adiabatic continuity argument, which states that if a Hamiltonian of a system is adiabatically transformed into another without closing the band band, the system must share the same topological class. On the other hand, during this transformation, if the band gap closes, there must be a topological phase transition. Based on this argument, for x = 1 we have the Bi2 Se3 that is a TI, whereas for x = 0 (Sb2 Se3 ) we have a trivial insulator. Thus, we can conclude that around a Bi concentration of 0.4 the system undergoes a trivial→TI phase transition [33]. The adiabatic continuity argument is a strong signature of a topological phase transition. Another indication is the band inversion due to the spin–orbit interaction [34]. For Bi2 Se3 there is an inversion of pz orbitals, while for Sb2 Se3 there is not, as shown in figures 3(a) and (d). Also, we present in figure 3(b) and (c) the projection of the pz orbitals for Bi
(8)
j
In 3D systems there are 8 TRIMs and 4 Z2 invariants ν0 ; (ν1 , ν2 , ν3 ). A strong TI characterised by ν0 = 1, can be obtained by: 8 (−1)ν0 = δi . (9) i=1
Ab initio programs can give us a generic gauge [35] on the m = eiζ (k) |ψm . In that sense the Berry states as follows, |ψ connection could change as: = A(k) − ∂ ζ (k) A(k) (10) ∂k For cyclic changes, such as in the Brillouin Zone (as defined for the name Berry phase) we have that: = P − (ζ (kf ) − ζ (ki )) P 2π
(11) 3
J. Phys.: Condens. Matter 27 (2015) 255501
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Figure 3. Band structure around the Fermi level of (Bix Sb1−x )2 Se3 for: (a) x = 0, (b) x = 0.375, (c) x = 0.459 and (d) x = 1. The projection in the pz orbitals of Bi/Sb are indicated by the blue dots, while the Se pz orbitals are indicated by the red dots.
Figure 4. Band structures of slabs of (Bix Sb1−x )2 Se3 : (a) x = 0 (Sb2 Se3 ), (b) x = 0.33, (c) x = 0.67, and (d) x = 1 (Bi2 Se3 ). The black lines corresponds to the projection in the bulk of the material and the red lines are the projection in its surface.
concentrations of 0.375 and 0.459. We find that for the former there is no such orbital inversion, whereas for Bi concentration of 45.9%, the energy positions of the Se pz orbitals at the band gap are inverted near the point. In order to provide further support to the trivial→TI transition, we calculate the Z2 number (equation (6)) for (Bix Sb1−x )2 Se3 as a function of the alloy concentration. We find Z2 = 0 for Bi concentration up to 0.4, while for concentrations larger than 40% we obtained Z2 = 1. Thus, based on the adiabatic argument, and the evaluation of the Z2 number we mapped with great precision the trivial→TI transition in (Bix Sb1−x )2 Se3 as a function of the alloy concentration. One of the main features in TIs, is the presence of protected metallic states on the surface. We calculated the slab band structure of (Bix Sb1−x )2 Se3 for several concentrations x between 0 and 1. The formation of those protected states can
be tuned by the alloy concentration. As shown in figure 4(a) there are no metallic states lying in the bandgap of Sb2 Se3 . By increasing the Bi concentration to x = 0.375, figure 4(b), the band gap of the system is smaller, but it is still absent of metallic surface states. Further increase in the Bi concentration (x > 0.4) topologically protected metallic arises, as depicted in figure 4(c) (x = 0.667). Here we have considered nine QLs to calculate the electronic band structure of the slab. These metallic states are preserved up to x = 1, figure 4(d). It is worth noting that, in agreement with previous reported results, the Dirac Point (DP) locates close in energy to the bulk valence band, figure 4(d). However, we find that for Bi concentrations between 0.4 < x < 0.7, the DP lies above the bulk valence band (figure 4(c)), that could suppress the electronic scattering process due to the bulk states. The upward displacement of the with respect to the bulk states, upon the presence of Sb 4
J. Phys.: Condens. Matter 27 (2015) 255501
L B Abdalla et al
Figure 5. Spin texture for the upper (upper panel) (lower (lower panel)) part of the Dirac cone in (a) Bi2 Se3 and (b) (Bi0.667 Sb0.333 )2 Se3 . The energy slice to obtain the spin texture was 0.1 eV for the unoccupied part and −0.01 for the occupied part. The color scale represents the amount of Sz .
atom in (Bix Sb1−x )2 Se3 alloys, can be attributed to the weaker SOC of Sb compared with Bi atoms [18]. Also, we observe a break of denegeracy between the surface states coming from the bottom and top surface, depicted in figure 4(c). This is due to a break of symmetry between the top and bottom surface. In addition we verify that not only can the energy dispersion of the electronic band structure be controlled by the alloy concentration, but also we find different pictures of the spin-texture near the DP of (Bix Sb1−x )2 Se3 as a function of the alloy concentration. In figure 5 we present the helical spin texture of (Bix Sb1−x )2 Se3 for x = 1 (left) and x = 0.667 (right). Here we took a slice of the surface band structure near the Fermi level. We note that in the pristine Bi2 Se3 case the Sx and Sy spin contributions are locked in the xˆ · yˆ plane, with a Sz contribution that oscillates around the Dirac Cone. This latter contribution has positive and negative values and goes through some nodes. The consequence is a S spin vector tilted from the xˆ · y, ˆ at some points of the Brillouin zone, by a certain angle going from ±1.15◦ for the conduction band and ±1.55◦ for the valence band. As the system is doped with Sb the Sz contribution gets more significant, but the oscillations have lesser nodes. For Bi concentration of 66% the energy slice plotted in figure 5(b) we see an S vector with angles from the plane varying from ±1.60◦ for the conduction band and ±5.38◦ for the valence band. From the figure 5 we also observe that the TRS is always preserved in the bulk Bi2 Se3 , and in the alloy.
4. Summary
In this paper we have performed a detailed and systematic investigation of the topological phases and the surface electronic properties of (Bix Sb1−x )2 Se3 . In order to provide an accurate description of the trivial→TI phase transition, the alloy system was described by using the SQS approach. The topological phases were determined through: (i) the bandgap closing, based on the adiabatic arguments, and (ii) explicitly calculation of the Z2 topological invariant number for all alloy concentrations considered in our study. In both procedures, (i) and (ii), we find that the trivial→TI phase transition occurs for Bi concentration x = 0.4. Different pictures of the energy dispersion of topologically protected surface states depending on the Bi/Sb ratio were observed. For the alloy concentration range of 0.4 < x < 0.7, the Dirac crossing point moves upward lying in the bandgap of the (Bix Sb1−x )2 Se3 alloy, being far from the bulk states. This result suggests a suppression of the scattering process ruled by bulk valence states.
Acknowledgments
We would like to acknowledge the Brazilian funding agencies: CAPES, CNPq and FAPESP. 5
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