1

Ternary Tetradymite Compounds as Topological Insulators

Lin-Lin Wang1

and Duane D. Johnson1,2§

1Division of Materials Science and Engineering, Ames Laboratory, Iowa State University, Ames, Iowa

50011; 2Department of Materials Science and Engineering, Iowa State University, Ames, IA 50011

Abstract

Ternary tetradymite compounds of Bi2Se2Te, Bi2Te2Se and Bi2Te2S are stable and found to be 3-

dimensional topological insulators (TI) via density functional theory calculations. As with their

binary counterparts Bi2Se3 and Bi2Te3, the ternary TI band inversion between group V and VI pz

orbitals in the bulk band structure was verified. With its key advantages, we identify Bi2Se2Te

with (111) surfaces (the cleavage plane) as a very good TI candidate to study responses of

massive Dirac Fermions to magnetic perturbations; this ternary has a surface-derived Dirac point

isolated in the bulk band gap at the Fermi level like Bi2Se3 and also a large spin texture

comparable to Bi2Te3. In contrast, for Bi2Te2Se and Bi2Te2S (111) surfaces, we find that the

Dirac point is below the Fermi level and buried among bulk bands. We also suggest that Bi2Te2S

offers large bulk resistivity that is needed for devices, and, similarly, for doped- Bi2Se2Te.

2

The materials that exhibit topological insulator (TI) behavior and quantum Hall Effect

reveal a novel quantum state for electrons,1,2

where the edge (surface) state of a two (three)

dimensional system is topologically protected against disorder by time-reversal symmetry and, as

a result, electrons experiences no backward scattering by non-magnetic impurities. The unique

feature of a 3-dimensional (3D) TI lies in its band structure, where surface bands connect the

valence and conduction bands and cross the Fermi level (Ef) an odd number of times along two

time-reversal equivalent k-points. Such a band feature is most commonly found in narrow gap

semiconductors with strong spin-orbit coupling (SOC). Unlike the interaction with an external

magnetic field, where time-reversal symmetry is broken, the SOC preserves time-reversal

symmetry. Since Bi2Se3 and Bi2Te3 were almost simultaneously observed3-5

and predicted6 to be

3D TI, an intensive search7-10

continues for other 3D TI systems. Notably, Bi2Se3 and Bi2Te3

belong to a class of line compounds called tetradymite. These compounds formed between group

V and VI elements have a quintuple-layered structure, in which group VI element occupies the

outmost and central (third) layer, and group V element occupies the second layer. The two group

VI positions are not equivalent. The primitive unit cell is rhombohedral (hR5) with the space

group of (No. 166) and five atomic layers stacked along . The stacking is similar to

fcc (ABCABC…), but the interlayer distances are different; in particular, the distance between

neighboring quintuple layer units is larger than the others, making it is easier to cleave with

group VI element exposed as the top surface (111) layer. Like Bi2Se3 and Bi2Te3, ternary

tetradymites, e.g., Bi2Se2Te, Bi2Te2Se and Bi2Te2S, are stable.11,12

We focus on the bulk and

surface band TI features of these ternary tetradymite compounds.

Upon band inversion and crossing of Ef due to SOC, a Dirac point (DP) is formed, the

connecting point between linearly dispersed branches from valence and conduction bands. DP

offers a platform to study a variety of interesting physics. Using angle-resolved photoelectron

spectroscopy (ARPES), Chen et al.13

have shown that massive Dirac Fermion can be produced

on Bi2Se3 (111) surface by breaking time-reversal symmetry via magnetic impurities. To do this,

the position of DP is critical; it must be in the gaps of both bulk and surface bands. For TI, the

DP is not always favorably located as such. For example, Bi2Se3 has a DP isolated in the bulk

band gap region making it a good candidate to study massive Dirac Fermion, whereas the DP for

Bi2Te3 is below Ef and buried among surface valence bands (Figure 4 in [6]). Also, Bi2Se3 has a

larger band gap compared to Bi2Te3, offering a means to adjust behavior at elevated temperature.

3

Another large difference between the binary tetradymites is the shape of Dirac cone. In contrast

to the perfect Dirac cone of Bi2Se3, the warped Dirac cone of Bi2Te3 shows an interesting spin

texture,14,15

suggested to form exotic charge and spin density waves. Thus, importantly, it is more

helpful for theory to predict not only TI from bulk band structure, but also to detail DP band

dispersion in surface band structure. Although bulk band structures of Bi2Te2Se and Bi2Te2S

have been calculated7 to be possible TI, with Bi2Te2Se confirmed in experiment,

16 the positions

of DP in the surface band structures have not been analyzed. Also, there has been no study on

Bi2Se2Te as a TI. From band features, we find that Bi2Se2Te is distinguished as a 3D TI with its

DP at Ef and isolated in the bulk band gap and its Dirac cone exhibits a large spin texture,

making it a better candidate to manifest (controllably) massive Dirac Fermions than Bi2Se3.

We present electronic band structures of Bi2Se2Te, Bi2Te2Se and Bi2Te2S calculated in

density functional theory17,18

(DFT). We analyze both the bulk and surface (via a slab model)

band structures to identify any 3D TIs along with the nature of the DP, especially its location and

shape of Dirac cone. One costly method to tell a TI from an ordinary band insulator is to

calculate the Z2 topological order.19

Yet, for structures with inversion symmetry, Fu et al.20

proposed inspecting the parity product of occupied bands on time-reversal equivalent k-points in

the bulk bands. Here we search for a band inversion and zero gap in bulk bands by tuning the

SOC (λ from 0 to 100%), and then, once verified, we calculate (slab) surface bands and assess

DP formation (including band inversion and the number of bands that cross Ef), an approach

used by Zhang et al.6 for semi-infinite surfaces of the binaries.

We use DFT with PW91exchange-correlation functional21

and plane-wave basis set with

projected augmented waves,22

as implemented in VASP.23,24

Bulk tetradymite can be represented

as a hexagonal lattice with fifteen atomic layers, which also gives the atomic basis in the slab

model for the (111) surface. Both the atomic structure and Brillouin zone for tetradymite bulk

and (111) surface have been shown before,6,25

so we do not repeat them. For bulk we use the

primitive rhombohedral cell of five atoms with 7×7×7 k-point mesh. We use a three quintuple-

layered slab along with no vacuum for bulk band projection and a 14 Å vacuum for

surface band calculations. The k-point meshes used are 10×10×2 and 10×10×1, respectively.

The kinetic cut off energy is 280 eV. We use experimental lattice constants (Table 1) throughout

our study. For tetradymite, the (111) surface (basal plane to the quintuple-layer stacking) is the

4

easiest to cleave. To show a TI surface band definitely connects between valence and conduction

bands, a semi-infinite surface is required,6 which we show, however, can be achieved with a

thick slab model. We find that three quintuple layers along are needed to model the semi-

infinite surface, with the two group VI terminated surfaces on top and bottom of the slab well

separated by at least 28 Å; we fix atoms in their bulk positions because relaxation is very small.

To validate our use of a thick slab to realize the surface band structure of a semi-infinite

surface, we show band structures of Bi2Se3 and Bi2Te3 in Figure 1. With full SOC, both bulk

Bi2Se3 and Bi2Te3 are semiconductors with a narrow indirect band gap (Δ) of 0.32 eV around Γ

and 0.14 eV around Z point, respectively. The bulk bands in Figure 1(a) and (c), and Δ are in

agreement with previous DFT results.6 The splitting into two nearby maximum at the Γ point is

due to SOC and gives a larger band gap at the Γ-point (ΔΓ) (Table 1). Due to the equivalency of

the two slab surface layers, each surface band in Figure 1(b) and (d) is doubly degenerate. To

show clearly surface bands, we include (as shaded area) the bulk bands projected in the (111)

direction. Clearly, some surface valence and conduction bands appear in the forbidden bulk gap

region and approach Ef at Γ point. At the thickness of three quintuple layers they are already

within 0.02 eV. By increasing the slab thickness toward semi-infinite limit, a DP forms. For

Bi2Se3 (111) surface, the DP is located at Ef and stands isolated in the bulk band gap. In contrast,

for Bi2Te3 (111) surface, the DP is below Ef and buried among nearby states along both

and . Such distinct DP relevant features in binary slab surface bands agree very well with

those calculated for semi-infinite surfaces.6

For tetradymite, the two group VI positions in the quintuple layer are not equivalent,

where central-layer substitution is common as observed in ternary Bi2Se2Te, Bi2Te2Se and

Bi2Te2S.11,12

For Bi2Se2Te, the Se in the central layer of the quintuple Bi2Se3 is replaced with Te,

resulting in a slightly larger lattice constant, see Table 1. It has an indirect gap of 0.22 eV. For

bulk, we calculate ΔΓ as a function of λ, with charge density fixed at that of λ=0. As seen in

Figure 2(d), the band gap becomes zero around the critical value λc =71%. The corresponding

band structure is shown in Figure 2(b). Compared to the band structure without SOC in Figure

2(a), the largest change is the lowering of the lowest conduction band and the rising of the

highest valence band to form jointly a DP at Γ point at Ef. Upon further increase of λ, Figure 2(c)

shows that the gap at Γ point is reopened and two nearby maximum are formed. Figure 2(e) plots

5

the projections of the highest valence band on Bi and Se pz orbital as a function of λ. Below

(above) 71% SOC, it is mostly composed of Se (Bi) pz components. Indeed, SOC causes the

band inversion in Bi2Se2Te bulk, similar to that observed in binary tetradymite, so Bi2Se2Te is

also a 3D TI. We hope the data in Table 1 will be useful for device development.

To see the surface band structure feature of Bi2Se2Te (111), we plot the band structure of

a three quintuple slab on top of the bulk projected bands in Figure 2 (f). Similar to Bi2Se3 (see

Figure 1(b)), we find the DP for Bi2Se2Te (111) surface is at Ef and not buried by other states

around Γ point. The two outmost atomic layers in the tetradymite quintuple-layered structure

mostly determine the position of DP. Thus, Bi2Se2Te is a good candidate to study massive Dirac

Fermion because its DP is located in both bulk and surface gaps.

Ternary tetradymite compounds Bi2Te2Se and Bi2Te2S are based on Bi2Te3, with the

central Te layer replaced with Se and S, respectively. In Figure 3 we show the bulk and surface

band structures for these two ternaries. By tuning the SOC strength, we find that the critical

value to achieve a DP at Ef (with band inversion) is 21% and 44%, respectively. Thus, both

compounds are 3D TIs. They both have an indirect gap of 0.28 eV, larger than that of Bi2Te3,

mostly due to a smaller lattice constant. Looking at slab surface bands, we find similar features

to that of Bi2Te3 in terms of the position of DP; i.e., the DP of Bi2Te2Se is buried even deeper

below Ef than that of Bi2Te3, and the DP of Bi2Te2S is at a similar height to that of Bi2Te3.

A significant difference between the surface band structures of Bi2Se3 and Bi2Te3 is the

shape of Dirac cone. Figure 4 shows the 3D band structure around Γ point for the conduction

band of each compound. Constant-energy contours (projected on bottom of figure) are drawn as

lines. As seen in Figure 4(a) and (c), the Dirac cone for Bi2Se3 remains perfect up to 0.4 eV,

while for Bi2Te3, it is only 0.2 eV, agreeing with previous experiments5,13

and theory.14,15

Beyond these energies, there is significant warping of the Dirac cone, as evidenced by the

change in the shape of contour lines on the bottom – first to a hexagon, then to a snowflake. Such

non-convex shapes produce more pairs of stationary points on the constant energy contours,

allowing scattering processes among different pairs of stationary points. This is in contrast to no

scattering for a convex Dirac cone. Such behavior has been observed in STM,26

where a line

defect on Bi2Te3 (111) surface suppresses scattering only in the energy range of circular constant

energy contours, not snowflake ones.

6

Figure 4 also shows the spin texture exhibit within the Dirac cone, i.e., the ratio of out-of-

plane to total electron spin moment color-mapped on the cone. For a perfect Dirac cone, the

electron spin always lies in the surface plane and is perpendicular to the wave vector. In contrast,

for a warped Dirac cone, Fu14

suggested that there should be a significant amount of out-of-plane

spin moments up to 60%, to maintain a Berry phase change of π in one circuit as required by

topological invariance. Such behavior can lead to interesting features, such as spin density

waves on a 3D TI surface and opening of DP by an in-plane magnetic field. For Bi2Te3 (see

Figure 4(c)), our results agree with previous calculation15

and show a large spin texture beyond

0.2 eV, except for . In contrast, Bi2Se3 (see Figure 4(a)) has a much smaller spin texture.

Compared to Bi2Se3, the up limit of the cone convexity for Bi2Se2Te (see Figure 4(b)) is

decreased to 0.3 eV and beyond that a large spin texture appears. Whereas, compared to Bi2Te3,

the up limit of the cone convexity for Bi2Te2Se and Bi2Te2S (see Figure 4(d) and (e)) are both

increased to 0.3 eV, and beyond that a small spin textures appear, with Bi2Te2S being smaller

than Bi2Te2Se. This shows that, even though the position of DP is only slightly affected by the

substitution of atom in the central layer, the Dirac cone warping and spin texture are greatly

affected by such substitution. In Bi2Te2Se and Bi2Se2S, the binding of the non-spin-orbit

coupling site in the central layer to two Bi sites increases hybridization and decreases spin-orbit

coupling. While in Bi2Se2Te, by replacing the central Se layer with Te, the spin-orbit coupling in

the Bi-Te-Bi trilayer is enhanced significantly and results in a large spin texture. Thus, in

Bi2Se2Te we have a superb TI candidate with (1) a DP standing alone in bulk band gap (just like

Bi2Se3) and (2) a large spin texture (just like Bi2Te3 and larger than Bi2Se3).

Lastly, any potential use of these materials as a 3D-TI device requires control over the

bulk resistivity. Ideal binary tetradymites Bi2Se3 and Bi2Te3 are semiconductors, but defects

(such as vacancy and Bi-Te-antisite) cause significant bulk conductivity, which overwhelms the

surface-state contribution. For example, Ren et al.16

have shown experimentally that the ternary

Bi2Te2Se has a much larger bulk resistivity than Bi2Te3 because the substitution of the central Te

site with Se reduces the formation of a Se vacancy and Bi-Te antisite. The same mechanism

should also be operative for Bi2Te2S because S is more electronegative and binds Bi stronger

than Se, preserving the stoichiometric structure even better than Bi2Te2Se. For example, the

energy cost to create a Bi-Te-antisite is increased by 0.7 eV when changing from Bi2Te2Se to

7

Bi2Te2S.27

Finally, doping Bi2Se2Te may offer further control over the resistivity for this “best in

class TI” tetradymite.

In conclusion, we find that ternary tetradymite compounds of Bi2Se2Te, Bi2Te2Se and

Bi2Te2S are bulk topological insulators, confirmed computationally by verifying band inversion

between group V and VI pz orbitals. We validated and then used band structures of a large (three

quintuple-layered) slab model to study surface band features, including the Dirac cone and its

associated spin texture. The Dirac point of Bi2Se2Te (111) surface lies at the Fermi level and

stands isolated in the bulk band gap, in contrast to Bi2Te2Se and Bi2Te2S (111) surfaces.

Moreover, Bi2Se2Te has a much larger spin texture on its Dirac cone than Bi2Se3, offering a good

material for experimental study of spin-texture-related opening of Dirac point. We suggest that

doped-Bi2Se2Te may offer controllable resistivity, along with its favorable isolated Dirac cone

and large spin texture. Due to defect formation, Bi2Te2S (also predicted to be a 3D TI) should

have a large bulk resistivity needed to realize a workable device.

Work at Ames Laboratory was supported by the U.S. Dept. of Energy, Office of Basic Energy

Sciences, Division of Materials Science and Engineering. Ames Laboratory is operated for DoE

by Iowa State University under Contract No. DE-AC02-07CH11358.

*llw@ameslab.gov

§ddj@ameslab.gov

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a (Å) c (Å) x1 x2 λc Δ (eV) ΔΓ (eV)

Bi2Se3 4.138 28.64 0.399 0.206 0.46 0.32 0.47

Bi2Te3 4.383 30.487 0.400 0.212 0.48 0.14 0.52

Bi2Se2Te 4.218 29.240 0.398 0.211 0.71 0.17 0.25

Bi2Te2Se 4.28 29.86 0.396 0.211 0.21 0.28 0.70

Bi2Te2S 4.316 30.01 0.392 0.212 0.44 0.28 0.53

Table 1. Experimental (Ref. 11 and 12) structural parameters (lattice constant a, c and internal

parameters x1 and x2) and DFT-PW91 results for bulk tetradymite compounds, i.e., critical

strength of SOC for band inversion at Γ point (λc), band gap (Δ) and band gap at Γ point (ΔΓ).

10

(a) (b)

(c) (d)

Figure 1. Band structures of Bi2Se3 [(a) bulk and (b) slab] and Bi2Te3 [(c) bulk and (d) slab] are

shown. Slab is three quintuple-layers along . In (b) and (d) the shaded region is the bulk-

projected bands in (111) direction.

11

(a) (b) (c)

(d) (e) (f)

Figure 2. Band structures of bulk Bi2Se2Te are shown with SOC strength (λ) of 0, 71 and 100%

in (a), (b) and (c), respectively. (d) Bulk band gap at Γ-point vs. λ. (e) Projection of the highest

occupied band at Γ-point on Bi and Se pz orbital vs. λ. (f) Band structure of a Bi2Se2Te slab.

12

(a) (b) (c)

(d) (e) (f)

Figure 3. Band structures of Bi2Te2Se and Bi2Te2S with full SOC are shown. For Bi2Te2Se, the

(a) bulk dispersion, (b) bulk band gap at Γ-point vs. λ, and (c) slab dispersion along . For

Bi2Te2S, (d), (e) and (f) are equivalent to (a), (b) and (c).

13

(a) (b) (c)

(d) (e)

Figure 4. (Color online) Spin texture color-mapped on the conduction-band Dirac cone, (a)

Bi2Se3, (b) Bi2Se2Te, (c) Bi2Te3, (d) Bi2Te2S and (e) Bi2Te2Se. Color indicates the amount of

out-of-plane electronic spin moment in percent. Bi2Se2Te exhibits the topology of the Bi2Se3

Dirac cone but the enhanced spin texture of Bi2Te3.