Topological phase transition in half-Heusler compounds HfIrX (X = As, Sb, Bi)

Computational Materials Science 124 (2016) 311–315

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Topological phase transition in half-Heusler compounds HfIrX (X = As, Sb, Bi) Guangtao Wang ⇑, JunHong Wei College of Physics and Information Engineering, Henan Normal University, Xinxiang, Henan 453007, People’s Republic of China

a r t i c l e

i n f o

Article history: Received 22 May 2016 Received in revised form 27 July 2016 Accepted 7 August 2016

Keywords: Topological insulator Weyl semi-metal Half-Heusler

a b s t r a c t We studied the lattice constant and atoms substitution tunable topological phase transition in the halfHeusler compounds HfIrX (X = As, Sb, Bi). At the equilibrium cubic crystal structure and excluding SOC, HfIrAs and HfIrBi are topological nontrivial semimetal, while HfIrSb is a trivial topological insulator. This is because that the ‘‘internal pressure” lifts the s-type C1 band above p-type C5 bands in HfIrSb. When SOC is included, HfIrAs and HfIrSb become topological insulator, and normal band insulator, respectively, while HfIrBi is still a topological semimetal. When we induce compressive stress in the ab-plane of HfIrBi, it becomes a Weyl semimetal, with eight Weyl-Points (WPS) at (
Kx, 0,
Kz), (0,
K y ;
Kz), Kx = Ky = 0.023 Å1, Kz = 0.108 Å1. Ó 2016 Elsevier B.V. All rights reserved.

According to the electronic structure, all bulk materials are divided into two classes: (1) metals, which have a finite electron density at the Fermi energy; (2) insulators, which show a finite band gap. Recently, two new classes of the so-called topological insulator and Weyl-semimetal have emerged in solid state physics. Topological insulator (TI) is a new class of materials, which have a full insulating gap in the bulk but contain topologically protected conducting states on edges or surfaces [1–6]. The surface states are chiral and inherently robust to external perturbations, which could be useful for future technological applications in spintronics and even quantum computing [8]. Since the first two-dimensional (2D) TI with quantum spin-Hall effect was predicted and realized in HgTe [3,9,10], many three-dimensional (3D) TIs have been proposed theoretically [11] and observed in experiments [12,13]. On the way to searching new TI materials, researchers find that many ternary half Hustler compounds are close to the border between the trivial and nontrivial topological insulators and will be the platform for topologically related materials [2–5,7]. Such material can be easily transformed from a trivial insulator to a nontrivial topological insulator and vice versa by a small variation of the lattice constant [5]. After the discovery of TI, a new type of topologically related materials are proposed in solid physics: Weyl semimetals [14– 28], which are three-dimensional crystalline systems where pairs of bands touch at points in momentum space, termed Weyl points, characterized by a definite topological charge: the chirality. Weyl ⇑ Corresponding author. E-mail address: [email protected] (G. Wang). http://dx.doi.org/10.1016/j.commatsci.2016.08.005 0927-0256/Ó 2016 Elsevier B.V. All rights reserved.

fermions were originally considered in massless quantum electrodynamics, but have not been observed until two groups [14–17,24] independently and simultaneously found the Fermi arcs on the (0 0 1) surface of TaAs single crystal [14,16]. Weyl semimetal has a band structure with double degenerated crossing bands at the Fermi level. Such crossing points are the Weyl points, where exists a linear dispersion relation in all three momentum space directions moving away from the Weyl point. The negative magnetoresistance in TaAS [16], anomalous Hall effect [25], and non-local transport properties have been attributed to the existence of Weyl points [26]. Fermi arc states on the surface are predicted to show novel quantum oscillations [27] in magneto-transport [21,28]. Because of the fundamental and practical interest in Weyl semimetals, it is crucial that robust candidate materials should be hunted and identified. It is theoretically known that Weyl fermions only exist in the crystal where time-reversal symmetry or inversion symmetry is broken. The first way to break the time-reversal symmetry is to induce magnetism into the compounds. But the best way to generate Weyl fermions is by breaking the spatial inversion symmetry, because of two reasons: (1) Nonmagnetic Welt semimetals are more easily studied by angle-resolved photoemission spectroscopy, since alignment of magnetic domains is no longer needed. (2) Magnetic related bands structure change can be ignored when measure the Berry curvature caused unusual transport properties. Because of these reasons, half-Heusler compounds may be the good platform to seek for new classes Weyl semimetals [29–33]. The crystal lattice of ternary half-Heusler compounds is described by the space group F43m, where Hf atom and Ir are

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G. Wang, J. Wei / Computational Materials Science 124 (2016) 311–315

4

a

3

Energy (eV)

2 Hf d 0.53 Ir d 0.14 As p 0.33

1 Γ

0

5

Γ

Γ

1

5

0 Hf s 0.29 Ir s 0.28 As s 0.25

1

-1

4

Γ

-1

1

2 1 0

-3

-4

-4

L 0

1

-1 Hf d 0.64

-3

W

2

4

-4

Γ

X

W

L 0

1

Energy (eV)

2 Hf d 0.64 Γ

Γ

Bi p 0.23

2 1

5

Hf s 0.30

-1

Γ

Ir s 0.28

-1 1

Bi s 0.22

-2

-2

-3

E E

0 1

3

d 3

5

0 Γ

2

2

Energy Difference (eV)

Hf-6s Hf-5d Ir-6s Ir-5d Bi-6s Bi-6p

3

-2

Dos (state/eV)

4

c

5

-3

L

3

Γ

Sb p 0.27

Dos (state/eV)

1

0

-1

-3

X

1

Sb s 0.30

Γ

5

-2

Γ

2

Γ

Ir s 0.25

1

3

1

Hf s 0.31 Γ

-2

L

Hf-6s Hf-5d Ir-6s Ir-5d Sb-5s Sb-5p

3

-2

-4

4

b

Hf-6s Hf-5d 3 Ir-6s Ir-5d As-4s 2 As-4p

Energy (eV)

4

1

1 5

E -E 1

5

0 c

-1

-3

-4 L

Γ

X

W

L 0

1

2

3

-4

Dos (state/eV)

-2 5.6

5.7

5.8

5.9

6

6.1

6.2

6.3

6.4

Lattice Constant (Å)

Fig. 1. Without spin-orbital coupling, the band structure and projected density of state of (a) HfIrAs, (b) HfIrSb, and (c) HfIrBi. (d) The band gap between C1 and of C5 changes with lattice constant of HfIrAs.

mutually tetrahedrally coordinated nearest neighbours. The Ir (4b) and As (4a) atoms form a zinc-blende-type sublattice, with Hf atoms filling in the remaining face-centered-cubic sites (4c). The number of valence electrons in the half-Heusler compounds HfIrX (X = As, Sb, Bi) is 18, with close shells as s2p6d10, leading to a semiconducting or semimetal state. Previous studies show that the alteration of the topological order by band inversion is a key ingredient of nontrivially topological material. Both spin-orbit coupling and lattice distortions are found to be essential for such band inversion [35,36]. So we studied the nontrivially topological properties of HfIrX (X = As, Sb, Bi), varying with lattice distortions and atoms substitution. The calculations were done with the STATE [37] code, in the ultra-soft pseudopotential plane wave method. The spin-orbit coupling (SOC) effect is self-consistently included. All the equilibrium lattice constant and the atomic positions were relaxed by with generalized gradient approximation [38] calculation, with the force tolerance of 105 Ry/Bohr. The equilibrium lattice constants of HfIrAs, HfIrSb, HfIrBi are 6.16, 6.34, and 6.49 Å, respectively, in agreement with the results of Gautier’s [42]. After carefully checking the convergence of calculated results with respect to the cutoff energy and the number of k-points, we adopted a cutoff energy of 500 eV and Monkhorst-Pack k-points generated with 16  16  16.

In order to study how the topological phase changes with substituting Sb atoms by As atoms, we used the virtual-crystal approximation (VCA) established by Bellaiche and Vanderbilt [39]. The berry curvatures were calculated by the maximally localized Wannier functions [34,40]. We first present the band structure and projected density of state of HfIrAs, HfIrSb, and HfIrBi without spin-orbit coupling in Fig. 1(a–c). Without SOC, the bands around the Fermi level consist of the s-type singlet C1 band and p-type triplet C5 bands at C-point. For both HfIrAs and HfIrBi, the C1 band is below C5 , but it is reverse for HfIrSb. It was called ‘‘band inversion” [32,34], If s-typed C1 band is below p-typed C5 bands. This means that, even without SOC, band inversion takes place in the HfIrAs and HfIrBi compounds. In Fig. 1a–c, the bands of C5 are mainly derived from p-d hybridized states, while the C1 band is mainly derived from the s states of Hf, Ir, and As (Sb, Bi) atoms. From HfIrAs to HfIrSb the atomic radius (at 4a) increase about 13% (from 1.21 Å of As to 1.41 Å of Sb), but the lattice constants only increase 3% (from 6.16 Å of HfIrAs to 6.34 Å of HfIrSb). So HfIrSb can be regarded as ‘‘compressed” HfIrAs, because their valance electrons are equal as s2p6d10. When HfIrAs is compressed by the external pressure, both energies of C1 and C5 bands will increased. But the energy of C1 band will increase fast than that of C5 , because that the orbital

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G. Wang, J. Wei / Computational Materials Science 124 (2016) 311–315

4

a

3

3

2

2 Γ7

1

Energy (eV)

Energy (eV)

4

Γ8

0 -1

b

1

Γ6

Γ8

0 -1

Γ6 -2

-3

-3

-4

Γ

L

4

X

W

-4

L

L

4

d

3

3

2

2

1

Energy (eV)

Energy (eV)

Γ7

-2

Γ7

0 Γ8

-1

-3

W

L

Γ

X

W

L

Γ8

-1

-3

X

L

0

-2

Γ

W

Γ7

Γ6

L

X

c

1

-2

-4

Γ

-4

Γ6

L

Fig. 2. With spin-orbital coupling, the band structure (a) HfIrAs, (b) HfIrSb, and (c) HfIrBi. (d) The projected band structure of HfIrAs, where the symbol size corresponding to the projected weight of the Bloch states onto the S-liked orbit.

Energy Difference (eV)

Energy (eV)

2

2

b

a

Γ7

1

0

Γ8

Γ8

-1 -2 L 2

Γ6

Γ7

0

1

Γ

c

-1

Γ

X L

X

d

E6 E8 E6-E8

1

Γ6

-2 2 1

0

0

-1

-1

-2 5.6

5.8

6

6.2

Lattice Constant (Å)

6.4 0

0.2

0.4

0.6

0.8

-2 1

HfIrAs xSb1-x(x=doping)

Fig. 3. (a) The bands structure of compressed HfIrAs with lattice constant a = 0.95a0 (a0 is the equilibrium lattice constant). (b) The bands structure of the expanded HfIrSb with lattice constant a = 1.05a0. The band energy of C6 and C8 and band gap varying with lattice constant (c) and (d) with Sb atoms substitution.

contributions to C1 is s orbits, which are more expand than p-d hybridized states, and more sensitive to the external pressure. At the equilibrium lattice constant (6.16 Å), HfIrAs has the inverted

bands structure as E1
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G. Wang, J. Wei / Computational Materials Science 124 (2016) 311–315 0.1

b

a Energy (ev)

0.05

0

-0.05

2e+07

Kx Ky

1e+07

Kz

0

z

Ω (K) (atomic units)

-0.1

Γ

Z’

c 0.04

8e+08

8e+08 7e+08 6e+08 5e+08 4e+08 3e+08 2e+08 1e+08 0

d

7e+08 0.02

6e+08 5e+08

0 -0.02

Berry 4e+08 3e+08 2e+08

-0.04

1e+08 0 -0.06

-0.04

-0.02

0

0.02

0.04

-0.04

-0.02

0

Kx

0.02

0.04

0 -0.05 -0.1 -0.15 0.06

0.15 0.1 0.05

Kz

Fig. 4. (a) The bulk Brillouin zone of xy-plane strained HfIrBi. There are four pairs of Weyl point in the bulk BZ, schematically shown as red (chirality ‘‘+1”) and blue (chirality ‘‘1”) circles. (b) The bands structure and Berry curvature along C-Z’ line, passing through one of the WPS. (c) Berry curvature from pairs of Weyl points in the KxKy plane at Kz = 0.108 Å. (d) The Berry curvature of KzKx plane. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

is due to the result that the SOC strength ðDSOC = E8-E7) in HfIrAs is stronger than that in HfIrBi. It seems confusing because that Bi atoms are heavier than As atoms, so HfIrBi is expected to have stronger SOC effect. But the SOC strength DSOC is not only determined by the p-states of As (or Bi), but also affected by the d-states of Hf and Ir atoms. In Fig. 1, the p-type C5 bands of HfIrAs come from 53% Hf-5d, 14% Ir-5d, and 33% As-4p, while the p-type C5 bands of HfIrBi come from 64% of Hf-5d and 23% of Bi-5p. From above datas, the numbers both 5d-states and 4p-state of HfIrAs are larger than those of HfIrBi. So the SOC strength of HfIrAs is stronger than that of HfIrBi. The similar phenomena has also been find in the HgTe/CdTe compounds, where the SOC strength of HgTe is smaller than that of CdTe compound [3,36], although Hg atom is heavier than Cd atom. The projected bands structure of HfIrAs (Fig. 3d) indicates that the states at C6 are mainly derived from s-orbit, while the states at C8 are mainly derived from p-d hybridized states. For the C7 band, it has p-orbital character at C-point, but the s-orbital ingredient increases quickly when it deflects from the C-point. The topological properties are determined by many facts, such as pressure and atoms substitution. So we studied how the bands order of C6 -C7 -C8 changed with the pressure, which can be induced by either lattice constants change or atoms substitution. We present the bands structure of compressed HfIrAs with a = 0.95a0 (a0 is the equilibrium lattice constant) in Fig. 3a and the expanded HfIrSb with lattice constant a = 1.05a0 in Fig. 3b. The bands structure of compressed HfIrAs is similar to that of equilibrium HfIrSb, i.e. HfIrSb seems like compressed HfIrAs. When we

released the pressure in HfIrSb by increasing its lattice constant to a = 1.05a0 in Fig. 3b, its bands structure show typically no-trivial topological insulator character, with C6 < C8 . In these compounds, the band gap E6-E8 is an important parameter, which determines their topological phases: in the nontrivial topological insulator or ordinary band insulator. In Fig. 3c, the band energy and band gap, varying with lattice constants, are summarized for HfIrAs. When the lattice constant is larger than 5.97 Å, E6 is smaller than E8 and the compound in the nontrivial topological insulator state. While it becomes band insulator, when it is compressed with lattice constant smaller than 5.97 Å. Not only reducing the lattice constant, but also substituting smaller As atoms with more larger Sb atoms can induce pressure into the compounds. In Fig. 3d, we fix the lattice constant to the equilibrium one of HfIrAs with a = 6.16 Å, and substitute As toms with more larger Sb atoms. At x = 1, i.e. HfIrAs, the compound has the invert bands structure. From x = 1 to x = 0.6, the band gap increases from 0.5 eV to zero, indicating the non-trivial topological insulating state. When we further increase the Sb atom substitution E6 becomes larger than E8, which means that the band inversion disappears and the compound enters into the ordinary band insulating state. In fact, substituting As atoms with larger Sb atoms with the fixed crystal volume is equal to acting pressure on the compounds. The C8 bands at the C point are fourfold degenerate as J = 3/2 multiplet. We induce an in-plane strain to lower the crystalline symmetry from (Cubic phase) Td to D2d (Tetragonal phase, a = 0.98a0, c = 1.04a0). The mirror symmetries in the kx =
kz and ky =
kz planes are broken, while those in kx =
ky planes survive.

G. Wang, J. Wei / Computational Materials Science 124 (2016) 311–315

As a results, the degenerated bands C8 (Fig. 3) split as Fig. 4(b). Such crossing point is Weyl point, which can be regarded as a singular point of Berry curvature. i.e. ‘‘magnetic monopole” in the momentum space [41]. Because of the crystalline symmetry, there are eight Weyl points in the whole Brillouin zone (see Fig. 4a). According to their chirality, these eight WPS are divided into two types: four ‘‘source” WPS in the KxKz-plane and four ‘‘Drain” WPS in the KyKz-plane in Fig. 4a. The bands structure and berry curvature along C-Z0 line are presented in Fig. 4b, where two bands crosse at the WP with its position as (0.023, 0, 0.108) (in unit Å1). At the same time, the berry curvature shows a sharp peak at this point in Fig. 4b. The ‘‘source” WPS and ‘‘Drain” WPS can be clearly seen in the KxKy-plane at Kz = 0.108 Å1, as shown in Fig. 4c, where two ‘‘source” WPS at (0.023, 0.0, 0.108) , (0.023, 0.0, 0.108) and two ‘‘Drain” WPS at (0.0, 0.023, 0.108) , (0.0, 0.023, 0.108). Such WPS can also be seen in Fig. 4d, where the berry curvature is plot in the KxKz-plane at Ky = 0. In summary, we studied the lattice constant and atoms substitution tunable topological phases change in the half-Heusler compounds HfIrX (X = As, Sb, Bi). At the equilibrium cubic crystal structure and excluding SOC, HfIrAs and HfIrBi are nontrivially topological semimetals, while HfIrSb is a trivial topological insulator. This is because that the ‘‘internal pressure” lifts up the s-type C5 band in HfIrSb. The band gap between s-type C1 and p-type C5 , E1-E5, is very sensitive to the lattice constants. When the lattice constant of HfIrAs is smaller than 5.97 Å, it has topologically trivial bands structure with C1 band above C5 band. But it shows topologically non-trivial bands structure when the lattice constant is larger than 5.97 Å. When SOC is include, HfIrAs and HfIrSb become non-trivial topological insulator, and ordinary band insulator, respectively. However HfIrBi is still topological semimetal. The SOC strength is larger in HfIrAS than in HfIrBi. When we induce compressive stress in the ab-plane of HfIrBi, we find eight WPS, with four ‘‘source” WPS in the KxKz-plane at (
Kx, 0,
Kz) and four ‘‘Drain” WPS in the KyKz-plane at (0,
K y ;
Kz), here Kx = Ky = 0.023 Å1, Kz = 0.108 Å1.

References

Acknowledgements

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We thank Dr. Suzuki for helpful discussions and Prof. Arita for providing the computing resource in Riken of Japan. The authors acknowledge the support from NSF of China (No. 11274095, No. 10947001) and the Program for Science and Technology Innovation Talents in Universities of Henan Province (No. 2012HASTIT009, No. 104200510014, and No. 114100510021). The calculations are supported by The High Performance Computing Center of Henan Normal University.

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