Phase Coexistence and Strain-Induced Topological Insulator in Two

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C: Physical Processes in Nanomaterials and Nanostructures

Phase Coexistence and Strain-Induced Topological Insulator in Two-Dimensional BiAs Tamiru Teshome, and Ayan Datta J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b05293 • Publication Date (Web): 13 Jun 2018 Downloaded from http://pubs.acs.org on June 19, 2018

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The Journal of Physical Chemistry

Phase Coexistence and Strain-Induced Topological Insulator in TwoDimensional BiAs Tamiru Teshome and Ayan Datta* Department of Spectroscopy, Indian Association for the Cultivation of Science, 2A, and 2B Raja S. C. Mullick Road, Jadavpur, Kolkata 700032, West Bengal, India. Email: [email protected] ABSTRACT: Two-dimensional (2D) binary compounds have been recently reported as promising materials for achieving topological insulator and dissipationless transport devices. Based on first-principle calculations combined with a tight-binding (TB) model, we investigate a new class of 2D BiAs polymorphs which are energetically and dynamically stable. The monolayers of α-, β- and γ-BiAs allotropes show significant direct band-gap, while the δ- and ε-BiAs phases display indirect band gaps. Using strain-engineering along with spin-orbital coupling (SOC), β-BiAs transforms from a normal insulator to nontrivial topological phase. Under tensile strain and SOC, the bands are inverted resulting in a topological phase transition with a sizable band-gap of 0.28 eV. The nontrivial topological state is explicitly confirmed by calculating topological invariant, Z2 = 1 and the characteristic of edge states which are topologically protected in a Dirac cone at the ᴦ-point. Hexagonal boron nitride is confirmed as an excellent substrate for supporting the β-BiAs film without perturbing the topological insulator state. The present results indicate promise for 2D TIs at room temperature. 1.

INTRODUCTION

have not been synthesized till date and would be extremely

Recently, topological insulators (TIs), namely quantum spin

difficult to fabricate devices apart from toxicity of the heavy

Hall (QSH) insulators1,2 have attracted interests as a new state

atoms.9-22 Among these systems, Bi4Br4 and ZrTe5/HfTe5 exist

of quantum materials. These, novel electronic states possess

as three dimensional (3D) layered materials and their

bulk insulating gap and conduct charge and spin in helical

nontrivial band-gaps are practically useful to realize TI at

edge states by time-reversal symmetry. Though first proposed

room temperature.

by Kane and Mele for graphene, a very small opening of band gap ~10-3 meV at the Dirac point remains practically

Several group-V elements in buckled and planar hexagonal

inconsequential due to weak SOC for typical devices.3

structures such as phosphorene,23 arsenene,9 antimonene15-18

Therefore, a major bottleneck in for TIs is to find new

and bismuthene24, 25 exhibit topological phase transition from

materials with sizable band-gap opening at the Dirac-cone due

NI to TIs. At present, Bismuth based systems are an excellent

to controlled SOC.

materials for topological insulator due to their strong SOC. For example; the two well-known binary compounds - Bi2Se3

Recently, several 2D TI materials have been predicted like

and Bi2Te3 are strong TIs.26-27 On the other hand, arsenene in

silicene, germanene and stanene.4 However, so far, only the

α- and β-phase is energetically stable and converts from NI to

5

Bismuthene on a SiC substrate, HgTe/CdTe and InAs/GaSb

TI under suitable strain modification.9, 28-32 Therefore, seeking

quantum wells have been experimentally confirmed at a very

suitable 2D TIs with significantly large band gap is an

low temperature (below 10 K) limited by their small band

important area of research.

gap.6-8 Unfortunately, the majority of these predicted materials

1

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Figure 1: Equilibrium structures for the polymorphs of Bismuth Arsenic (BiAs) in the top and side views (a) α-BiAs, (b) βBiAs, (c) γ-BiAs, (d) δ-BiAs and (e) ε-BiAs. The shaded regions were shown to be unit-cell of the polymorphs; pink spheres represent Bismuth (Bi) atoms and green Arsenic (As) atoms. Hence, a NI to TI crossover can be in particular interesting in

were performed to verify the stabilities at room temperature.

2D materials system for quantum devices where NI/TI

Wannier

Charge

Center

(WCC)

implemented

in

38

switchability can be tuned under electric or superconducting

WANNIER90 package to identify the topological invariance

fields. To the best of our knowledge, the electronic and

characterized by the Z2 number, which is implemented in Z2

geometric

pack.

phase-diagram

of

a

nontrivial

topological

compound (BiAs) with appreciable band-gap with respect to

3.

its unique polymorphs has not been elucidated till date.

Results and Discussion

In this article, we have investigated 2D BiAs polymorphs and their electronic structures based on first principle

2.

calculation, and these new materials labeled as α-, β-, γ -,

Computational Details

The calculations were performed using the Vienna ab initio simulation package (VASP).

33

δ-, and ε-BiAs forms in their stable structures are shown in

The exchange-correlation term

Fig. 1(a-e). These five different phases arise from relative

is described within the generalized gradient approximation

connectives between the atoms in the 2D lattices. There

(GGA) parameterized by the Perdew-Burke-Ernzerhof (PBE)

are four atoms per unit cell in the α-BiAs and γ-BiAs as

functional.34 A vacuum space of 20 Å is set to avoid the

shown in Fig. 1(a, c), while there are eight atoms per unit

interaction between layers caused by the periodic boundary

cell in the δ-BiAs and ε-BiAs phase as shown in Fig. 1(d,

condition. The kinetic-energy cutoff plane-wave expansion is

e). On the other hand, β-BiAs with a hexagonal structure

set to be 500 eV. All the atoms in the unit cell are fully relaxed

contains two atoms per unit cell and the buckling height,

until the force on each atom are less than 0.01 eV/Å and

lattice constant and bond length were found to be 1.56 Å,

-5

convergence threshold was set to be 10 eV of the energy.

2.75 Å and 4.03 Å, respectively as shown in Fig. 1(b). The

Furthermore, Monkhorst-Pack k-point grid of 11 × 11 × 1 was

structural details of all the polymorphs are reported in

35

adopted. The strain modification was described as 𝜀 =

Table 1.

∆𝑎/𝑎! , where 𝑎! is the equilibrium lattice and ∆𝑎 + 𝑎! is the 3.1 Dynamical Stabilities and Electronic properties of

strained modified lattice. SOC effects were included in SCF 36

BiAs polymorphs

electronic structure calculations. The phonon calculations are carried out using the PHONOPY

37

To evaluate the relative stability of 2D-BiAs polymorphs, we

code combined with

calculated the cohesive energies as the following expression:

density functional perturbation theory (DFPT) method in

𝐸!"! = (𝐸!"!!"! − 𝑛𝐸!" − 𝑛𝐸!" )/𝑛

VASP and ab-initio molecular dynamics (MD) simulations

2


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where 𝐸!"!!"! , 𝐸!" and 𝐸!" being the total energies of BiAs

The relative cohesive energies of 2D BiAs polymorphs with

allotropes, single Bi atom and single As atom, respectively.

respect to that of α-BiAs are illustrated in Fig. 2(f) and clearly, β-BiAs is the most stable structure. The dynamical stability of the BiAs allotropes was further verified by calculating their vibrational spectra as shown in Fig. 2(a-e). The absences of imaginary frequencies show that all the five BiAs polymorphs are dynamically stable. Additionally, molecular dynamics (MD) simulations were performed on the five BiAs polymorphs for 10 ps with a time step of 1.0 fs at 300 K. In all polymorphs, the T vs time graph remains the unchanged for fluctuations between 290 and 310 K are shown in Fig. S2. The MD simulations indicate that these BiAs polymorphs are also thermally stable at room temperature.

To understand how

significantly the band-gaps depend on the structure of the

Figure 2: Phonon spectrum of (a) α-BiAs, (b) β-BiAs, (c) γ-

polymorph, the band structures are calculated for α-, β-, γ -, δ-,

BiAs, (d) δ-BiAs, (e) ε-BiAs and (f) relative cohesive energy

and ε-BiAs monolayers as shown in Fig 3(a-e) without SOC

with respect to α-BiAs.

and Fig. 3 (f-j) with SOC effect.

Figure 3: Electronic band structures of BiAs polymorphs without and with SOC (a, f) α-BiAs, (b, g) β-BiAs, (c, h) γ-BiAs, (d, i) δBiAs, (e, j) ε- BiAs, respectively.

3



Page 4 of 11

Table 1: Equilibrium BiAs polymorphs: Lattice parameters a and b (Å), bond length; d (Å), bond-angle; θ (°), dihedral angle; ɸ (°), buckling height; Δh (Å), g band gap value calculated PBE without and with SOC; (eV), Cohesive energy; Ecoh (eV/atom) and ΔE!"! = 𝐸!"! − 𝐸!"! (α-BiAs); ΔEcoh (eV/atom) relative stability of polymorph with respect to α-BiAs. Model

a

b

d

θ

ɸ

α-BiAs β-BiAs γ-BiAs δ-BiAs ε-BiAs

4.17 4.03 6.91 5.94 6.34

4.81 4.03 4.03 6.14 7.89

2.75 2.79 2.80 2.75 2.74

97.29 92.31 92.99 96.52 98.93

82.03 87.59 87.29 9.76 0.38

Δh

gPBE

gsoc+PBE

2.71 1.56 1.74 2.72 2.11

1.27 1.02 1.01 0.61 0.75

0.58 0.69 0.48 0.45 0.51

Ecoh

ΔEcoh

-1.81 -2.46 -0.88 -0.76 -1.38

0.00 -0.65 0.93 1.10 0.43

Figure 4: Electronic band structures of β-BiAs without (a-d) and with SOC (e-h) by applying different values of tensile strains. The Fermi level set to be 0 eV. We found that α-BiAs, β-BiAs and γ-BiAs are direct band-gap

3.2 Topological Phase Transition in Strained β-BiAs

semiconductors at ᴦ-point, while others δ-BiAs and ε-BiAs are

Strain-engineering is a powerful approach for tuning the

indirect semiconductors regardless of whether SOC effects are

electronic and topological properties of 2D materials.21 When

included or not. For α-BiAs, the VBM and CBM are located at

a biaxial tensile strain is induced, Bi-As is stretched and the

ᴦ-point with the band gap of 1.27 eV and 0.58 eV without and

lattices are relaxed consequently (Fig. S3). We critically

with SOC. Similarly, for β-BiAs, the direct band-gaps are 1.02

evaluate the band structure evolution under biaxial tensile

eV (0.62 eV at HSE06 level, see supp. Inf. Fig. S4) without

strain for β-BiAs as it is the most stable phase as shown in Fig.

and 0.69 eV with SOC. For γ-BiAs without SOC the VBM

4(a-h). It is also interesting to explore if a nontrivial

and CBM are located at ᴦ-point resulting in band-gap = 1.01

topological

eV (0.48 eV with SOC). Clearly, α, β and γ polymorphs of

configuration of β-BiAs under a suitable biaxial tensile strain.

BiAs have reasonable direct band-gaps which make them

The variation of band-gaps under tensile strain with and

prospective candidate for TIs under external perturbations like

without SOC for β-BiAs is shown in Fig. 5(a). We started

strain or electric field.

from the semiconducting phase β-BiAs (ε = 0 %) having a

phase

transition

occurs

for

the

buckled

direct band gap of 1.02 eV at its equilibrium structure (Fig. S4). Till a tensile strain applied, ε < 12 %, it behaves as a

4


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The Journal of Physical Chemistry !

semiconductors without and with SOC effect as shown in Fig.

𝜀!" = −8.70 𝑒𝑉.40 The optimal values for β-BiAs are

4(a-h). Further increasing the strain, the conduction band

𝑉!!" = −1.162 𝑒𝑉,

progressively shifts downward to the Fermi level. On the other

and 𝑉!!" = −0.672 𝑒𝑉, respectively. With the above set of

hand, the valance band shifts upwards leading to semi-metal at

parameters, the TB bands are in reasonable agreement with the

ε = 12 % without SOC and hence, no band inversion observed

DFT results as shown in Fig. 5(b-d). The p-orbitals near the

as shown in Fig. 5(c). Clearly, at ε > 12 % without SOC, β-

Fermi level split into two groups (𝑝! and 𝑝! ) orbitals

BiAs is a semi-metals. As the strain is increased further a band

𝑉!"# = 1.122 𝑒𝑉,

𝑉!!" = 2.689 𝑒𝑉

and 𝑝! orbital, respectively. Projecting 𝑝! , 𝑝! and 𝑝! bands

inversion occurs at ᴦ-point in presence of SOC (Eg = 0.28 eV).

onto different atomic orbitals, we found that the energy

Therefore, with SOC and ε > 12 %, one observes a phase

spectrum of β-BiAs near the Fermi level is modified mostly to

transition for a normal insulator to topological insulator (Fig.

𝑝! and 𝑝! orbitals.

5(a)). Our predicted nontrivial band topology is in excellent agreement with the results previously reported for Bismuth Oxide film.39 In fact, both Arsenic and Bismuth have strong SOC effects and hence, band inversion can be achieved when a large strain is applied.9, 32To identify the mechanism of the band inversion, we propose a tight-binding model of 𝑝! , 𝑝! and 𝑝! orbitals. The effective Hamiltonian is considered as:

Here,

𝑐!!!

!

𝑡!" ( 𝑐!!! 𝑐! + ℎ. 𝑐. ) !,! ,!,!

!,!

𝜀!! ,

!"

𝜀!! 𝑐!!! 𝑐!! +

𝐻!" =

and

𝑐!!

represent the on-site energy, creation,

and annihilation operators of an electron at the α-orbital of the !"

i-th atom. The parameter 𝑡!" is the nearest-neighbor hopping energy of an electron between α-orbital of i-th atom and βorbital of j-th atom, α, β ∈ (𝑝! , 𝑝! , 𝑝! ), which can be

Figure 5: (a) The band gaps of β-BiAs without and with SOC

performed by fitting the DFT results as shown in Fig. 5 (b, c

at the different strain. For ε < 12 % without and with SOC are

and d). According to tight-binding theory, the hopping

normal insulators (Cyan color region), [12, 16] % with SOC

energies can be represented as:

are topological insulators (pink color region) and [12, 16] %

!! 𝑡!! = 𝑉!!"

without SOC are semi-metal materials (yellow region). DFT-

!!

𝑡!" ! = 𝑉!"# × 𝑐𝑜𝑠𝜃

PBE and MLWF fitted band structures at (b) equilibrium state

!!! 𝑡!"

= 𝑉!"# × 𝑐𝑜𝑠𝜑

and (c, d) without and with SOC effect at 12 % strain,

! !!

= 𝑉!!" × 𝑐𝑜𝑠 ! 𝜃 + 𝑉!!" × 𝑠𝑖𝑛! 𝜃

respectively. The TB model data are indicated in red dotted

!! !!

= 𝑉!!" × 𝑐𝑜𝑠 ! 𝜑 + 𝑉!!" × 𝑠𝑖𝑛! 𝜑

𝑡!"! 𝑡!"

!! !!

𝑡!"

line and blue color refers to DFT-PBE (b, c, and d).

= 𝑉!!" − 𝑉!!" ×𝑐𝑜𝑠𝜃 ×𝑐𝑜𝑠𝜑

Topological phase transition in β-BiAs is confirmed by

Where θ and 𝜑 are the angles of the vector pointing from i-th

calculating the topological invariant Z2 index based on the

atom to j-th atom with respect to x and y axis. The on-site

U(2N) non-Abelian Berry connection proposed by Rui Yu and

! = energies of s and p orbitals are set to the values of 𝜀!"

co-worker.41

−17.68 𝑒𝑉,

!

𝜀!" = −8.30 𝑒𝑉,

! 𝜀!" = −15.01 𝑒𝑉,

and

5



Page 6 of 11

Figure 6: (a) The calculation of Wilson loop (Wannier Charge Center) along ky for β-BiAs monolayer at 12 % strain yielding Z2 = 1, (b) Dirac edge state of β-BiAs at 12 % biaxial strain and (c) Edge spins polarization.

Figure 7: (a) Side and top view of epitaxial growth β-BiAs of TI on three layers of h-BN substrate, (b) band structure without and (c) with SOC at 12 % strain. The Z2 topological invariant is related by counting the number

utilizing the Green's function method with the tight-binding

of crossing between any arbitrary horizontal reference line and

models from VASP and wanniner90 package based on

evaluation of θ mod 2π, where the odd and even numbers

maximally localized wannier function (MLWFs) to calculate

denoted Z2 = 1 and 0 nontrivial and trivial topological

the edge state. In Fig. 6(b) we show the edge bands perfectly

phase, respectively. Figure 6(a) shows that the WCC

connected the conduction band and valence band with 1D

evolution along ky for the time-reversal plane of β-BiAs at 12

gapless edge states at ᴦ-point. The counter propagating edge

% strain when SOC is considered to yield TI with topological

states exhibit spin up and spin down polarization contribution

invariant, Z2 = 1. Additionally, to confirm the topological

in edge spectral function as shown in Fig. 6(c). We recognize

character of β-BiAs at 12 % strain with SOC, one needs to

that the spin-momenta of those Dirac-type edge states are

also show the existence of helical edge states protected by

locked at the ᴦ-point within the bulk gap thereby confirming

time-reversal symmetry, which is an essential characteristic of

the existence of nontrivial topology leading to absolutely

2D TIs. We have performed the band structure of the ribbon

polarized conductive channels.

6


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The Journal of Physical Chemistry dynamically under such large tensile strain up to 16 % as

Nevertheless to have practical application, it is highly

presented in Fig. S1(a-c) with the onset of the topological

desirable to experimentally realize a substrate to support 2D

phase transition at 12 % strain. However, for ε = 17.2 % the

films. Here, we choose three layers of hexagonal boron nitride

material shows small imaginary frequency are observed (υ = -

42-45

0.0928 THz) which indicates that the β-BiAs becomes

with a good dielectric constant for the epitaxial growth for β-

unstable beyond such strains. Up to now, graphene have been

BiAs. A lattice mismatch of 0.09 Å and the interlayer

subjected to a large strain by laser shock-induced straining

distance=4.02 Å, results in stable van der Waals (vdW)

process on flexible polydimethylsiloxane (PDMS) substrate,

heterostructure. The binding energy is calculated to be -0.028

and also the indentation method carried out with an atomic

eV per unit cell indicating weak dispersion interactions.

force microscope (AFM).53 Kim et al. obtained under large

Interestingly,

β-BiAs/h-BN

tensile strain of 18.7 % for stretched graphene.54 In an AFM

heterostructure without and with SOC at strain 12 % remains

indentation experiment, MoS2 monolayer could be strained

unperturbed as shown in Fig. 7 (b and c), respectively. There

upto 11%.55 For the purpose of application based on

is no charge transfer between β-BiAs and h-BN as confirmed

electronics and spintronics device technology (spin-controlled

by Bader charge analysis, thus the 𝑝! and 𝑝! orbitals near the

phase transition can be enhanced in spin transport) tunability

fermi level are dominantly contributed by Bi and As atoms

of the spin-splitting by applying large tensile strain has been

and the charge density difference being calculated as shown in

realized in many 2D materials.56-59 In this context, we believe

Fig. S6. Recently, 2D TIs materials such as bismuthene5,

that our prediction of TI in β-BiAs under a tensile strain could

plumbene,46

be realized experimentally.

(h-BN) as shown in Fig. 7(a), which has large band gap

the

band

germanene

structure

and

of

stanene

bilayer,47

SiGe,21

arsenene,32 arsenene oxide48 and Bi monolayer49 have been successfully

grown

on

different

substrates

4.

materials.

In conclusion, we predicted novel of 2D BiAs polymorphs,

Evidently, the β-BiAs/h-BN heterostructure is a robust

which are dynamically and thermally stable. α-, β- and γ-BiAs

topological insulator who’s the band inversion is not perturbed

are direct band-gap semiconductors, while the δ-, and ε-BiAs

by substrate.

phases display indirect band gap. Interestingly, β-BiAs has a nontrivial phase with significantly large band-gap. We expect

Generally, most of the 2D nontrivial topology proposed till

that the excellent TI behavior of β-BiAs with significant band-

date with sizable bulk gaps is composed of heavy metals,

gap can result in QSH devices at room temperature. For the

which have strong SOC interaction, including Pb, Hg, Sn, Sb,

practical application, we also proposed h-BN as excellent

and Bi. Although, 2D TIs has also achieved in some light

candidate substrate for supporting the β-BiAs film without

elements but their bulk gaps are very small induced by weak

perturbing the topological insulator.

SOC, such as, the gaps are ~10-3 meV for graphene,48,49 silicene (1.55-2.9 meV),4 germanene (Eg = 23.9-108 meV),4, 5052

Conclusions

ASSOCIATED CONTENT

arsenene (Eg = 696 meV),9 few-layered black phosphorus (5

Supporting Information

meV),22 BiNH2 (Eg = 0.83 eV)49 and Arsenene Oxide (Eg =

Phonon dispersion, Details of AIMD simulations, variation

232meV).48 In comparison, our first principle calculation

of strain respect to angle, lattice constant and bond-length

showed that 2D β-BiAs, a newly predicted binary compound

for

has a sizable gap up to 0.28 eV, suggesting it as a promising

β-BiAs,

structures

at

equilibrium

state

contribution of orbitals, PBE and HSE06, Partial density of

nontrivial topological insulator at room temperature. As a

states (PDOS) at ε = 12 % CBM and VBM, without and

result of buckling geometry the β-BiAs can be stable

7

Band



Page 8 of 11

with SOC, respectively, β-BiAs monolayer deposited on h-

Transition in HgTe Quantum Wells, Science 2006, 314,

BN substrate and charge density difference.

1757-1761 9.

Zhang, H. J.; Ma, Y.; Chen, Z. F. Quantum Spin Hall

AUTHOR INFORMATION

Insulators in Strain-Modified Arsenene, Nanoscale 2015,

Corresponding Author

7, 19152-19159. 10. Wang, Z. F.; Su, N. H.; Liu, F. Prediction of a Two-

E-mail: [email protected]. Tel.: +91-33-24734971. Notes

Dimensional Organic Topological Insulator, Nano Lett.

The authors declare no competing financial interest.

2013, 13, 2842-2845. 11. Zhou, L. J.; Kou, L. Z.; Sun, Y.; Felser, C.; Hu, F. M.;

ACKNOWLEDGEMENT

Shan, G. C.; Smith, S. C.; Yan, B. H.; Frauenheim, T.,

We acknowledge The World Academy of Sciences (TWAS) -

New Family of Quantum Spin Hall Insulators in Two-

Indian Association for the Cultivation of Science (IACS) (FR

Dimensional

number: 3240280472) financial support and AD thanks, DST,

Nontrivial Band Gaps, Nano Lett. 2015, 15, 7867-7872.

INSA and BRNS for partial funding.

Transition

Metal

Halide

with

Large

12. Ma, Y. D.; Kou, L. Z.; Li, X.; Dai, Y.; Smith, S. C.; Heine, T. Quantum spin Hall Effect and Topological

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Phase Coexistence and Strain-Induced Topological Insulator in Two

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