Accepted Manuscript

Influence of lattice expansion on the topological band order of InAsxSb1-x (x=0,

0.25, 0.5, 0.75, 1) alloys

Shirin Namjoo, Amir S.H. Rozatian, Iraj Jabbari

PII: S0925-8388(14)03059-X

DOI: http://dx.doi.org/10.1016/j.jallcom.2014.12.131

Reference: JALCOM 32920

To appear in: Journal of Alloys and Compounds

Received Date: 22 July 2014

Revised Date: 13 December 2014

Accepted Date: 23 December 2014

Please cite this article as: S. Namjoo, A.S.H. Rozatian, I. Jabbari, Influence of lattice expansion on the topological

band order of InAsxSb1-x (x=0, 0.25, 0.5, 0.75, 1) alloys, Journal of Alloys and Compounds (2014), doi: http://

dx.doi.org/10.1016/j.jallcom.2014.12.131

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1

Influence of lattice expansion on the topological band order of InAsxSb1-x (x=0, 0.25, 0.5, 0.75, 1)

alloys

Shirin Namjooa, Amir S. H. Rozatiana, and Iraj Jabbarib

aDepartment of Physics, University of Isfahan, Hezar Jarib Street, Isfahan 81746-73441, Iran

bFaculty of New sciences and technology, University of Isfahan, Hezar Jarib Street, Isfahan 81746-73441, Iran

Abstract

The topological band structures of InAsxSb1-x (x=0, 0.25, 0.5, 0.75, 1) alloys have been investigated

using density functional theory by utilizing the Wien2k package. These alloys are in a topologically

trivial phase in their unstrained states and exhibit a small band gap. Since in small band-gap cubic

semiconductors the nontrivial topological phase can be achieved by lattice expansion, we investigate

the effect of hydrostatic and biaxial lattice expansion on band inversion strength and band order of

these alloys. It is found that under reasonable hydrostatic lattice expansion, InAsxSb1-x (x=0, 0.25,

0.75, 1) alloys with cubic symmetry and InAs0.5Sb0.5 alloy with tetragonal symmetry, are converted to

nontrivial topological semiconductors with zero band gap and non-zero band gap, respectively. In

order to convert InAsxSb1-x (x=0, 0.25, 0.75, 1) alloys into topological semiconductors with non- zero

band gap, we let the systems of InAsxSb1- x (x=0, 0.25, 0.75, 1) alloys undergo a biaxial lattice

expansion. Thus by breaking the cubic symmetry in these alloys, not only they are converted to

topologically nontrivial phase but also a small band gap is opened at Γ point.

2

1- Introduction

Topological insulators are materials which have a bulk band gap generated by strong spin-orbit

coupling, but contain gapless surface states [1]. There are several criteria that have been proposed to

identify the topological insulator state. For materials with inversion symmetry, the parity criteria

developed by Fu and Kane can be readily applied [2]. For crystals that do not have inversion

symmetry, the topological invariants can be calculated directly from the bulk band structure. Based on

the unique electronic surface states of topological insulator, it is predicted that these materials could

be useful for future technological applications in spintronics and quantum computing [3]. Thus, the

search for topological insulators with exotic metallic surface states has increased greatly in the fields

of condensed matter physics and material science. So far many families of topological insulators have

been theoretically predicted or/and experimentally confirmed [4-11]. Some materials are not in a

topological nontrivial phase in their native states, but under reasonable lattice expansion they become

topological insulator. For instance, Feng et al. [7] calculated the band structure of InSb under

hydrostatic and biaxial lattice expansion. Using a tight-binding model [12, 13], they analyzed that

lattice expansion decreases the coupling potentials which induces a topological phase transition. In

particular, they found that InSb as a topological trivial semiconductor can become a nontrivial

topological semiconductor under proper hydrostatic or biaxial lattice expansion. InSb crystallizes in

the cubic zinc blende structure; it is also a narrow band gap semiconductor. At the Γ point, the energy

bands of this compound near the Fermi-level split into Γ6 (twofold degenerate), Γ7 (twofold

degenerate), and Γ8 (fourfold degenerate) states [14], because of the zinc-blende crystal symmetry and

strong spin-orbit interaction. The twofold-degenerated Γ6 state is located above the fourfold-

degenerated Γ8 state, forming the normal band order. In compounds with cubic symmetry the band

inversion strength is defined as ΔΕ=EΓ6-EΓ8. Materials with negative ΔΕ are in the topologically

nontrivial phase while those with positive ΔΕ are in the topologically trivial phase.

InAs exhibits the same crystal structure as InSb. It has a narrow band gap of about 0.42 eV [15],

moreover the InAsxSb1-x alloys have the lowest band gap among all III-V semiconductors. Therefore

InAs and the InAsxSb1-x alloys have the potential to become topological insulators or semiconductors

under proper lattice expansion. InAs, InSb and their ternary alloys InAsxSb1- x have been extensively

3

used in infrared optoelectronic devices including laser, photo detectors and optical gas sensors [16-

20]. The studies of the basic properties are very important for the understanding of the device

characteristics and improvement of their performance. InAs and InAsxSb1-x (x=0, 0.25, 0.5, 0.75)

alloys show a normal band order in their native states. Since the nontrivial topological phase may be

realized by lattice expansion, in this work we study two types of lattice distortions. First, we increase

the lattice constants of InAs and InAsxSb1-x (0.25, 0.5, 0.75 and also x=0 for comparison with

previous study [7] ) equally along all the three axes, and, second, to mimic the cubic- symmetry

breaking, we let these systems undergo a nonhydrostatic strain, using the lattice expansion in the ab-

plane by leaving the c-axis free to relax. For exchange and correlation effects, we use the local density

approximation (LDA) [21, 22] potential and also a generalized gradient approximation (GGA)

potential according to Wu and Cohen [23] to determine the equilibrium value c. The band structures

of the relaxed cells, and thus the band inversion strength (ΔΕ), are computed within the modified

Becke-Johnson exchange potential which is a meta-GGA and has been shown to yield band gaps of

various semi-conductors and insulators in excellent agreement with experiment. The outline of this

paper is as follows. A brief description of the calculation method is given in section 2. The results are

presented and discussed in Section 3. The article ends with the ‘‘Conclusions” section.

2- Calculation method

The calculations presented in this work were performed within the framework of density functional

theory (DFT) by utilizing the full potential linearized augmented plane wave method [24, 25] as

implemented in the Wien2k code [26]. Spin-orbit coupling is included by a second-variational

procedure [24], where states up to 9 Ry above Fermi energy are included in the basis expansion. The

modified Becke-Johnson exchange potential together with local-density approximation for the

correlation potential (mBJLDA) [27] was used to obtain the band structures. The mBJLDA

(mBJGGA) potential combines the modified Becke-Johnson exchange potential and the local density

approximation (generalized gradient approximation) correlation potential. In the solid-state

community, the vast majority of electronic structure calculations are done using the Kohn-Sham

equations. To solve the Kohn-Sham equations, an explicit expression for exchange- correlation is

needed. The local density approximation (LDA) and the generalized gradient approximation (GGA)

4

for the exchange-correlation potential [22], reproduce rather well the band structure of even

complicated metallic systems, but fail to reproduce the band gap of semiconductors. The mBJLDA

(mBJGGA) potential, however, offers an improvement over LDA or GGA in describing band gaps of

many materials, including semiconductors with the zinc-blende structure. More importantly, the

mBJLDA (mBJGGA) potential has been shown to mimic very well the behavior of an orbital

dependent potential around the band gap. Thus it is expected to obtain accurate positions of states near

the band edge. Since there is no exchange and correlation energy term from which the mBJLDA

potential can be deduced, a direct optimization procedure to get the lattice parameters in a consistent

way is not possible. Therefore, the equilibrium lattice constants were determined by total energy

minimization within GGA approach using the scheme of Wu and Cohen. The convergence parameter

(the product of the smallest of the atomic sphere radii RMT and plane wave cut off parameter Kmax),

which controls the size of the basis sets in these calculations, was set to 9. The maximum l quantum

number for the wave function expansion inside the atomic sphere was confined to lmax =10. The Gmax

parameter was taken to be 12.0 Bohr-1. Brillouin-zone (BZ) integrations within the self-consistency

cycles were performed via a tetrahedron method [28], using 286 k points for binary compounds and

220 k points for ternary alloys in the irreducible BZ. All of these values were chosen so as to ensure

convergence of the results. The force on each atom after relaxation decreased to less than 0.5

mRyd/a.u.

3- Results and discussions

3-1 Band structure of InAsxSb1- x alloys in native states and under hydrostatic lattice expansion

By total energy minimization within the Wu-Cohen generalized gradient approximation, the lattice

constant of InAs is obtained about 6.0995 Å, which is in excellent agreement with the experimental

value of 6.0584 Å [15]. Based on this lattice constant, the mBJLDA calculation produces a band gap

about 0.47 eV. This value for the band gap deviates from the experimental value (0.42 eV) by less

than 12%. Thus, our prediction for the gap is in excellent agreement with the experimental value in

contrast to earlier theoretical work that predicts this compound to be a zero band gap semiconductor

[29], underestimating the band gap in this earlier work is the normal result of first-principle GGA

calculation of band structure of semiconductors and insulators.

5

Our results for the band structure of InAs are shown in Fig. 1. According to Fig. 1, the s-like state Γ6

lies above the fourfold degenerate p-like Γ8 state showing that this compound is a topological trivial

semiconductor in its unstrained state. As mentioned in the Introduction, the nontrivial topological

phase can be generally realized by applying hydrostatic lattice expansion. Thus we stretch the lattice

constant of InAs to a=a0+n%a0 (n=1 to 5) leading to the band structures of InAs under hydrostatic

lattice expansion from 1% to 5% shown in Fig. 2.

As shown in Figs. 2(a), 2(b), and 2(c), for values of n=1,2,3 ΔΕ is positive because the s-like Γ6 state

sits above the p-like Γ8 state, hence the trivial topological phase is retained. However, according to

Fig. 2(d), starting from a value of n=3.2% the Γ6 state is located below the Γ8 state and ΔΕ becomes

negative. Thus under hydrostatic lattice expansion somewhere between 3% and 3.2% the trivial

topological semiconductor is converted into the nontrivial topological semiconductor. This trend

continues under higher percentages of lattice expansion [Figs. 2(e) and 2(f)].

Similar to InAs, also InSb is a narrow band gap semiconductor. Feng et al. [7] have calculated the

band structure of InSb under lattice expansion. Using a tight-binding approach [12, 13], they analyzed

that lattice expansion decreases the coupling potentials and then induces a topological phase

transition. Due to the similarity between the band structure of InSb and InAs, and according to the fact

that the topological nontrivial phase can be driven by proper lattice expansion and also by alloying,

we investigate the topological band order of alloys. Note that alloy crystal structures are defined by

replacing Sb atoms by As atoms in InSb compound with values of x as 0.25, 0.5, 0.75, 1 and also 0 for

comparison with previous study [7]. For InAs0.25Sb0.75 and InAs0.75Sb0.25, the simplest structure is an

eight-atom simple cubic lattice. The cations with the lower concentration form a regular simple cubic

lattice. For InAs0.5Sb0.5, the smallest ordered structure is a four-atom tetragonal cell. The calculated

band structures show that all considered alloys InAsxSb1-x (x= 0.25, 0.5, 0.75, 1 and also x=0) are

narrow band gap semiconductors. The mBJLDA calculation produces the band gap of InAs0.25Sb0.75,

InAs0.5Sb0.5, InAs0.75Sb0.25 and InSb about 0.082 eV, 0.0484 eV, 0.196 eV and 0.12 eV respectively.

According to these values of band gaps, InAsxSb1-x (x=0, 0.25, 0.5, 0.75) alloys have the potential to

become topological semiconductor under smaller lattice expansion as compared to InAs. The

calculated band structures and band inversion strength (ΔΕ) of InAsxSb1-x (x=0, 0.25, 0.5, 0.75, 1)

6

alloys in their native states and under hydrostatic lattice expansion, ranging from 1% to 5%, are

shown in Fig. 3.

According to this figure, InAsxSb1-x (x=0, 0.25, 0.5, 0.75) alloys in their native states are all narrow

band gap semiconductors with normal band order while under hydrostatic lattice expansion they are

converted to semiconductors with inverted band order. For these alloys the band inversion strength

(ΔE) decreases with increasing the hydrostatic lattice expansion. Since InAs0.25Sb0.75, InAs0.5Sb0.5, and

InSb have a smaller band gap than InAs0.75Sb0.25 and InAs, they become topological semiconductors

under lower values of hydrostatic lattice expansion. The band inversion for InAs0.25Sb0.75, InAs0.5Sb0.5

and InSb happens somewhere between their native states and 1% hydrostatic lattice expansion while

for InAs0.75Sb0.25 the band inversion takes place between 1% and 2% hydrostatic lattice expansion. As

mentioned before, the smallest ordered structure for InAs0.5Sb0.5 is a four-atom tetragonal cell. Since

the fourfold degeneracy of the Γ8 states is protected by the cubic symmetry, the tetragonal

InAs0.5Sb0.5 lifts the fourfold-degenerate Γ8 . Here, with only a 0%-1% hydrostatic lattice expansion,

the band inversion takes place between the s-like Γ6 state (below the Fermi level) and the Γ8+ state

(above the Fermi level) and a topologically nontrivial state forms while a band gap is opened at Γ

point.

The present work predicts that under hydrostatic lattice expansion InSb is converted to a nontrivial

topological semiconductor somewhere between its native state and 1% hydrostatic lattice expansion

while the previous study [7] predicted that the band inversion is in a place between 1% and 2% lattice

expansion. The difference between the results is likely due to the different lattice constants used in

these studies.

3-2 Band structure of InAsxSb1- x (x=0, 0.25, 0.75, 1) under biaxial lattice expansion

As shown in Fig. 3, a 0%-1% hydrostatic lattice expansion converts InAs0.5Sb0.5 into the topologically

nontrivial state with a band gap at the Γ point leading to the fact that the fourfold degeneracy of the Γ8

state is protected by the cubic symmetry. Although hydrostatic lattice expansion has thus been

demonstrated to convert the normal band order of InAsxSb1- x (x=0, 0.25, 0.75, 1) into the inverted

band order, a nonhydrostatic strain is still needed to break the cubic symmetry and create a band gap

at Γ point. Thus, in this section we allow the InAsxSb1- x (x=0, 0.25, 0.75, 1) alloys to undergo a

7

biaxial lattice expansion. To this end, we expand the crystal structures in ab plane and leave the c axis

free to relax. The equilibrium value c is determined by total energy minimization with fixed a using

GGA (Wu-Cohen) and also LDA potentials. The calculated band structures of InAsxSb1-x(x=0, 0.25,

0.75, 1) alloys with equilibrium value c, that is obtained using GGA (Wu-Cohen) and also LDA

potentials, are given in Fig. 4.

As can be seen in Fig. 4 when comparing the unstrained cubic structure with the in-plane strained

tetragonal structure, the original Γ8 state splits into states with Γ8+ symmetry (higher energy) and

Γ8- symmetry (lower energy). Note that in these systems with tetragonal symmetry, the band

inversion strength (ΔΕ) is defined as energy difference between the s-like Γ6 state and p like Γ8+ states

at the Γ point. The calculated topological band inversion strength (ΔΕ) as a function of lattice

expansion for InAsxSb1-x(x=0, 0.25, 0.75, 1) alloys is shown in Fig. 5. According to this figure, for

InAsxSb1-x (x=0, 0.25, 0.75, 1) alloys, band inversion strength (ΔΕ) changes from positive values (the

s-like Γ6 state sits above the Fermi level) to negative values (the s-like Γ6 state sits below the Fermi

level). Under biaxial lattice expansion, we observe that InAsxSb1-x (x=0, 0.25, 0.75, 1) alloys are not

only converted to topological nontrivial phase, but also observe the opening of a small band gap

appears at Γ point due to the breaking of the cubic symmetry. Comparing the results for structural

optimizations with LDA and GGA we can conclude that although the topological band inversion of

InAsxSb1-x (x=0, 0.25, 0.75, 1) alloys takes place under slightly different biaxial lattice expansions

with different potentials (LDA and GGA (Wu -Cohen)), the behavior of band inversion strength (ΔΕ)

with respect to biaxial lattice expansion is very similar and in both functionals Δ decreases with

lattice expansion. Since, as a rule of thumb, it is known that the LDA underestimates the lattice

parameters and GGA overestimates them, according to this fact we conclude that the band inversion

happens somewhere between 1% and 2% for InAs0.25Sb0.75, between 2% and 3% for InSb and

InAs0.75Sb0.25, and also between 5% and 6% for InAs Thus, the transition point, in which the

topological phase transition takes place, is increased in the same order as the value of band gap of

these alloys. In alloys with higher value of band gap, the band inversion occurs for larger values of

lattice expansion.

8

According to our work, InSb is converted to nontrivial topological semiconductor under the biaxial

lattice expansion of 2% - 3%. This result is in excellent agreement with the previous study [7]. There

is no theoretical results for band order of InAsxSb1-x (x=0.25, 0.5, 0.75, 1) alloys to compare with.

These results can be used as a reference for future investigations.

4- Conclusions

We have systematically investigated the band structure of InAsxSb1- x (x=0, 0.25, 0.5, 0.75, 1) alloys

using the mBJLDA exchange-correlation potential. Our calculations show that topological band order

of InAsxSb1- x (x=0, 0.25, 0.5, 0.75, 1) alloys can be tuned using hydrostatic or biaxial lattice

expansion. Our results further demonstrate that these alloys are all narrow band gap semiconductors

with the potential to become topological semiconductors under reasonable lattice expansion. Since

InAs0.25Sb0.75 (Eg= 0.082 eV), InAs0.5Sb0.5 (Eg= 0.0484 eV) and InSb (Eg=0.12 eV) have smaller band

gaps than InAs0.75Sb0.25 (Eg= 0.196 eV) and InAs (Eg=0.42eV), they become topological

semiconductor under lower hydrostatic lattice expansion. Due to the tetragonal symmetry in

InAs0.5Sb0.5, a topological nontrivial state forms at only 0%-1% hydrostatic lattice expansion while in

addition a band gap opens at Γ point. To convert InAsxSb1- x (x=0, 0.25, 0.75, 1) alloys with cubic

symmetry to nonzero band gap semiconductors with inverted band order, we let the systems of

InAsxSb1- x ( x=0, 0.25, 0.75, 1) undergo biaxial lattice expansion by expanding the crystal structures

in ab plane and relaxing in the c axis using LDA and also GGA (Wu-Cohen) potentials. This can be

achieved by growing the sample on a substrate with a larger lattice constant. The band inversion takes

place upon biaxial lattice expansion, somewhere between 1% and 2% for InAs0.25Sb0.75, between 2%

and 3% for InSb and InAs0.75Sb0.25, and also between 5% and 6% for InAs. Our work shows how the

topological nontrivial phase can be driven by lattice expansion.

Acknowledgments

The authors would like to thank the Office of Graduate Studies of the University of Isfahan for their

support.

9

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Figure Caption:

FIG. 1. The band structures of InAs.

Fig. 2. The calculated band structures of InAs under 1% (a), 2% (b), 3% (c), 3.2% (d), 4% (e), and 5% (f) hydrostatic lattice

expansion.

Fig. 3. The calculated band structure (right) and the band inversion strength (ΔΕ) (left) of InAsxSb1-x (x=0, 0.25, 0.5, 0.75, 1)

alloys in their native states and under hydrostatic lattice expansion ranging from 1% to 5%.

Fig. 4. The calculated band structure of InAsxSb1-x (x=0, 0.25, 0.75, 1) alloys under biaxial lattice expansion by lattice

optimization along the c-axis using GGA (Wu- Cohen) potential (right) and also LDA potential (left).

Fig. 5. Topological band inversion strength (ΔΕ) of InAsxSb1-x (x=0, 0.25, 0.75, 1) alloys under biaxial lattice expansion as a

function of lattice expansion, by lattice optimization along the c-axis using both LDA and GGA (Wu -Cohen) potentials, The

inversion strength in these systems (ΔΕ) is defined as energy difference between the s-like Γ6 state and p like Γ8+ states at Γ

point.

11

L Λ Γ Δ Χ

Γ6

Γ8

Γ7

Ene

rgy

(eV

)

-4

-2

0

2

4

Γ6

Γ8

Γ7

FIG. 1.

12

L Λ Γ ΧΔ

a=a0+ 2%a

0

L Λ Γ ΧΔ

a=a0+ 3%a

0

L Λ Γ ΧΔ

a=a0+1%a

0

EN

ER

GY

(ev

)

-3

-2

-1

0

1

2

3

L Λ Γ ΧΔ

a=a0+3.2%a0a=a

0+5%a

0a=a

0+4%a

0

L Λ Γ ΧΔ L Λ Γ ΧΔ

(a) (b) (c) (d) (e) (f)

Fig. 2.

13

Ene

rgy

(eV

)

-1.0

-0.5

0.0

0.5

1.0

ΔΓ ΧL ΔΓ ΧL Γ ΧL Λ Δ Γ ΧL Λ Δ ΧL Λ ΔΓ L ΧΛ ΔΓ

x=0 x=0 x=0 x=0 x=0 x=0

ΛΛ

Ene

rgy

(eV

)

-1.0

-0.5

0.0

0.5

1.0

ΓR Χ Λ Δ Γ Χ Λ Δ ΓR Χ Λ Δ ΓR Χ Λ Δ ΓR Χ Λ Δ ΓR Χ Λ Δ

x=0.25 x=0.25 x=0.25 x=0.25 x=0.25 x=0.25

Δ R Γ ΧΛ R ΧΛ Γ R Γ ΧΛ

R Γ ΧΛ R Γ ΧΛ R Λ ΧΓ Δ Δ Δ Δ Δ

x=0.5x=0.5x=0.5x=0.5x=0.5x=0.5

Γ Χ R Λ Δ Γ Χ R Λ Δ Γ Χ R Λ Δ Γ Χ R Λ Δ Γ Χ R Λ Δ Γ Χ R Λ Δ

x=0.75 x=0.75 x=0.75x=0.75 x=0.75x=0.75

L Λ Γ ΧΔ

a=a0+ 2%a

0

L Λ Γ ΧΔ L Λ Γ ΧΔ

a=a0+1%a

0

L Λ Γ ΧΔ L Λ Γ ΧΔ-1.0

-0.5

0.0

0.5

1.0

L Λ Γ ΧΔ

a=a0

x=1 x=1 x=1x=1 x=1

x=1

Ene

rgy

(eV

)

-1.0

-0.5

0.0

0.5

1.0

x=0.5

-1.0

-0.5

0.0

0.5

1.0

x=0.75 x=0.75x=0.75x=0.75 x=0.75 x=0.75 x=0.75

x=1 x=1

a=a0+ 3%a

0a=a

0+ 4%a

0

x=1

x=0 x=0 x=0 x=0 x=0

a=a0+ 5%a

0

R

Ene

rgy

(eV

)E

nerg

y (e

V)

Fig. 3.

ΔΕ ( e

V)

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

x=0

x=0.25

x=0.5

0 1 2 3 4 5 6

x=1

x=0.75

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.40.6

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.40.6

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.40.6

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.40.6

ΔΕ ( e

V)

ΔΕ ( e

V)

ΔΕ ( e

V)

ΔΕ ( e

V)

a=b=c (%)

14

Fig. 4.

a=b (1%)

Ene

rgy

(eV

)

-2

-1

0

1

2

Γ7

Ν Γ ΡΣ Λ Ν Γ ΡΣ Λ Ν Γ ΡΣ Λ Ν Γ ΡΣ Λ Ν Γ ΡΣ Λ

a=b (2%) a=b (3%) a=b (4%) a=b (5%)

x=0 x=0x=0x=0x=0

Ene

rgy

(eV

)

-2

-1

0

1

2

R Λ Γ Δ Χ

Γ8

R Λ Γ Δ Χ R Λ Δ ΧΓ R Λ Γ Δ Χ-2

-1

0

1

2

R Λ Γ Δ Χ

x=0.25 x=0.25x=0.25x=0.25x=0.25

Λ Γ ΧΔ Λ Γ ΓR RΧ ΧΔΛ Δ ΓR ΧΛ

Ene

rgy

(eV

)

-2

-1

0

1

2

Γ ΔR Λ R

x=0.75 x=0.75 x=0.75 x=0.75 x=0.75

x=1

Ene

rgy

(eV

)

-2

-1

0

1

2

Λ Σ ΡΓΝ Λ Σ ΡΓΝ

Γ7

Λ Σ ΡΓΝ-2

-1

0

1

2

Λ Σ ΡΓΝ Γ-2

-1

0

1

2

Λ Σ ΡΝ

x=1 x=1 x=1 x=1

Χ Δ

a=b(1%)

Ene

rgy

(eV

)

-2

-1

0

1

2

Ν Σ Λ Γ P Ν Σ Λ Γ P Ν Σ Λ Γ P Ν Σ Λ Γ PΝ Σ Λ Γ P

a=b(2%) a=b(3%) a=b(4%) a=b(5%)

x=0 x=0 x=0 x=0 x=0

x=0.25

Ene

rgy

(eV

)

-2

-1

0

1

2

ΓR Χ Λ Δ ΓR Χ Λ Δ ΓR Χ Λ Δ ΓR Χ Λ ΔΓR Χ Λ Δ

x=0.25 x=0.25 x=0.25 x=0.25

Ene

rgy

(eV

)

-2

-1

0

1

2

R Γ ΧΛ Δ R Γ ΧΛ Δ R Γ ΧΛ Δ Γ ΧΛ ΔR Γ ΧΛR Δ

x=0.75 x=0.75 x=0.75 x=0.75 x=0.75

Ene

rgy

(eV

)

-2

-1

0

1

2

Ν ΓΣ Λ P Ν ΓΣ Λ P Ν ΓΣ Λ P Ν ΓΣ Λ P Ν ΓΣ Λ P

x=1 x=1x=1 x=1 x=1

15

x=1

a=b(%)

0 1 2 3 4 5 6 7

ggalda

x=0.25

0 1 2 3 4 5 6

gga

lda

a=b(%)

x=0

a=b (%)

0 1 2 3 4 5 6

ΔΕ ( e

V)

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

gga

lda

x=0.75

a=b (%)

0 1 2 3 4 5 6

ggalda

Fig. 5.

16

Highlights

1- We systematically investigated the band structure of InAsxSb1- x ( x=0, 0.25, 0.5, 0.75, 1) alloys using MBJLDA exchange-correlation potential.

2- We investigated the effect of hydrostatic lattice expansion on band inversion strength and band order of InAsxSb1-x (x=0, 0.25, 0.5, 0.75, 1) alloys.

3- We also studied the effect of biaxial lattice expansion on band inversion strength and band order of InAsxSb1- x (x=0, 0.25, 0.75, 1) alloys .