I–II–V and I–III–IV half-Heusler compounds for optoelectronic applications: Comparative ab initio study

Journal of Alloys and Compounds 587 (2014) 451–458

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I–II–V and I–III–IV half-Heusler compounds for optoelectronic applications: Comparative ab initio study S. Kacimi, H. Mehnane, A. Zaoui ⇑ Modelling and Simulation in Materials Science Laboratory, Djillali Liabès University of Sidi Bel-Abbès, Sidi Bel-Abbès 22000, Algeria

a r t i c l e

i n f o

Article history: Received 29 August 2013 Received in revised form 3 October 2013 Accepted 4 October 2013 Available online 24 October 2013 Keywords: DFT APW+lo Electronic structure Optical properties Half-Heusler

a b s t r a c t We have investigated structural, electronic and optical properties of 96 half-Heusler materials, including compounds of I–II–V and I–III–IV types using first-principles calculations based on the density functional theory. The calculated lattice constants and band gaps are used as basis informations to select candidate materials favorable for specific optoelectronic applications. The band gap trend in the selected XYZ materials is found to be similar to the one in the zinc-blende III–V compounds. The assignment of the structures in the optical spectra and band structure transitions are investigated in detail. The predicted values of the dielectric constants for selected half-Heusler systems are close to those of the III–V binary compounds. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction The search for new semiconductor materials with structural and electronic properties for the advancement of optoelectronic applications, such as thin-film solar cells or laser diodes, constitutes one of the central challenges in materials science. II–VI and III–V compound semiconductors have long been viewed as promising materials for the device applications in electronic and optoelectronic technologies. Many of these binary semiconductors crystallize in a cubic zinc-blende structure. The limited applications of some binary semiconductors in various fields have prompted scientists to look for alternative axes of promising materials. A particular interesting class of materials belong to half-Heusler compounds or ‘‘Nowotny–Juza,’’ [1] with a chemical composition XYZ. For example in III–V compounds, the crystal structure of Nowotny–Juza compounds can be derived by transmuting the group III atom into an isovalent pair I + II, and are found to crystallize mostly in cubic zinc-blende-type related structures [1–3]. So, if the elements X, Y, and Z have a total number of eight valence electrons, they form a particularly stable groundstate structure; and the most electropositive element X donates n valence electrons to the more electronegative elements Y and Z. Therefore, the class of eight-electron half-Heusler compounds includes a large number of semiconductors, whose band gaps vary in a wide range [4]. Similar stable eight-electron states cause the band gaps in binary semiconductors of the III–V or II–VI types. ⇑ Corresponding author. Tel.: +213 778 090 975; fax: +213 48 54 11 52. E-mail address: [email protected] (A. Zaoui). 0925-8388/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jallcom.2013.10.046

Up to now only a small number of the huge class of possible eight electron half-Heusler compounds of I–II–V and I–III–IV types have been synthesized [1,5–22], and have been also investigated with ab initio calculations [4,23–33]. The similarity of physical properties and informations obtained for both of the half-Heusler compounds and of the binary semiconductors helped us to find new half-Heusler semi-conductors favorable for optoelectronic applications. In this work, we study a set of 96 half-Heusler compounds using density-functional theory calculations. The selection of compounds focuses on materials that are particularly relevant for industrial applications. Therefore, we have eliminated the strongly toxic, dangerous, rare, expensive and heavy elements from this selection. An accurate determination of band gaps is still a challenge. Of great help to predict band gaps sufficiently well is the modified version of the exchange potential proposed by Becke and Johnson [34], we use here. The plan of the manuscript is as follows: The next Section 2 gives a brief description of the method used of the calculations; Section 3 presents the analyses of the calculated results and finally, the conclusion is given in Section 4. 2. Methodology For the DFT calculations we use the Wien2k code [35], where the so called full-potential (linear) augmented plane-wave plus local orbitals method [36] has been implemented. In this method, wave functions, charge density, and potential are expanded in spherical harmonics inside no overlapping muffin-tin spheres,

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and plane waves are used outside in the remaining interstitial region of the unit cell. In the code, core states are treated differently within a multi-configuration relativistic Dirac-Fock approach, while valence states are calculated in a scalar relativistic approach. For the exchange-correlation energy we used the generalized gradient approximation PBE-GGA [37] and a modified version of the exchange potential TB-mBJ proposed by Becke and Johnson [34]. From the total energy, we computed equilibrium lattice constants and bulk moduli by fitting energy versus volume to Murnaghan’s equation [38]. The total energy was determined using a set of 56 k-points in the irreducible sector of Brillouin zone, equivalent to an 11  11  11 Monkhorst–Pack [39] and appropriate for all compounds. A value of 9 for the cutoff energy was used. MT radii between 1.5 and 2.5 bohr depending on the elements were chosen.

3. Results and discussion We have studied the electronic structure of 96 half-Heusler chosen from the I–II–V and I–III–IV types. We used the PBE-GGA approach for the structure optimization and TB-mBJ one for the band structures. For comparison we examined several III–V and II–VI compounds in the zinc-blende structure. As the first step, total energy (Etot) versus cell volume calculations were carried out to determine the equilibrium structural parameters. Our results of

structural parameters (lattice constants) and electronic ones (band gaps) are listed in Table 1. The calculated lattice constant and bulk modulus of zinc-blende III–V and II–VI semiconductors are in good agreement with the experimentally measured values [40–52]. Due to the close relation between the zinc-blende and the half-Heusler structure, structural and electronic properties should be directly comparable. We consider the lattice parameter (the lattice constants do not fit and the epitaxial growth of half-Heusler compounds on silicon substrate) and the energy band gap (gap > 1 eV) as two basic criteria to exclude several compounds and consequently to select half-Heusler compounds similar to III–V and II–VI semiconductors. The calculated lattice constants of XYZ systems differ only slightly from the experimental ones [9,10,17–22] (see Table 1). Also we have obtained quite accurate band gaps for the semiconductors with an orbital-independent exchange-correlation potential TB-mBJ which depends however on semilocal quantities compared to the local one PBE-GGA used for the lattice constants determination. The highest direct band gap of 4.25 eV was obtained for LiBeN, whereas very small gaps and almost touching valence and conduction bands were found for several ternary compounds of the I–II–V type: KCdAs, KCdN, NaCdN, and of I–III–IV type: KAlGe, KGaC, KGaGe, KGaSi, KInC, KInGe, KInSi, LiInGe, NaGaGe, NaInC, and NaInGe]. From Table 1, the percentages of semiconducting materials of all half-Heusler types can be deduced. One finds 86% of these compounds are semiconducting while only

Table 1 Equilibrium lattice constants acalc and calculated band gap values Egcalc of cubic III–V and II–VI semiconductors and of 96 half-Heusler materials using both PBE-GGA and TB-mBJ approaches and compared to the available experiment data. Comps.

GGA

MBJ

aexp II–IV CaO CaS CaSe CaTe CdO CdS CdSe CdTe MgO MgS MgSe MgTe ZnO ZnS ZnSe ZnTe III–V AlAs AlN AlP AlSb GaAs GaN GaP GaSb InAs InN InP InSb ScAs ScN ScP ScSb I–II–V KBeAs KBeN KBeP

5.818 [40] 6.052 [40] 6.480 [40] 5.660 [41] 5.890 [41] 6.360 [41] 5.409 [40] 5.668 [40] 6.089 [40] 5.660 4.370 5.467 6.136 5.649 4.500 5.450 6.095 5.058 4.980 5.868 6.479

[42] [43] [42] [42] [42] [44] [45] [43] [42] [46] [42] [42]

acalc

Egcalc

Nat./Gap

5.248 6.305 6.591 7.108 5.142 5.944 6.209 6.626 4.616 5.700 5.998 6.511 4.643 5.470 5.757 6.208

3.265 3.629 3.172 3.088 0 1.010 0.475 0.557 3.455 3.356 2.527 2.324 0.635 1.939 1.087 0.980

X–U X–U L–U L–U U–U U–U U–U U–U U–U U–U U–U U–U U––U U–U U–U U–U

5.731 4.407 5.507 6.228 5.753 4.581 5.517 6.224 6.198 5.078 5.970 6.643 6.022 4.894 5.932 6.612

1.500 3.297 1.629 1.227 0.126 1.405 1.522 0 0 0 0.401 0 1.510 2.419 1.627 1.413

U–X U–X U–X U–L U–U U–U U–U U–U U–U U–U U–U U–U X–X X–W X–X X–X

6.598 5.672 6.414

0.508 0.355 1.109

U–U X–U U–U

Egexp.

0.840 2.550 1.900 1.920

[47] [40] [40] [40]

4.500 3.600 3.470 3.440 3.800 2.960 2.710

[48] [48] [48] [47] [40] [40] [40]

2.950 [49] 2.520 1.680 1.420 3.300

[50] [51] [51] [52]

0.820 [50] 0.420 [50] 1.350 [49] 0.240 [50]

Egcalc

Gap/nat.

6.135 5.269 4.725 4.011 1.638 2.559 1.877 1.646 5.861 5.023 3.982 3.431 2.536 3.537 2.584 2.183

X–U X–X W–X L–X U–U U–U U–U U–U U–U U–U U–U U–U U–U U–U U–U U–U

2.202 4.826 2.364 1.836 1.226 2.603 2.311 0.503 0.323 0.464 1.342 0.052 2.049 3.623 2.180 1.786

U–X U–X U–X U–X U–U U–U U–X U–U U–U U–U U–U U–U X–X X–W X–X X–W

1.287 1.304 1.982

U–U X–U U–U

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S. Kacimi et al. / Journal of Alloys and Compounds 587 (2014) 451–458 Table 1 (continued) Comps.

GGA aexp

KBeSb KCaAs KCaN KCaP KCaSb KCdAs KCdN KCdP KCdSb KMgAs KMgN KMgP KMgSb KZnAs KZnN KZnP KZnSb LiBeAs LiBeN LiBeP LiBeSb LiCaAs LiCaN LiCaP LiCaSb LiCdAs LiCdN LiCdP LiCdSb LiMgAs LiMgN LiMgP LiMgSb LiZnAs LiZnN LiZnP LiZnSb NaBeAs NaBeN NaBeP NaBeSb NaCaAs NaCaN NaCaP NaCaSb NaCdAs NaCdN NaCdP NaCdSb NaMgAs NaMgN NaMgP NaMgSb NaZnAs NaZnN NaZnP NaZnSb I–III–IV KAlC KAlGe KAlSi KGaC KGaGe KGaSi KInC KInGe KInSi KScC KScGe KScSi LiAlC LiAlGe

6.23 [17] 6.100 [10] 6.180 [18] 4.955 [19] 6.005 [18] 5.940 [6] 4.910 [20] 5.765 [21]

5.912 [22]

5.960 [9]

MBJ acalc

Egcalc

Nat./Gap

6.992 7.385 6.288 7.212 7.812 7.066 6.197 6.891 7.448 7.030 6.005 6.846 7.444 6.861 5.962 6.674 7.254 5.578 4.364 5.368 6.042 6.673 5.544 6.493 7.118 6.345 5.387 6.137 6.744 6.205 5.010 6.010 6.666 5.980 4.928 5.765 6.411 6.038 4.970 5.845 6.464 6.960 5.855 6.784 7.397 6.633 5.702 6.442 7.031 6.555 5.448 6.368 6.995 6.348 5.362 6.145 6.763

0.547 1.233 0.659 1.522 1.501 0 0 0 0 0.469 0.103 0.965 0.599 0 0 0 0 0.990 2.829 1.179 0.818 1.836 2.294 1.957 1.608 0 0 0.555 0.176 1.374 2.291 1.542 1.201 0.491 0.516 1.353 0.550 1.308 1.826 1.315 1.218 1.598 1.154 1.940 1.694 0 0 0.011 0 0.833 0.762 1.481 1.013 0 0 0.460 0

U–U X–U X–U X–U L–U U–U U–U U–U U–U U–U X–U U–U U–U U–U U–U U–U U–U U–X U–X U–X U–X U–X X–U U–X U–X U–U U–U U–U U–U U–X U–U U–X U–X U–U U–U U–X U–U U–U U–U U–X U–X L–U X–U L–X U–X U–U U–U U–U U–U U–U U–U U–U U–U U–U U–U U–U U–U

6.010 6.866 6.756 6.090 6.906 6.783 6.325 7.133 7.022 6.165 7.133 7.005 7.254 4.920

0 0 0 0 0 0 0 0 0 0.212 0.429 0.377 0.814 0.053

U–U U–U U–U U–U U–U U–U U–U U–U U–U X–X X–X X–X U–X U–X

Egexp.

0.850 [4] 2.380 [33] 3.230 [16] 2.430 [18] 1.510 [6] 1.910 [20] 2.040 [21]

Egcalc

Gap/nat.

1.182 2.557 2.421 2.667 2.375 0 0.020 0.719 0.375 1.4609 1.3774 2.031 1.405 0.390 0.297 0.949 0.258 1.614 4.252 1.782 1.294 2.537 3.671 2.719 2.156 0.789 0.202 1.504 0.910 2.069 3.677 2.274 1.746 1.581 1.719 1.986 1.427 2.215 3.083 2.154 1.806 2.673 2.843 2.813 2.322 0.464 0 1.082 0.555 1.922 2.193 2.62 1.873 0.897 0.749 1.687 0.813

U–U X–U X–U X–X X–X U–U U–U U–U L–U U–U X–U U–U U–U U–U U–U U–U U–U U–X U–X U–X U–X U–X X–U U–X U–X U–U U–U U–U U–U U–X U–U U–X U–X U–U U–U U–X U–U U–X U–U U–X U–X L–X X–U L–X U–X U–U U–U U–U U–U U–U U–U U–U U–U U–U U–U U–U U–U

0.293 0 0.688 0 0 0 0 0 0 1.062 0.950 0.948 1.639 0.520

U–U U–U U–U U–U U–U U–U U–U U–U U–U X–X X–X X–X U–X U–X (continued on next page)

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Table 1 (continued) Comps.

LiAlSi LiGaC LiGaGe LiGaSi LiInC LiInGe LiInSi LiScC LiScGe LiScSi NaAlC NaAlGe NaAlSi NaGaC NaGaGe NaGaSi NaInC NaInGe NaInSi NaScC NaScGe NaScSi

GGA

MBJ

aexp

acalc

Egcalc

Nat./Gap

5.930 [22]

6.020 4.998 6.040 5.911 5.429 6.401 6.281 5.321 6.306 6.267 5.396 6.386 6.294 5.455 6.419 6.289 5.791 6.712 6.595 5.680 6.645 6.605

0.118 1.022 0 0.216 0 0 0.241 0.980 0.805 0.717 1.128 0.029 0.462 0 0 0 0 0 0 0.668 0.681 0.639

U–X U–X U–U U–X U–U U–U U–X X–X X–X X–X U–U U–U U–X U–U U–U U–U U–U U–U U–U X–X X–X X–X

6.060 [9] 6.300 [13]

14% are semi metallic. Also one notices that almost all compounds of the I–II–V type are semiconductors, while only 69% of the I–III– IV compounds have a band gap Eg (TB-mBJ) > 0. The calculated band gaps cover the range from 0 to 4.30 eV. We notice that replacing N with the heavier pnictogens P, As, or Sb was found to narrow the band gap due to the increasing band width of both the valence and conduction bands. This trend ensured, when we replace C by Si or Ge atoms. In addition, if the X and Z ions are held constant, as in LiMgP and LiCdP, we observe that replacing the more ionic Mg by the softer Cd also results in a narrowing of the band gap. Thus, eight-electron half-Heusler compounds can be used for several applications and the suitable ones can be found for practically all desired band gaps. To examine also for comparison for several III–V and II–VI semiconductors, the calculated lattice parameters and energy band gaps of half-Heusler compounds are illustrated in Fig. 1. In Table 2, we limited our research to the selected substances, which possess comparable physical properties to those of III–V and II–VI semiconductors with only a ±0.25 Å variation in lattice parameter, and a ±0.2 eV variation in band gap. This is due to the same structural

Egexp.

Egcalc

Gap/nat.

0.514 1.712 0.485 0.595 0.562 0 0.609 1.634 1.052 0.948 1.944 0.834 0.999 0.535 0 0.597 0 0 0.266 1.431 1.046 1.003

U–X U–X U–U U–X U–U U–U U–X X–X X–X X–X U–X U–U U–X U–U U–U U–U U–U U–U U–U X–X X–X X–X

environment of both binary and ternary compounds and to the efficiency of the approximation used in this study. Other criteria are also used in this study such as, chemical structure (zinc-blende), static optical properties (the same order) and the nature of band gap (direct or indirect), which are a great help to predict new XYZ semiconductors. In addition, we omitted toxic, dangerous, rare and expensive elements. With all above mentioned informations, we select pure main group of half-Heusler compounds: (LiZnAs  GaAs), (LiZnP, NaAlC  GaP), (LiZnAs  InP), (NaGaSi, KAlC  InAs), (NaGaSi  GaSb) and (NaInSi  InSb). In Fig. 2, the band structures of XYZ compounds are shown and compared with zinc-blende semiconductors at high symmetry points using both PBE-GGA and TB-mBJ approaches. One can see that band structures of half-Heusler compounds show great overall similarities to those of binary compounds. Close to the band gap, the band structures of the selected compounds are also similar to that of III–V semiconductors, suggesting that (YZ) may be viewed as III–V semiconductors. For all semiconductor that possess indirect gaps with the conduction band minimum at X point,

Fig. 1. Diagram of lattice constants a and calculated band gaps Eg of half-Heusler compounds in the range of 5 < a < 7 Å and Eg < 3 eV. A ±0.25 Å variation in lattice parameter and a ±0.2 eV variation in band gap are found, comparing with the binary compounds.

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Table 2 Results obtained with calculations of cubic III–V semiconductors and a selection of half-Heusler materials. Shown are equilibrium lattice constants acalc and calculated band gap Egcalc using both PBE-GGA and TB-mBJ approaches. Numbers in bold denote our results. aPBE (Å)

EgTB-mBJ (eV)

Directivity

e1(0)TB-mBJ

n(0)TB-mBJ

The first zone LiScC NaAlC NaScC KScC LiBeP LiBeAs LiBeSb KBeN LiZnP LiZnAs LiCdP NaZnP KMgN GaAs

5.32 5.40 5.68 6.17 5.37 5.58 6.04 5.67 5.77 5.98 6.14 6.15 6.01 5.75

1.63 1.94 1.43 1.06 1.78 1.61 1.29 1.30 1.97 1.58 1.50 1.69 1.38 1.23

X–X C–X X–X X–X U–X U–X U–X X–U U–X U–U U–U U–U X–U U–U

9.09 7.16 9.08 10.50 7.44 7.91 9.62 6.05 7.91 9.00 8.10 7.47 9.29 10.11

3.02 2.68 3.01 3.24 2.73 2.81 3.10 2.46 2.81 3.00 2.85 2.73 7.71 3.18

The second zone NaMgN LiZnP NaAlC GaP

5.45 5.77 5.40 5.52

2.19 1.99 1.94 2.31

U–U U–U U–U U–U

49.96 7.91 7.16 8.29

7.08 2.81 2.68 2.88

The third zone LiBeSb LiZnAs NaScC KBeN NaZnP LiCdP KScC InP

6.04 5.98 5.68 5.67 6.15 6.14 6.17 5.97

1.29 1.58 1.43 1.30 1.69 1.50 1.06 1.34

U–X U–U X–X X–U U–U U–U X–X U–U

9.62 9.00 9.08 6.05 7.47 8.10 10.50 8.47

3.10 3.00 3.01 2.46 2.73 2.85 3.24 2.91

The fourth zone LiInSi NaGaSi LiAlSi LiGaGe KAlC KZnN LiGaSi LiInGe KInC KGaC KCdN InAs

6.28 6.29 6.02 6.04 6.01 5.96 5.91 6.40 6.33 6.09 6.20 6.20

0.61 0.60 0.51 0.49 0.29 0.30 0.60 0.00 0.00 0.00 0.02 0.32

U–X U–U U–X U–U U–U U–U U–X U–U U–U U–U U–U U–U

16.56 15.86 13.03 18.75 11.71 8.82 15.78 21.76 24.19 22.99 16.14 11.30

4.07 3.98 3.61 4.33 3.42 2.79 3.97 4.67 4.93 4.80 4.03 3.36

The fifth zone LiInSi NaGaSi NaAlSi LiGaGe KAlC KZnN LiCdAs NaZnAs NaAlGe GaSb

6.28 6.29 6.29 6.04 6.01 5.96 6.35 6.35 6.39 6.22

0.61 0.60 1.00 0.49 0.29 0.30 0.79 0.90 0.83 0.50

U–X U–U U–X U–U U–U U–U U–U U–U U–U U–U

16.56 15.86 12.07 18.75 11.71 8.82 9.68 9.08 13.87 13.58

4.07 3.98 3.47 4.33 3.42 2.79 3.11 3.01 3.72 3.68

The sixth zone KZnAs NaCdAs NaInSi NaInGe KGaSi KAlGe KGaGe NaGaGe LiInGe InSb

6.86 6.63 6.60 6.71 6.78 6.87 6.91 6.42 6.40 6.64

0.39 0.46 0.27 0.00 0.00 0.00 0.00 0.00 0.00 0.05

U–U U–U U–U U–U U–U U–U U–U U–U U–U U–U

10.21 9.81 17.10 25.79 20.99 17.65 29.99 23.63 21.76 15.27

3.20 3.13 4.14 5.08 4.58 4.21 5.49 4.86 4.67 3.91

exchanging X+ (for example Na by Li) leads to direct semiconductors with the conduction band minimum at C. In the III–V binary analogous compounds the d bands are deeper in energy compared to those in the ternary ones; this makes the

p–d interaction stronger in the latter ones. This p–d coupling, which is included in our calculations, affects the electronic and structural properties of Nowotny–Juza compounds by decreasing the gap and lattice parameter. For example, whereas the Zn atom

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Fig. 2. Band structures along the principal high-symmetry directions in the BZ for the selected XYZ half-Heusler compounds and their analogous III–V semiconductors.

possesses an additional electronic shell than the Mg atom, the lattice parameter of LiZnN (4.91 Å) [20] is smaller than that of LiMgN (4.95 Å) [16], and its band gap is also smaller 1.91 eV [20] for LiZnN and 3.23 eV [16] for LiMgN. From these results, the electronic structure of half-Heusler, which can be manipulated by the insertion of the different elements in the tetrahedral sites,

offers a remarkable advantage to the half-Heusler compared to III–V and II–VI semiconductor. Our band structure calculations indicate that all studied half-Heusler compounds possess a direct band gap at C point except for LiZnP and NaAlC, where the valence band maximum is at C point and the minimum of the conduction band is at X. Finally, we can deduce that the inclusion of TB-mBJ

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457

Fig. 3. The imaginary and real parts of dielectric function of the selected XYZ half-Heusler compounds and their analogous III–V semiconductors.

opens a band gap at EF at the C point leading to typical dips in the band structure for all compounds. The optical properties of the material at all photon energies E = ⁄x are determined by dielectric function e(x), which is given by

Eform ðXYZÞ ¼ Etot ðXYZÞ  ½Etot ðXÞ þ Etot ðYÞ þ Etot ðZÞ;

eðxÞ ¼ e1 ðxÞ þ ie2 ðxÞ The imaginary part of the frequency-dependent dielectric function e2(x) of the material is determined mainly by the transition between the valence and conduction bands, according to the perturbation theory [53], e2(x) is expressed as

e2 ðxÞ ¼

4p2 e2 X 3m2 x2 l;n

Z BZ

2 ð2pÞ3

3

d kjPnl j2  d½El ðkÞ  En ðkÞ  hx

e2(x) is strongly related to the joint density of states (DOS) and momentum matrix element. The real part of dielectric function e1(x) is obtained from e2(x) using the Kramers–Kronig relation [54]

e1 ðxÞ ¼ 1 þ

2

p

Z 0

x

At the end of this study, the stability of the investigated half-Heusler compounds is also estimated. For this purpose we calculate their formation energies, to check if these systems can be easily synthesized, which are defined as:

x0 e2 ðx0 Þ 0 dx x02  x2

The calculated real e1(x) and imaginary e2(x) dielectric function of XYZ compounds and its analogue III–V semiconductors are shown in Fig. 3. The imaginary part e2(x) is obtained directly from the electronic structure calculations. From the result, the real part e1(x) is determined using the Kramers Kronig dispersion relation. At first glance, dielectric functions of all selected half-Heusler compounds and III–V semiconductors are almost similar. In the 0–15 eV photon energy range, there are the same peaks of e2(x) for both ternary and binary compounds represented in Fig. 3 which can be assigned to the same transitions of the specific points in the Brillouin Zone (BZ). Our calculated e1(x) and the refractive index n(x) values at zero frequency for XYZ compounds shown in Table 2, are well comparable to those of III–V semiconductors. These results provide another promising voice for future optoelectronic applications.

where Etot are the total energies of the corresponding substances at their optimized geometries as obtained in our FLAPW-GGA calculations. The formation energy values of LiZnAs, LiZnP, NaAlC, NaGaSi, NaInSi, and KAlC compounds are –15.327764, –9.614752, –7.376041, –10.176339, –21.547059 and –6.812134 eV/atom respectively. A negative Eform indicates that it is energetically favorable to form stable half-Heusler compounds.

4. Conclusions In summary, we have presented a first-principles study of structural, electronic and optical properties in zinc-blende structure to demonstrate semiconducting half-Heusler compounds by comparing with III–V and II–VI semiconductors. In our calculations the APW+lo in the PBE-GGA and TB-mBJ scheme has been used. The lattice parameters and energy band gaps were calculated and are compared with the available results. We note that we have obtained very accurate band gaps of semiconductors with an orbital-independent exchange-correlation potential which depends solely on semilocal quantities (TB-mBJ). Through this study, we have chosen several criteria such as, lattice parameters, energy band gaps, static optical properties and other to identify interesting materials. So, we have succeeded to select six compounds (LiZnAs, LiZnP, NaAlC, NaGaSi, NaInSi, and KAlC) which fulfill these criteria. All XYZ compounds are found to be semiconductors with a direct band gap at C point except for LiZnP and NaAlC, where the valence band maximum is at C point and the minimum of the conduction band is at X.

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References [1] H. Nowotny, K. Bachmayer, Monatsch. Chem. 81 (1950) 488. [2] H. Nowotny, B. Glatzl, Monatsh. Chem. 82 (1950) 720. [3] R. Juza, F. Hund, Z. Anorg. Chem. 257 (1) (1948); R. Juza, W. Dethlefesen, H. Seidel, K. Benda, Anorg. Allg. Chem. 356 (1968) 253. [4] C. Kandpal, C. Felser, R. Seshadri, J. Phys. D 39 (2006) 776. [5] R. Juza, K. Langer, K.V. Benda, Angew. Chem., Int. Ed. Engl. 7 (1968) 360. [6] K. Kuriyama, T. Kato, K. Kawada, Phys. Rev. B 49 (1994) 11452. [7] K. Kuriyama, R. Taguchi, K. Kushida, K. Ushiyama, J. Cryst. Growth 198–199 (1999) 802. [8] K. Kuriyama, Y. Yamashita, T. Ishikawa, K. Kushida, Phys. Rev. B 75 (2007) 233204. [9] H. Nowotny, F. Holub, Monatsch. Chem. 91 (1960) 877. [10] R. Bacewicz, T.F. Ciszek, Appl. Phys. Lett. 52 (1988) 1150. [11] L. Spina, Y.-Z. Jia, B. Ducourant, M. Tillard, C. Belin, Z. Kristallogr. 218 (2003) 740. [12] W. Blase, G. Cordier, R. Kniep, Z. Anorg, Allg. Chem. 619 (1993) 1161. [13] W. Bockelmann, H.-U. Schuster, Z. Anorg, Allg. Chem. 410 (1974) 241. [14] U. Eberz, W. Seelentag, H.U. Schuster, Z. Naturforsch., B 35 (1980) 1341. [15] R. Marazza, D. Rossi, R. Ferro, J. Less-Common Met. 138 (1988) 189. [16] K. Kuriyama, K. Nagasawa, K. Kushida, J. Cryst. Growth 237–239 (2002) 2019– 2022. [17] R. Bacewicz, T.F. Ciszek, Mater. Res. Bull. 23 (1988) 1247. [18] K. Kuriyama, K. Kushida, R. Taguchi, Solid State Comm. 108 (1998) 429. [19] K. Kuriyama, R. Taguchi, K. Kushida, K. Ushiyama, J. Cryst. Growth 802 (1999) 198. [20] K. Kuriyama, T. Kato, T. Tanaka, Phys. Rev. B 49 (1994) 4511. [21] K. Kuriyama, T. Katoh, Phys. Rev. B 37 (1988) 7140. [22] P. Villars, L.D. Calvert, Pearson’s Handbook of Crystallographic Data for Intermetallic Phases, second ed., ASM International, Materials Park, OH, 1996. [23] H. Mehnane, B. Bekkouche, S. Kacimi, A. Hallouche, M. Djermouni, A. Zaoui, Superlattices Microstruct. 51 (2012) 772–784. [24] F. Kalarasse, B. Bennecer, J. Phys. Chem. Solids 67 (2006) 1850. [25] G. Jaiganesh, T.M.A. Britto, R.D. Eithiraj, G. Kalpana, J. Phys.: Condens. Matter 20 (2008) 085220. [26] L.H. Yu, K.L. Yao, Z.L. Liu, Y.S. Zhang, Phys. Lett. A 367 (2007) 389. [27] S.-H. Wei, A. Zunger, Phys. Rev. Lett. 56 (1986) 528. [28] S. Curtarolo, G.L.W. Hart, M.B. Nardelli, N. Mingo, S. Sanvito, O. Levy, Nat. Mater. 12 (2013) 181. [29] S. Ogut, K.M. Rabe, Phys. Rev. B. 51 (1995) 10443. [30] D. Kieven, R. Lenk, S. Naghavi, C. Felser, T. Gruhn, Phys. Rev. B. 81 (2010) 075208. [31] T. Gruhn, Phys. Rev. B. 82 (2010) 125210.

[32] F. Casper, R. Seshadri, C. Felser, Phys. Status Solidi A 206 (2009) 1090. [33] A. Roy, J.W. Bennett, K.M. Rabe, D. Vanderbilt, Phys. Rev. Lett. 109 (2012) 037602. [34] F. Tran, P. Blaha, Phys. Rev. Lett. 102 (2009) 226401. [35] P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka, J. Luitz, Wien2k, An Augmented Plane Wave Plus Local Orbitals Program for Calculating Crystal Properties, Vienna University of Technology, Vienna, 2001. [36] E. Sjöstedt, L. Nordstrom, D.J. Singh, Solid State Commun. 114 (2000) 15. [37] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) 3865. [38] F.D. Murnaghan, Proc. Natl. Acad. Sci. USA. 30 (1944) 5390. [39] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1976) 5188. [40] O. Zakharov, A. Rubio, X. Blase, M.L. Cohen, S.G. Louie, Phys. Rev. B 50 (1994) 10780. [41] H. Okuyama, K. Nakano, T. Miyajima, K. Akimoto, J. Cryst. Growth 117 (139) (1992); A. Waag, H. Heinke, S. Scholl, C.R. Becker, G. Landwehr, J. Cryst. Growth 131 (607) (1993); L. Konczenwicz, P. Bigenwal, T. Cloitre, M. Chibane, R. Ricou, P. Testuo, O. Briot, R.L. Aulombard, J. Cryst. Growth 159 (117) (1996). [42] K. H Hellwege, O. Madelung, Landolt–Brönstein New Series Group III, Vol. 17a, Springer, Berlin, 1982. [43] J.H. Edgar, Properties of Group III Nitrides (Electronic Materials Information Service (EMIS) Datareviews Series), Institution of Electrical Engineers, London, 1994. [44] T. Lei, T.D. Moustakas, R.J. Graham, Y. He, S.J. Berkowitz, J. Appl. Phys. 71 (1992) 4933. [45] M. Levinstein, S. Rumyantsev, M. Shur, Handbook Series on Semiconductor Parameters, Vols. 1–2, World Scientific, London, 1999. [46] S. Strite, J. Ruan, D.J. Smith, J. Sariel, N. Manning, H. Chen, Bull. Am. Phys. Soc. 37 (1992) 346. [47] A. Schleife, F. Fuchs, J. Furthmüller, F. Bechstedt, Phys. Rev. B 73 (2006) 245212. [48] H. Okuyama, K. Nakano, T. Miyajima, K. Akimoto, Jpn. J. Appl. Phys., Part 230, L1620 (1991); S. G. Parker, A. R. Reinberg, J. E. Pinnell, and W. C. Holton, J. Electrochem. Soc. 118 (1971) 979. [49] S. Adachi, J. Appl. Phys. 61 (1987) 4869. [50] I. Vurgaftman, J.R. Meyer, L.R. Ram-Mohan, J. Appl. Phys. 89 (2001) 5815. [51] N.W. Ashcroft, N.D. Mermin, Solid State Physics, Holt Rinehart and Winston, New York, 1976. [52] W.R.L. Lambrecht, b. Segall, S. Strite, G. Martin, A. Agarwal, H. Morkoc, A. Rockett, Phys. Rev. B 50 (1994) 14155. [53] M.A. Khan, A. Kashyap, A.K. Solanki, T. Nautiyal, S. Auluck, Phys. Rev. B 23 (1993) 16974. [54] F. Wooten, Optical Properties of Solids, Academic Press, New York, 1972.