Superlattices and Microstructures 51 (2012) 772–784

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Superlattices and Microstructures

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . co m / l o c a t e / s u p e r l a t t i c e s

First-principles study of new half Heuslerfor optoelectronic applications

H. Mehnane a, B. Bekkouche b, S. Kacimi a, A. Hallouche a, M. Djermouni a,A. Zaoui a,⇑a Modelling and Simulation in Materials Science Laboratory, Djillali Liabès University of Sidi Bel-Abbès, Sidi Bel-Abbès 22000, Algeriab Signals and Systems Laboratory, Abdelhamid Ibn Badis University of Mostaganem, Mostaganem 27000, Algeria

a r t i c l e i n f o

Article history:Received 1 February 2012Received in revised form 13 March 2012Accepted 20 March 2012Available online 29 March 2012

Keywords:DFTElectronic structureOptical propertiesSemiconductors

0749-6036/$ - see front matter � 2012 Elsevier Lthttp://dx.doi.org/10.1016/j.spmi.2012.03.020

⇑ Corresponding author. Tel.: +213 778090975.E-mail addresses: alzaoui@gmail.com, ali_zaoui

a b s t r a c t

We report investigations of the structural, electronic and opticalproperties of 36 half-Heusler compounds in comparison with II–VI semiconductors using the first-principles calculations based onthe density functional theory. In this work, we demonstrate thesimilarity in the electronic structure of these materials with thatof II–VI semiconductors through the analysis of lattice parameters,band gaps and static dielectric constants at ambient pressure. Theevolution of these properties under pressure is also necessary topredict new candidates for the optoelectronic devices.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

II–VI semiconductors have attracted much interest in fundamental research in various areas. Forexample, the study of diluted magnetic semiconductor (DMS) and manganese-based alloys [1], theemission of single photons in the visible range with CdSe quantum boxes [2], or Bose condensationof excitants in CdTe-based microcavities [3] or again Aharonov–Bohm effect [4]. II–VI semiconductors,such as ZnS, ZnSe and CdTe, appear to be promised for optoelectronic applications [5–16]. The limita-tion of some II–VI semiconductors in various fields makes the search of new semiconductors a majorchallenge for materials science. The first work on the chalcopyrites was carried out by Hahn et al. [17];later, studies on this family of compounds have been largely motivated by their potential in differentapplications. The similarity between the zinc-blende and chalcopyrite structure has pushed the

d. All rights reserved.

@yahoo.fr (A. Zaoui).

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scientists to focus their attention on this compounds family. Effectively, Zunger and his collaborators[18,19] have explained the formation of ternary compounds made by a substitution of atoms in a tet-rahedral empty site.

Another particularly interesting class of materials is named half-Heusler compounds or ‘‘Nowotny–Juza,’’ [20] with a chemical composition XYZ. So, the half-Heusler compounds, which have eight va-lence electrons, including a large number of semiconductors with energy gaps vary in a wide range[21]. In general, the half-Heusler materials with eight valence electrons can be of II–VI, I–II–V, I–III–IV, II–II–IV and III–II–III type. In addition, the ferromagnetic behavior was observed by de Grootet al. [22] in half-Heusler compounds. Researchers have resumed the studies on these materials thatcan show: the topological properties [23–25], they can also be used as spintronic devices [26] and asthermo-electrics with high-performance [27–29]. Recently, Roy et al. [30] have predicted the piezo-electric response and their associated properties in half-Heusler compounds. Another significant the-oretical advancement came in 1985, and first principle calculations were presented by Wood et al. [18]who have predicted the energy gaps of about 1 eV in Li–half-Heusler compounds. After experimentaland theoretical observations [31,32] on the NiSnZr half-Heusler, a lot of additional semiconductor sys-tems have been identified [21,33,34]. Some of these compounds, such as LiMgN and LiMgP, have largeenergy gaps that make them suitable for optoelectronic applications [35–37]. In this work, we are par-ticularly interested by I–II–V half-Heusler type and our main purpose is to predict new semiconduc-tors suitable for optoelectronic applications comparing with II–VI compounds.

This paper is planned as follows: Section 2 gives a brief description of the method used in the cal-culations; Section 3 presents the analyses of the calculated results and in Section 4 we are going tosummarize what has been done in the previous section.

2. Methodology

The calculations have been performed within DFT implemented in the WIEN2K code [38]. Theatoms were represented by hybrid full-potential (linear) augmented plane-wave plus local orbitals(L/APW + lo) method [39]. In this method wave functions, charge density, and potential are expandedin spherical harmonics within no overlapping muffin-tin spheres, and plane waves are used in theremaining interstitial region of the unit cell. In the code, the core and valence states are treated differ-ently. Core states are treated within a multiconfiguration relativistic Dirac–Fock approach, while va-lence states are treated in a scalar relativistic approach. The exchange–correlation energy wascalculated using the generalized gradient approximation PBE–GGA [40]. Very carefully step analysisis done to ensure convergence of the total energy in terms of the variational cutoff-energy parameter.At the same time we have used an appropriate set of k points to compute the total energy. We havecomputed equilibrium lattice constants and bulk moduli by fitting the total energy versus volume tothe Murnaghan equation [41].

The total energy was minimized using a set of 56 k-points in the irreducible sector of Brillouin zonefor the cubic- F-43m and the value of 9 Ry for the cutoff energy were used. We have adopted the valuesof MT radii between 1.5 and 2.5 for all elements.

3. Results and discussion

We have studied the electronic structure of 36 half-Heusler with X = Li, Na, and K, Y = Mg, Ca, Znand Cd, and Z = N, P and As, respectively using APW + lo calculations. For comparison we examinedeight II–VI compounds in the zinc-blende structure. Results from the calculations are summarizedin Table 1. The calculated lattice constant and bulk modulus of zinc-blende II–VI semiconductorsare in good agreement with the experimentally measured values [43] and other theoretical results[33,42,49–51]. Due to the close relation between the zinc-blende and the half-Heusler structure, struc-tural and electronic properties are directly comparable. For example, the CdS is a semiconductor witha lattice parameter of the order of 5.944 Å and an energy gap of �1 eV. So, to replace this material wemust find an analogue half-Heusler with the same lattice parameter and the same energy gap. In thiscontext, we have considered the lattice parameter and the energy band gap as two basic criteria to

Table 1Structural parameters of II–VI semi-conductors and I–II–V half-Heusler compounds calculated by the GGA and compared to othertheoretical and experimental results.

aWien2k (Å) BWien2k (GPa) B0Wien2k aexp (Å) aTheo (Å)

CdO 5.143 92.345 4.693 – 5.148 [42]CdS 5.944 56.079 4.131 5.818 [43] 5.948 [33]CdSe 6.21 46.806 3.96 6.052 [43] 6.216 [49]CdTe 6.627 36.059 4.256 6.480 [43] 6.573 [50]ZnO 4.643 134.888 4.764 – –ZnS 5.471 71.86 4.286 5.409 [43] 5.352 [51]ZnSe 5.758 59.352 4.12 5.668 [43] 5.627 [51]ZnTe 6.208 46.205 4.369 6.089 [43] 6.183 [50]KCaN 6.289 40.358 4.065 – 6.286 [33]KCaP 7.066 29.077 4.102 – 7.201 [33]KCaAs 7.386 27.155 3.875 – –KMgN 6.005 46.156 4.267 – 6.005 [33]KMgP 6.847 33.322 4.134 – 6.847 [33]KMgAs 7.03 29.087 4.131 – –KZnN 5.963 46.374 4.283 – 5.959 [33]KZnP 6.674 32.521 4.146 – 6.673 [33]KZnAs 6.861 28.996 4.002 – –KCdN 6.197 44.406 4.153 – –KCdP 6.891 31.514 4.281 – –KCdAs 7.066 29.026 4.102 – –LiCaN 5.544 90.911 4.19 – 5.553 [33]LiCaP 6.138 54.965 3.953 – 6.501 [33]LiCaAs 6.673 36.684 3.995 – –LiMgN 5.01 97.11 3.667 4.955 [44] 5.010 [36]LiMgP 6.011 53.929 3.694 6.005 [37] 6.022 [33]LiMgAs 6.205 46.408 3.953 6.180 [37] 6.218 [21]LiZnN 4.929 114.945 3.987 4.910 [45] 4.924 [33]LiZnP 5.766 65.231 4.204 5.765 [46] 5.766 [33]LiZnAs 5.98 55.91 3.9 5.940 [47] 5.980 [34]LiCdN 5.388 90.911 4.19 – –LiCdP 6.137 54.965 4.069 6.100 [48] 6.130 [34]LiCdAs 6.345 46.265 4.042 – –NaCaN 5.855 55.213 4.341 – 5.858 [33]NaCaP 6.785 36.753 4.055 – 6.785 [33]NaCaAs 6.961 33.04 3.995 – –NaMgN 5.448 71.976 4.058 – 5.443 [33]NaMgP 6.369 45.337 4.021 – 6.375 [33]NaMgAs 6.556 39.245 3.994 – –NaZnN 5.363 80.355 4.272 – 5.358 [33]NaZnP 6.146 50.76 3.896 – 6.149 [33]NaZnAs 6.349 42.761 4.145 – –NaCdN 5.703 70.997 3.867 – –NaCdP 6.443 45.534 4.055 – –NaCdAs 6.634 39.239 4.113 – –

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select candidates similar to II–VI semiconductors. The calculated lattice constants vary between 4.924(LiZnN) and 7.386 Å (KCaAs) which have the same tend with experiment data (LiMgN = 4.955 Å [44]and LiMgAs = 6.18 Å [37]). Fig. 1 includes all lattice parameters of binary and ternary semiconductors.We see clearly that the nitrides have smaller lattice constants and when we change N by P and As, thelattice constant increases. This increase can be ascribed to the larger ion radius of phosphor and ar-senic. This observation has already been confirmed by several studies [21,33]. As shown in this figure,the lattice parameter of ZnS is about 5.47 Å and the equivalent half-Heusler (LiCaN = 5.544 Å, LiC-dN = 5.388 Å, NaMgN = 5.448 Å and NaZnN = 5.363 Å) have almost the same values. So the latticeparameter criterion allowed us to select some materials and eliminate others. From Table 1, the ob-tained values of B0 for all half-Heusler semiconductors show that nitrides have higher bulk moduli val-ues than phosphorides and arsenides.

Fig. 1. Lattice parameters of II–VI semiconductors and XYZ half-Heusler.

Fig. 2. Energy gaps of II–VI semiconductors and XYZ half-Heusler.

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The second criterion is the energy band gap. Fig. 2 shows the gap energy of each half-Heusler com-pound compared with those of II–VI semiconductors. These quantities are obtained by calculating theelectronic band structure. The band structure of LiZnP was successfully explained as the electronicstructure of an hypothetical (ZnP)� semiconductor, for which the conduction bands are modified bythe presence of the Li+ cation [18]. From Table 2, the energy band gap ordering of all materials isEg(LiYZ) > Eg(NaYZ) > Eg(KYZ). This means that the widening of band gap due to the replacement of a heavyelement by a light one acts effectively for the band gap modification as predicted theoretically by an‘‘interstitial insertion rule’’ [18]. This rule suggests that when one inserts Li+ at the interstitial (½, ½, ½)site of the hypothetical zinc-blende (YZ)– structure, the Xc

3 conduction band of the Brillouin zone isshifted to higher energies more than the other band, exposing the Cc

1 point as the conduction mini-mum that explains the change of the band gap directness in this series of materials. So, replacing Nwith the heavier pnictogens P or As was found to narrow the band gap due to the increasing band

Table 2The energy gap Eg, the real static dielectric function e1(0), the static refractive index n(0) and the pressure coefficients a and b of II–VI semi-conductors and I–II–V half-Heusler compounds calculated by the GGA and compared to other theoretical andexperimental results.

Ega (eV) Eg

b (eV) Egc (eV) Nature/gap e1(0) n(0) a (GPa) b (GPa)

CdO 0 0.840 [42] 0 [42] Direct 6.261 2.502 0 0CdS 1.01 2.550 [43] 1.370 [43] Direct 5.171 2.274 3.313 �5.4CdSe 0.476 1.900 [43] 0.760 [43] Direct 5.937 2.437 3.696 �6.1CdTe 0.558 1.920 [43] 0.800 [43] Direct 6.598 2.569 6.643 �1ZnO 0.636 3.440 [42] 0.730 [43] Direct 3.974 1.994 1.046 �1.1ZnS 1.939 3.800 [43] 2.370 [43] Direct 5.748 2.398 5.313 �6.4ZnSe 1.087 2.960 [43] 1.450[43] Direct 6.048 2.459 5.832 �7ZnTe 0.98 2.710 [43] 1.330 [43] Direct 6.976 2.641 8.445 �11.6KCaN 0.66 – 0.680 [33] Indirect 4.934 2.221 5.855 �7KCaP 1.522 – 1.540 [33] Indirect 5.439 2.332 7.4 �13.5KCaAs 1.234 – – Indirect 5.481 2.341 7.271 �13.4KMgN 0.104 – 0.130 [33] Indirect 6.311 2.512 5.486 �8KMgP 0.966 – 0.960 [33] Direct 6.661 2.581 11.426 �15.9KMgAs 0.469 – – Direct 6.276 2.505 11.136 �15.8KZnN 0 – 0 [33] Direct 7.644 2.761 �0.661 4.5KZnP 0 – 0 [33] Direct 7.472 2.733 5.924 1.7KZnAs 0 – – Direct 0.822 2.886 0.492 13.3KCdN 0 – – Direct 7.7 2.775 0 0KCdP 0 – – Direct 7.242 2.691 4.348 �0.6KCdAs 0 – – Direct 8.012 2.831 �0.647 11.8LiCaN 2.295 – 2.210 [33] Indirect 5.685 2.384 5.517 �6.2LiCaP 1.958 – 1.950 [33] Indirect 6.292 2.508LiCaAs 1.836 – – Indirect 6.532 2.556 0.347 �0.9LiMgN 2.292 3.230 [36] 2.280 [33] Direct 3.986 2.000 6.391 �4.9LiMgP 1.543 2.430 [37] 1.550 [33] Indirect 6.475 2.545 �1.695 1LiMgAs 1.375 2.380 [30] 1.150 [52] Indirect 6.944 2.635 �1.774 1.4LiZnN 0.517 1.910 [45] 0.520 [33] Direct 5.928 2.435 3.671 �3.1LiZnP 1.354 2.040 [46] 1.350 [33] Indirect 6.708 2.59 �2.304 2.1LiZnAs 0.491 1.510 [47] 1.100 [33] Direct 8.427 2.903 7.806 �9.4LiCdN 0 – – Indirect 6.898 2.626 0 0LiCdP 0.556 0.850 [21] 0.510 [52] Direct 7.568 2.751 5.812 �8.5LiCdAs 0 – – Direct 8.09 2.844 3.722 �2.4NaCaN 1.154 – 1.150 [33] Indirect 4.372 2.091 5.514 �7.1NaCaP 1.94 – 1.950 [33] Indirect 6.081 2.466 �0.28 �0.6NaCaAs 1.598 – – Direct 6.16 2.482 9.234 �13.4NaMgN 0.763 – 0.770 [33] Direct 5.076 2.253 6.174 �5.7NaMgP 1.482 – 1.470 [33] Direct 6.204 2.491 9.565 �12.3NaMgAs 0.834 – – Direct 6.551 2.56 9.487 �12.6NaZnN 0 – 0.0 [33] Direct 6.376 2.525 �0.655 5.1NaZnP 0.46 – 0.440 [33] Direct 8.604 2.933 8.211 �10.2NaZnAs 0 – – Direct 8.22 2.867 4.028 1.6NaCdN 0 – – Direct 7.347 2.71 0 0NaCdP 0.011 – – Direct 7.345 2.71 6.301 �9.3NaCdAs 0 – – Direct 8.255 2.873 �0.497 8.4

a Our results.b Experiments.c Other calculations.

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width of both the valence and conduction bands. If the X and Z ions are held constant, as in LiMgP andLiCdP, we observe that replacing the more ionic Mg by the softer Cd also results in a narrowing of theband gap. It is also important to note that many materials have zero energy band gaps comparable tothat of CdO. This underestimation of the band gap, which is caused by the DFT, makes us incapable toclass these materials (semiconductors with zero band gap or semi-metals). Therefore, experimentalexamination is needed to correctly describe the electronic structure of these compounds.

By comparing our results with experimental observations, we noticed that there is a significantunderestimation of energy gaps in the II–VI and half-Heusler due to the application of the DFT.

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Next, we have collected our two criteria: lattice parameter and energy gap in Fig. 3, firstly to facil-itate comparison, and in the other hand for better visualize and select the half-Heusler compoundsthat are comparable to II–VI semiconductors. Indeed, we have succeeded to predict ten half-Heuslerthat satisfy these two criteria: (LiCaN � ZnS), (LiZnP, NaCaN � ZnSe, CdS), (NaMgAs � ZnTe), (NaM-gAs � CdTe), (LiZnAs, LiCdP, NaZnP, KCaN � CdSe) and (NaZnN, LiCdN � CdO).

Next, we have presented the band structures of some candidate compounds and their analogues II–VI in Fig. 4. It is clear that, from these figures, the studied compounds are semiconductors and theirband structures show great overall similarities to those of II–VI compounds. In the II–VI binary anal-ogous compounds the d bands are deeper in energy compared to those of the ternary ones, this makesthe 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 gapand lattice parameter. For example, whereas the Zn atom possesses an additional electronic shell thanthe Mg atom, the lattice parameter of LiZnN (4.91 Å) [45] is smaller than that of LiMgN (4.95 Å) [36],

Fig. 3. Energy gaps and lattice parameters of the different compounds I–II–V and II–VI.

Fig. 4. Energy band structures of I–II–V and II–VI.

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and its band gap is also smaller 1.91 eV [45] for LiZnN and 3.23 eV [36] for LiMgN. From these results,the electronic structure of half-Heusler, which can be manipulated by the insertion of the differentelements in the tetrahedral sites, offers a remarkable advantage to the half-Heusler compared to II–VI semiconductor.

Fig. 5 shows the partial density of states of the NaMgAs compound and its analogue ZnTe. We notethat this compound is among the ten selected materials sited above. We focus our attention especiallyin the vicinity of the Fermi level. This figure shows that the maximum of the valence band is domi-nated by p states of the anions in the two compounds with a small contribution of the different statesof all atoms. The conduction band of ZnTe compound is dominated by s states of the Zn cation, while p-As states occupy this area in NaMgAs compound. The difference in results between I–II–V and II–VIsemiconductors can be explained by the absence of d states in the half-Heusler and the existence ofNa electropositive element in the NaMgAs.

The distribution charge density of our NaCaN half-Heusler and that of CdS compound in (110)plane are plotted in Fig. 6. We note that the Ca–N bond in the NaCaN compound is similar to the

Fig. 5. Partial densities of states of NaMgAs half-Heusler and ZnTe semiconductor.

Fig. 6. Total electron density contours of NaCaN and CdS in (110) plane.

Fig. 7. The imaginary and real parts of dielectric function of NaMgAs and ZnTe compounds under hydrostatic pressure.

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covalent Cd–S bond in the CdS compound. In addition, the Na–N bond is purely ionic due to the elec-tronegativity difference between sodium and nitrogen. NaCaN compound can be considered as both acovalent and ionic material. Half-Heusler materials XYZ can be viewed as a zinc-blende structure of(YZ)� with X+ interstitials, where Y and Z are covalently bound.

Table 3The energy gap Eg, the real static dielectric function e1(0), the static refractive index n(0) of II–VI semi-conductors and I–II–V half-Heusler compounds under hydrostatic pressure.

Compounds Eg (eV) e1(0) n(0)

CdO Pressure P = 5 GPa 0 6.251 2.500CdS 1.174 5.089 2.256CdSe 0.658 5.856 2.420CdTe 0.878 6.633 2.576ZnO 0.687 3.946 1.986ZnS 2.201 5.931 2.435ZnSe 1.372 6.225 2.495ZnTe 1.405 7.358 2.713KCaN 0.961 5.169 2.274LiCaN 2.493 5.777 2.404LiZnP 1.242 8.092 2.845LiZnAs 0.875 8.599 2.932LiCdN 0 6.712 2.591LiCdP 0.839 7.702 2.775NaCaN 1.428 5.102 2.259NaMgAs 1.302 6.746 2.957NaZnN 0 6.053 2.460NaZnP 0.865 7.613 2.759CdO Pressure P = 10 GPa 0 6.278 2.506CdS 1.291 5.105 2.259CdSe 0.788 5.842 2.417CdTe 1.124 6.626 2.574ZnO 0.730 3.930 1.983ZnS 2.412 6.094 2.469ZnSe 1.604 6.392 2.528ZnTe 1.723 7.777 2.789KCaN 1.195 5.075 2.253LiCaN 2.515 5.649 2.377LiZnP 1.144 7.937 2.817LiZnAs 0.946 8.491 2.914LiCdN 0 6.569 2.566LiCdP 1.060 8.188 2.862NaCaN 1.641 5.193 2.279NaMgAs 1.534 6.816 2.611NaZnN 0 5.792 2.407NaZnP 1.186 7.766 2.787CdO Pressure P = 15 GPa 0 6.296 2.509CdS 1.379 5.135 2.266CdSe 0.886 5.850 2.419CdTe 1.296 6.222 2.494ZnO 0.765 3.913 1.978ZnS 2.583 6.127 2.475ZnSe 1.799 6.557 2.561ZnTe 1.911 8.027 2.833KCaN 1.242 5.644 2.376LiCaN 2.534 5.727 2.393LiZnP 1.057 8.047 2.837LiZnAs 0.854 8.271 2.876LiCdN 0 6.226 2.495LiCdP 1.181 7.990 2.827NaCaN 1.814 5.279 2.298NaMgAs 1.482 6.650 2.579NaZnN 0 5.612 2.369NaZnP 1.449 7.673 2.770CdO Pressure P = 20 GPa 0 6.336 2.517CdS 1.449 5.169 2.273CdSe 0.964 5.868 2.422CdTe 1.434 6.716 2.592ZnO 0.798 3.901 1.975

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Table 3 (continued)

Compounds Eg (eV) e1(0) n(0)

ZnS 2.741 5.462 2.376ZnSe 1.968 6.673 2.583ZnTe 1.804 7.899 2.811KCaN 1.227 5.757 2.340LiCaN 2.548 5.087 2.255LiZnP 0.979 7.759 2.786LiZnAs 0.768 8.208 2.865LiCdN 0 6.346 2.519LiCdP 1.125 7.889 2.809NaCaN 1.959 5.371 2.317NaMgAs 1.434 6.540 2.557NaZnN 0.053 5.955 2.244NaZnP 1.438 8.102 2.846

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The optical properties of the material at all photon energies E ¼ �hw are also determined by dielec-tric function e(x), which is given by,

eðxÞ ¼ e1ðxÞ þ ie2ðxÞ ð1Þ

The imaginary part of the frequency-dependent dielectric function e2(x) of the material is deter-mined mainly by the transition between the valence and conduction bands, according to the pertur-bation theory [53], e2(x) is expressed as:

e2ðxÞ ¼4p2e2

3m2x2

Pl;n

ZBZ

2

ð2pÞ3d3kjPnlj2 � d½ElðkÞ � EnðkÞ � �hx� ð2Þ

Fig. 8. The total density of states of NaCaN and ZnSe compounds under hydrostatic pressure.

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

e1ðxÞ ¼ 1þ 2p

Z x

0

x0e2ðx0Þx02 �x2 dx0 ð3Þ

The calculated real e1(x) and imaginary e2(x) parts of the dielectric function of NaMgAs and itsanalogue ZnTe are shown in Fig. 7. The imaginary part e2(x) is obtained directly from the electronicstructure calculations. From the result, the real part e1(x) is determined using the Kramers Kronig dis-persion relation. At first glance, the dielectric functions of NaMgAs and ZnTe are almost similar. Theoptical gap value and the optical inter-band transitions are very similar. In the 0–10 eV photon energyrange, there are the same peaks of e2(x) for NaMgAs and ZnTe represented in Fig. 7 which can be as-signed to the same transitions of the specific points in the BZ. At zero frequency, our calculated e1(x)for NaMgAs is equals 6.746 as shown in Table 3, which is well comparable to the ZnTe value 7.358.These results provide another voice promising researchers to focus their attention on these com-pounds for future optoelectronic applications.

The application of hydrostatic pressure as an external control parameter contributes to the coher-ent understanding of the evolution of different properties under pressure in half-Heusler compoundsand II–VI semiconductors. According to our literature research, there is no experimental or theoreticalreference on the pressure effect on the electronic structure of this series of compounds. This is whatour work is considered as a first evaluation. We have applied also an hydrostatic pressure only on theten selected candidates and we have checked the effect of this stress on the electronic properties.Among these ten candidate materials, we have chosen the NaMgAs compound as a prototype materialcompared to ZnTe semiconductor as it is shown in Fig. 8. From this figure, the contributions of statesremain almost unchanged under hydrostatic pressure with a slight change in energy positions of d-orbitals. The topological similarity between the NaMgAs and ZnTe is permanent until the pressure

Fig. 9. Fundamental energy gaps of NaMgAs and ZnTe compounds versus the pressure.

Fig. 10. Pressure coefficients and the optical properties of I–II–V and II–VI compounds.

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20 GPa, with a significant change in the polarity of ZnTe compared to that of the NaMgAs. We noticethat, Zn and Te valence bands become wider under pressure.

From Fig. 9, an electronic transition is localized. At a given pressure, the direct band gap C ? C be-comes indirect C ? X in both compounds ZnTe and NaMgAs. The transition pressure of the NaMgAs isabout 8.7 GPa, while it is of about 15.2 GPa for ZnTe. This result proves that NaMgAs is more sensitiveto hydrostatic pressure than the ZnTe.

We have calculated the pressure coefficients from the following equation:

EgðPÞ ¼ Egð0Þ þ aP0 þ bP00

Eg(P) represents the energy gap at a defined pressure.Eg(0) represents the energy gap at zero pressure.P0 and P0 0: the first and second pressure derivative with respect to the energy gap.a and b: pressure coefficients.

These coefficients transmit the response of the band structure of a compound under pressure.The effect of hydrostatic pressure on the optical properties is also weak (see Table 3). The previous

figure (Fig. 7) shows that from 15 GPa the optical properties and in particular, the dielectric functionwill reach their maximum values. We have also presented in the Fig. 10 the pressure coefficients, thedielectric functions and the refractive index of the ten half-Heusler with their analogue II–VI semicon-ductors. We note that only three half-Heusler compounds LiCaN, NaCaN and NaMgAs have the samecharacteristics comparing with II–VI semiconductors ZnS, CdS, and ZnTe, respectively and they may begood candidates for optoelectronic applications.

4. Conclusion

To conclude, we have carried out DFT-based first principles calculations to investigate the elec-tronic structure of XYZ half Heusler compounds in comparison to II–VI semiconductors. The calcula-tions make predictions about physical properties that can be obtained from the electronic bandstructure, density of state, charge density and optical properties. From the similarity in all obtainedphysical properties which are considered as basic criteria for the comparison, we have succeededto found new half-Heusler semi-conductors for optoelectronic applications. So, we select threecompounds (LiCaN, NaCaN and NaMgAs) among all materials studied in this work which fulfill these

784 H. Mehnane et al. / Superlattices and Microstructures 51 (2012) 772–784

criteria. Future experimental work aimed to use this series of materials as semi-conductors promisingfor optoelectronic devices should benefit from the results presented here.

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