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Journal of Alloys and Compounds 439 (2007) 37–54

Electronic structure and chemical bonding in half-Heusler phases

Laila Offernes ∗, P. Ravindran, A. KjekshusDepartment of Chemistry, University of Oslo, PO Box 1033, Blindern, N-0315 Oslo, Norway

Received 16 August 2006; accepted 30 August 2006Available online 23 October 2006

bstract

The electronic structure and chemical bonding in half-Heusler phases have been systematically investigated using first-principles, self-consistentight-binding linear-muffin-tin-orbital calculations within the atomic-sphere approximation (TB-LMTO-ASA). The density-of-states profiles forhe mainstream half-Heusler phases exhibit very similar features originating from the cubic, positional-parameter-free AlLiSi-type structuralrrangement and resemblances in composition. The electronic structures of these half-Heusler phases are accordingly very suitable for rigid-and considerations and predictions made on this basis are tested against actually calculated data. The nature of the chemical bonding haseen systematically explored for the large transition-metal branch of the half-Heusler family using density-of-states, charge-density, charge-ransfer, electron-localization-function, and crystal-orbital-Hamilton-population plots. The study has laid stress on the remarkable consistency inonding behaviour among the considered intermetallic phases even though the properties range from non-magnetic metals and semiconductors viaerromagnetic metals to half-metallic ferromagnetic metals, brought about by large spread in element combinations and valence-electron content.

he typical half-Heusler phase exhibits an appreciable covalent contribution to the bonding independent of the electronic state at the Fermi-level.he empirical rule that the (numerous) half-Heusler phases with valence-electron content of 18 are semiconducting and more stable than those

metallic) with a higher or lower valence-electron content is substantiated.2006 Elsevier B.V. All rights reserved.

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eywords: Half-Heusler phases; Intermetallics; Electronic band structure; ELF

. Introduction

During earlier studies [1–3] of the half-Heusler phasesuMnSb and AuMnSn, a number of interesting features came tour attention. In this and a couple of forthcoming papers, we willddress some of these aspects, the present contribution focusingn electronic structure and chemical bonding in the half-Heusleramily. AuMnSb and AuMnSn are metallic, soft ferromagnetsith large magnetic moments and poor electric conductivity.tructurally these phases belong to the large Heusler family,hich comprises the so-called full-Heusler (X2YZ) phases in

ddition to the half-Heusler (XYZ) variants. There is no sin-le set of properties that characterizes the entire Heusler family.

owever, from a structure-chemical point of view the Heusler

amily are described by only two variables, viz. compositionelement combination and to some extent stoichiometry) and the

∗ Corresponding author. Tel.: +47 22 85 55 60; fax: +47 22 85 54 41.E-mail addresses: laila.offernes@kjemi.uio.no (L. Offernes),

rne.kjekshus@kjemi.uio.no (A. Kjekshus).

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925-8388/$ – see front matter © 2006 Elsevier B.V. All rights reserved.oi:10.1016/j.jallcom.2006.08.316

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ubic lattice parameter. The Heusler family trademark is a sim-le structural framework that can accommodate a vast amount ofifferent element combinations, resulting in phases with a largeiversity in physical properties. In the half-Heusler phases, X isypically a heavy transition metal (T), Y is a light transition metalr a rare-earth metal (R; this branch of the family will, however,nly be briefly considered here), while Z is a late main-group ele-ent (M), most frequently Sb or Sn. The resulting total number

f valence electrons per formula unit, referred to as the valence-lectron content (VEC), varies to a great extent. Fig. 1 illustrateshis variation and gives an impression of the multitude of knownalf-Heusler phases. Gold-containing phases only constitute amall branch of the large half-Heusler family so the scope of thenvestigation was extended to most of the half-Heusler family.

The crystal structure of the half-Heusler phases is of thelLiSi type (Fig. 2(a); space group F43m; see Ref. [2]), which

an be regarded as an ordered ternary version of the CaF2-type

tructure. Referring to the general formulae XYZ, X takes aoordination number of 8 (four X–Y bonds in tetrahedral config-ration and another four X–Z bonds in identical configuration).he coordination numbers of Y and Z are 10, both with capped

38 L. Offernes et al. / Journal of Alloys an

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ig. 1. Histogram for reported half-Heusler phases as a function of valence-lectron content (VEC). The total number of phases included in this represen-ation is 102.

by 4 X) octahedral geometry, amounting to 4 Y–X and 6 Y–Zonds and 4 Z–X and 6 Z–Y bonds, respectively. The structureas as mentioned no variable positional parameters, leaving theully ordered, stoichiometric XYZ phases with the lattice param-ter, a, as the only structural variable.

The other family branch (viz. the X2YZ full-Heusler phases,amed after German chemist Fritz Heusler) is probably still bestnown for the appearance of ferromagnetism in phases of tradi-ionally non-magnetic elements [4] and localized magnetism in

etallic phases [5]. However, full-Heusler phases are nowadaystudied with great interest owing to potential utilization in giantagnetoresistance (GMR) [6] and magnetic shape memory [7]

evices. The full-Heusler alloys take the Cu2MnAl-type struc-ure (Fig. 2(c)) from which the AlLiSi type formally derivesy leaving half of the X sub-lattice empty in an ordered pat-

ern. The, thus, created voids in the structure give rise to lessrbital overlap and consequently to a larger degree of local-zation and presence of gaps in the density-of-states (DOS).hese distinctions give the half-Heusler phases electronic and

asas

ig. 2. (a) One unit cell of the AlLiSi-type structure adopted by the half-Heusler phaepresents the section for which computational results are presented in following illusabelled according to the general designations X, Y, and Z used in the text.

d Compounds 439 (2007) 37–54

agnetic properties different from the full-Heusler phases. Thelectronic structure of the half-Heusler phases varies somewhatystematic as manifested by changes from metallic to semicon-ucting, through metallic to half-metallic (see below), and backo metallic behaviour as VEC increases [8,9]. The semiconduct-ng characteristics are associated with VEC = 18, and as seenrom Fig. 1, VEC = 18 is a highly preferred configuration, overthird of the half-Heusler phases belonging to this category.

In general the phases with VEC = 18 are either narrow-gapemiconductors or semimetals (defined as materials with lowumber of electrons at the Fermi-level, N(EF)). Phases withEC = 18 have been investigated for potential use in thermo-

lectric devices [10] and some of the R-containing phases areound to display GMR features [11]. So-called half-metallic fer-omagnetic (HMF) materials are found among the phases withEC = 22 [12]. In the HMF phases, the majority-spin channel

xhibits metallic characteristics, while the minority-spin channelxposes a semiconductor-like gap at the Fermi-level (EF). Thishould theoretically result in 100% spin-polarized electroniconduction [13–15]. Highly spin-polarized materials, like theMFs, are technological important in the growing field of spin-

ronics since on adding the spin degree of freedom the data pro-essing speed is imagined increased by orders of magnitude, thelectric power consumption correspondingly decreased, and theverall size of electronic devices can maintain the continuouslyhrinking trend [14,16,17]. HMF materials are, e.g., incorpo-ated in magnetic multilayers, which, due to the spin-dependentcattering of electrons, exhibit GMR. These multilayers are con-idered important in the development of magnetic recordingechnology. Some of the HMF half-Heusler phases also shownteresting magneto-optical properties, like the large magneto-ptical Kerr effect (MOKE) found for PtMnSb [18]. Materialsith high MOKE are used in read/write applications in data-

torage technology [19].Several of the ferromagnetic half-Heusler phases with VEC

round 22 are not half-metallic, but exhibit a gap in the minority-pin channel in the vicinity of EF. Some of these phases also havenarrow solid-solution range [1,20–22], which does not neces-

arily include the exact 1:1:1 stoichiometry as, e.g., established

ses. (b) Extract of the (1 1 0) plane in the AlLiSi-type structure. The dashed cuttrations. (c) The Cu2MnAl-type structure of the full-Heusler phases. Atoms are

L. Offernes et al. / Journal of Alloys an

Fig. 3. Histogram for reported half-Heusler phases, which contain 3d elements.Note that a phase which contains two 3d elements, e.g., NiMnSb will appearti

ftsdt

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bRbotdotefiaaeptimlb

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lfawtIreported). Fig. 4 shows that the phases with VEC = 18 or 22

wice in this representation. The total number of unique phases included in thellustration is 74.

or AuMnSn and AuMnSb [1,20]. However, since only a few ofhe ternary systems which contain half-Heusler phases has beenystematically mapped in the form of phase diagrams, a furtheriscussion of trends in non-stoichiometry is postponed to one ofhe forthcoming papers [23].

The uncovering of interesting magnetic properties has puthe focus on half-Heusler phases, which contain 3d elements.ig. 3 shows how the present knowledge distributes such phaseslong the 3d series. More than half of them contain either Mn ori. Especially, the Mn-based full- and half-Heusler phases are

onsidered to be true local moment phases with the magneticoment essentially confined to the Mn sites of the structure

24–26]. The occurrence of the unoccupied sites in the structuref the half-Heusler phases relative to that of the full-Heuslerhases gives rise to narrower bands and enhances the local-zed character of the magnetic moments. In fact, the magnetic

oments in these Mn-based ferromagnetic half-Heusler phasesre approximately integers, specified by the excess amount oflectrons (VEC − 18) compared with the 18-valence-electronemiconductors. In view of the large separation of the magnetic

n(Y) atoms (Mn–Mn distances exceeding 4 A), the ferromag-etism is thought to be established through indirect exchangenteractions, rather than originating from direct d–d overlap25] between Mn atoms. It has been suggested that the mag-etic coupling in the full- and half-Heusler alloys is domi-ated by Ruderman–Kittel–Kasuya–Yoshida-like (RKKY [27])xchange interaction between localized moments and itinerantlectrons [28], in which the magnetic information is transferredhrough local spin polarization of the conduction electrons. Theesulting co-operative magnetic state should then exhibit ferro-agnetic or antiferromagnetic alignment of the moments largely

epending on the interatomic distances; a consequence of oscil-

ating variations in the conduction-electron density. Other com-eting exchange interactions may, e.g., involve superexchangehrough the s and p electrons of the Z atoms, RKKY indirect s–d

(tw

d Compounds 439 (2007) 37–54 39

xchange, and d–d exchange through polarization and interbandixing (hybridization) [29–32].The bonding situation in the Heusler family has certainly

een discussed in the literature on earlier occasions (see, e.g.,efs. [6,12,29,33]), but the focus has mainly been targeted on theand gaps in semiconducting and HMF phases. The occurrencef semiconducting phases with VEC = 18 has been explained byhe “18-electron rule” using an ionic electron counting proce-ure [33]. The Z atoms constitute the most electronegative partf the half-Heusler phases, followed by the X atoms, leavinghe Y element as the electron donating constituent. The formallectron accounting scheme leads to the hypothetic (ionic) con-gurations X−a(d10), Y+b(d0), and Z−c(s2p6), where trivially+ c = b. However, this and similar approaches do not conveyny information about the bonding situation in the material, thelectron counts gathered are merely useful for accounting pur-oses. All experimental and “theoretical” findings suggest thathe half-Heusler phases are largely covalent bonded. The gapn the minority-spin channel in the HMF phases are, e.g., com-only accepted as an effect of covalent hybridization, which

eads to bonding and antibonding minority-spin states separatedy a gap [29].

The bonding situation in intermetallic materials is messynd the field is in need of new impulses [34]. The traditionalmpirical-based approach has made use of concepts like elec-ronegativity and atomic size. The elements, which enter aypical half-Heusler phase belong to a fairly narrow range inhe electronegativity scale where the tabulated values moreoverre burdened with various uncertainties. Strict electronegativityonsiderations must accordingly be considered unproductive inhis case. The simple rigid-sphere packing model is not appli-able for the half-Heusler phases either. As evident from Fig. 2,he (position-parameter-free) AlLiSi-type structure requires thathe X–Y and X–Z bond distances are equal, which in turn wouldmply that rigid-sphere Y and Z objects should have identicalize. Such a strict size criterion can obviously not be fulfilled forll element combinations, which occur within the half-Heusleramily. At least one of the atomic constituents is, therefore,equired to exhibit a certain diffuse and polarizable character,hich will allow the atom in question to enter the structural

ramework as a non-spherical object. The Z constituent (lateain-group element) of the half-Heusler phases will probably

e the softer partner in this case. Although meaningful bond-ength data cannot be extracted and evaluated for individualalf-Heusler phases, it is still possible to obtain qualitative infor-ation from trends in the structure data.Fig. 4 depicts the relationship between experimentally estab-

ished interatomic X–Y (= X–Z) distances and the Z constitutentor series of half-Heusler phases with Z = Sn and Sb. Note thatll phases with available and relevant data are treated and thate have taken the liberty to include a few phases with Z = Ga

o indicate the continuation of the relationships to Z from groupII of the periodic table (no half-Heusler phases with Z = In are

viz. representatives with filled or half-filled orbital configura-ion) exhibit a relatively shorter X–Y bond length than thoseith VEC �= 18 or 22. Comparing the four series in the lower

40 L. Offernes et al. / Journal of Alloys an

Fig. 4. The X–Y = X–Z bond length (and lattice parameter a) as function of Zcomponent for series of half-Heusler phases. Legends to the symbols used todistinguish the different series are shown on the illustration. The main purposeof the illustration is to visualize that phases with VEC = 18 or 22 (emphasizedbe

peaFtfitdcptSipSewfls

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2

tgra

t(totsAatmBsoftuepotdt(fcmsgdif[

lT[tptcdethe electron redistribution in the crystal lattice (compared to

y thicker solid lines) has a relatively shorter X–Y bond length than the trendstablished for phases with VEC �= 18 or 22.

art of the illustration it is seen that there occurs a relative short-ning of the X–Y bond on going from Sn to Sb in the NbCoZnd NiTiZ series (with VEC = 18 at Z = Sn) and the CoTiZ andeVZ series (with VEC = 18 at Z = Sb). This is as expected from

he shrinkage in size introduced by shell contraction on electronlling. However, the fact that the shrinkage is much smaller in

he two former series (than in the two latter series) suggest aecrease in bond strength on going from the Sn phases to theorresponding Sb phases. In the AuMnZ series (see the upperart of Fig. 4) this behaviour is manifested even clearer sincehe X–Y bond length increases from AuMnSn (VEC = 22) to itsb counterpart. In line with the above reasoning this behaviour

s not seen for the PtMnZ and PdMnZ phases where the Sbhases (VEC = 22) has the shorter bond length. Likewise, then phases of the IrMnZ, TiRhZ, CuMgZ, and PtMnZ seriesxhibit similar bond lengths (VEC �= 18 or 22 for all phases),hereas the Sb phases of the three latter series (VEC = 18

or TiRhSb and CuMgSb; VEC = 22 for PtMnSb) show aarger contraction in bond length than found for the IrMnZeries.

In the rest of this paper, we will explore connections betweenhysical properties and bonding by means of density-functional-heory (DFT) calculations for “typical” XYZ phases with vary-ng VEC. This paper is sectioned so that, after a brief outline ofhe computational details, results and discussion are presentedogether. First, a general overview of the DOS are given for allhases. Then, we turn to the findings for a typical VEC = 18epresentative where the different examination tools are alsoriefly described. In order to evaluate the bonding interactions,alence-charge-density analyses have been preformed for differ-nt crystal planes as well as for the three-dimensional structuralrrangement. Since the (1 1 0) plane of the AlLiSi-type struc-

ure conveniently cuts through the centres of the X, Y, and Zonstituents (Fig. 2(b)) most analyses will be presented for thislane only. Finally, a full comparison of phases with different

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d Compounds 439 (2007) 37–54

EC is attempted, with emphasis on differences and similarities,hich have potential impact on bonding.

. Computational details

The density-functional-theory (DFT) calculations of the elec-ronic structure are performed within the framework of theeneralized-gradient approximation (GGA; with exchange cor-elation according to Perdew et al. [35]) and the local-densitypproximation (LDA).

First-principles, self-consistent, tight-binding linear-muffin-in-orbital calculations within the atomic sphere approximationTB-LMTO-ASA) [36] were performed for all phases subjectedo this study. These calculations are semi-relativistic (i.e., with-ut spin–orbit coupling, but all other relativistic effects included)aking also into account combined correction terms. The basisets consisted of 6s, 6p, and 5d orbitals for 5d elements such asu and Pt, 4s, 4p, and 3d orbitals for 3d elements such as Mn

nd Ni, and 5s, 5p, and 5d orbitals for Sb and Sn. The integra-ion over the Brillouin zone (BZ) was made by the tetrahedron

ethod, sampling a grid of 245 k points in the irreducible part ofZ (4096 in the whole zone). The crystal lattice is divided into

pace-filling, slightly overlapping spheres centred on each of theccupied atomic sites. An empty sphere is included at the siteormally vacated on going from the Cu2MnAl- to the AlLiSi-ype structure (Fig. 2(a and c)). The Wigner–Seitz-sphere radiised are scaled so that the sum of the volume of all the spheresquals the volume of the unit cell. The experimental latticearameters listed in Table 1 are used in the calculations. Volumeptimization was performed for some of the main phases, buthe outcome of these calculations did not indicate any significanteviation (actual deviations ranging between 0.8 and 1.2%) fromhe experimental values. Full potential linear-muffin-tin-orbitalFLMTO) band-structure calculations [44,45] were performedor selected phases. The FLMTO method includes spin–orbitoupling and is evidently more accurate than the TB-LMTOethod, but although there are certain numeric differences and

mall shifts in energy, the DOS obtained by the former methodave no additional, qualitative information which justifies aetailed account in this report. All band structures documentedn the succeeding section refer to the TB-LMTO calculations. Aew results from the FLMTO calculations are reported in Ref.13].

Crystal-orbital-Hamilton population (COHP) and electron-ocalization-function (ELF) plots are calculated according to theB-LMTO code as implemented in the TBLMTO-47 package

46]. Charge-density (CD) analyses have been performed forhe three-dimensional unit cell as well as for different crystallanes. The analyses include charge transfer (CT), which ishe self-consistent valence-electron density of the phase underonsideration in a particular region, minus the valence-electronensity which the corresponding free atom would havexhibited in the same region. This enables visualization of

he electron distribution of the involved free atoms placed onhe same locations) caused by the bonding interactions. Theresent analyses also included the spin-resolved valence-charge

L. Offernes et al. / Journal of Alloys and Compounds 439 (2007) 37–54 41

Table 1Experimental lattice parameter (quoted from Refs. [1,2,8,9,37–39]) electronic classification according to calculationsa, and calculated and experimental (quoted fromRefs. [1,2,8,9,38,40–43]) magnetic moments

VEC Phase XYZ a (A) Classification Moment (μB, f.u.−1) ICOHP Optimum COHP ELF

Calculated Experimental X–Y X–Z Y–Z Location (eV) VECmismatch

X–Y [X–Z]

8 MgLiSb 6.62 SC 0 – 0.22 1.70 0.18 0 – [0.88]

16 FeTiSn 6.056 MET ∼10−4 – 1.00 0.92 0.43 +0.38 ∼2 0.56

17 FeTiSb 5.997 F 0.96 – 1.04 1.01 0.48 +0.03 ∼0.2 0.58CoTiSn 5.957 MET ∼10−4 – 1.00 0.90 0.44 +0.17 ∼1 0.54

18 NiTiSn 5.941 SC 0 0 1.68 1.72 0.95 0 – 0.51CoVSn 5.98 SC 0 0 0.95 0.86 0.46 0 – –CoTiSb 5.832 SC 0 0 1.07 1.03 0.51 0 – 0.58CuMgSb 6.164 M 0 0 0.39 0.86 0.35 0 – 0.46ZnLiN 4.877 SC 0 0 0.05 0.06 1.19 0 – 0.27

19 NiTiSb 5.872 MET ∼10−6 – 0.97 0.87 0.52 −0.80 ∼1 0.49CoVSb 5.766 MET ∼10−6 0.18 1.16 1.01 0.54 −0.25 ∼1 –IrMnAl 5.992 F 1.15 0.4 1.28 0.96 0.29 +0.40 ∼1.5 0.48 [0.70]

20 IrMnSn 6.182 F 3.44 2.25 1.16 1.04 0.35 +0.45 ∼1 0.45

21 IrMnSb 6.164 F 3.08 3.1 1.21 1.10 0.40 +0.15 ∼0.2 0.49RhMnSb 6.145 F 3.27 3.35 1.01 0.95 0.47 +0.32 ∼0.5 –PtMnSn 6.264 F 3.53 3.37 1.04 0.97 0.39 +0.45 ∼0.5 –

22 NiMnSb 5.909 HMF 3.91b 3.65 0.86 0.96 0.53 0 – 0.45PtMnSb 6.201 HMF 4.00 4.14 1.05 1.01 0.41 0 – 0.45AuMnSn 6.323 F 4.00 3.8 0.86 0.88 0.39 +0.01 ∼0.2 0.40

23 AuMnSb 6.379 F 4.56b 4.2 0.78 0.85 0.38 −0.47 ∼0.5 0.41CuMnSb 6.088 AFc – 0 (AF) 0.56 0.79 0.52 −0.59 ∼1 –

Calculated values are obtained by the TB-LMTO package. ICOHP values are specified for interaction between all atom pairs together with the deviation from the“optimum” COHP and a crude estimate of the corresponding VEC mismatch (from DOS integration; in parenthesis). The ELF column gives the highest ELF for theappropriate attractor.

agnet

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a Abbreviated as SC, semiconductor; MET, metal; HMF, half-metallic ferromb From FLMTO calculations.c From exp. only [39].

ensity. All these tools have been used to explore the bondingroperties.

. Results and discussion

.1. Electronic structure—density-of-states

Spin-polarized DOS were calculated for about 30 half-eusler phases with VEC between 16 and 23, the number ofhases for each VEC reflecting the shape of the histogram inig. 1. To facilitate comparisons and detection of trends wehose phases with similar element combinations for the DOSnalyses (Table 1). Since phases with the same VEC turnedut to give similar DOS profiles, the following phases are pre-ented as representative for our findings (VEC in parenthesis):eTiSn (16), FeTiSb (17), CoTiSn (17), CoTiSb (18), NiTiSn18), NiTiSb (19), IrMnSn (20), IrMnSb (21), PtMnSn (21),tMnSb (22), AuMnSn (22), and AuMnSb (23). Fig. 5 showschematic DOSs for VEC values ranging from 16 to 23, while

ig. 6 shows the actual TB-LMTO-calculated total DOS forome of the just mentioned phases. Total and partial DOSs foriTiSn (18) and AuMnSn (22) are presented in Fig. 7. A fewhases that should be considered somewhat different, namely

h

fb

; F, ferromagnet; AF, antiferromagnet.

gLiSb (8), ZnLiN (18), CuMgSb (18), and IrMnAl (19), havelso been investigated and the finding for these will be discussedeparately (Fig. 8).

Generally, the DOSs in Figs. 6 and 7 show very similar fea-ures as are to be expected because of the identical (AlLiSi-type;ee above) structural arrangement and the resemblances in chem-cal composition (please note that the phases presented here astypical”, make up only one branch of the half-Heusler fam-ly). The DOS profiles have peaked features, with valleys andseudo-gaps. The s states are mainly originating from Z and lieow in energy (down to about −12 eV for the Sb phases and downo about −10 eV for the Sn phases). The s states are separatedrom the p and d states by an energy gap of 1–4 eV. The there-fter following band is essentially composed of X-d states withome admixture of Z-p and Y-d states. The Y-d states are usu-lly (depending on VEC) found above EF. For phases with onlymall or no spin polarization, the Y-d states are mainly unoccu-ied, while for the electron-rich, highly spin-polarized phaseshe majority Y-d states are found below EF, well separated from

igher-lying minority Y-d states.

The semiconducting VEC = 18 configuration is highly pre-erred (see Fig. 1). This is to be expected since filling of theands up to the gap which then separates filled and empty states

42 L. Offernes et al. / Journal of Alloys and Compounds 439 (2007) 37–54

Fig. 5. Schematic partial density-of-states (DOS) diagrams for half-Heusler phases with different valence-electron content (VEC). State-character labels are indicatedon the illustration. (a) Weakly spin-polarized DOSs for VEC = 16 or 17. EF is indicated by the vertical dotted line. (b) Non-spin-polarized DOSs for VEC = 17,1 ctiveli

(2bstpbdlticdraTc

tdTtaascaieao

8, and 19, EF being indicated by dotted, solid, and dashed vertical line, respendicated by dotted, solid, and dashed vertical line, respectively.

Eg) corresponds to a closed-shell configuration with exactlyelectrons in the s band, 6 in the p bands and 10 in the d

ands. Filling of all bonding states while leaving all antibondingtates empty, separates bonding and anti-bonding states, lowershe energy and adds stability. On inspection of the DOS for ahase with VEC = 18, e.g., NiTiSn (Fig. 7(a)), some featuresecome evident. This phase is in a non-spin-polarized semicon-ucting state. The s states, with principally Sn character, layow in energy (down to about −10 eV) and are separated fromhe p and d states by a gap of more than 2 eV. The delocal-zed electrons of the p states also have a large degree of Snharacter and extend up to about −5 eV. The more localizedstates have mainly Ni character and are also found in this

ange, the eg and t2g states are separated by a pseudo-gap atround −2 eV. These states are separated from the unoccupiedi-d states by a band gap Eg ≈ 0.5 eV (recalling that TB-LMTO-alculated band gaps are likely to be underestimated compared to

aptH

y. (c) Strongly spin-polarized DOSs for VEC = 20 or 21, 22, and 23, EF being

he actual Eg value [47]). The partial DOSs in Fig. 7(a) unveil aefinite degree of d–d hybridized interaction between Ni andi, suggesting covalent-like bonding along the 〈1 1 1〉 direc-

ions of the structure. The extent of this hybridization variesccording to the energy separation between the d states of thetoms in question, e.g., in the CoTiSb phase (and even moreo in the CoVSn phase) the d states of the two 3d elementsomes energetically closer together than in the NiTiSn phasend, in turn, this gives rise to a larger degree of d–d mixingn CoTiSb than NiTiSn (see also Section 3.2). As mentionedarlier the VEC = 18 configuration can formally be imagined asrising on complete filling of the d orbitals of X and s and prbitals of Z and this situation is schematically illustrated with

n idealized DOS in Fig. 5(b). In this simple ionic-motivatedicture, the Y component poses as cation, donating electronso the more electronegative X and Z constituents. In the half-eusler phases, the bonding is certainly not pure ionic, but an

L. Offernes et al. / Journal of Alloys and Compounds 439 (2007) 37–54 43

Fig. 6. Spin-polarized total density-of-states (DOS) for selected phases withVEC ranging between 16 and 23 (VEC value in parenthesis). Schematic ver-sions of these DOS profiles are found in Fig. 5. Profile a corresponds to Fig. 5(a),profiles b, c, and d to Fig. 5(b) with dotted, solid, and dashed marking of EF,ria

it

VTthcFVae

Faar

Fig. 8. Total and partial spin-polarized density-of-states (DOS) for (a) MgLiSb;(b) ZnLiN; (c) CuMgSb; (d) IrMnAl; VEC values in parenthesis. Total DOSprofiles are marked by solid lines, while X, Y, and Z partial DOS profiles aremdi

eOrsaersabNff

espectively, profiles e/f, g, and h to Fig. 5(c) with dotted, solid, and dashed mark-ng of EF, respectively. Locations of EF according to the theoretical calculationsre indicated by dashed vertical lines.

onic description can still be used as a simple electron countingool.

The half-Heusler structure does allow deviations from theEC = 18 configuration. For the phases where X and Y both areelements the VEC mainly varies between 16 and 23. According

o the simple ionic description VEC = 23 corresponds to filling ofalf of the d orbitals of the Y constituent, giving rise to electrononfigurations which exhibit co-operative magnetic properties.

or the phases where X is a T element, while Y is a R element,EC mainly varies between 25 and 32, which corresponds tovariation between half-filled and filled f orbitals for the R

lement. Concentrating on the phases where X and Y both are T

ig. 7. Total and partial spin-polarized density-of-states (DOS) for (a) NiTiSnnd (b) AuMnSb. Total DOS profiles are shown by solid lines, while X, Y,nd Z partial DOS profiles are marked by dashed, dot-dashed, and dotted lines,espectively. EF is marked by dashed vertical lines.

wt[bponmS1iaooaTtlt

omst

arked by dashed, dot-dashed, and dotted lines, respectively. EF is marked byashed vertical lines. Note that the scale on the DOS axis varies between thellustrations.

lements, two possibilities should be considered for VEC �= 18.ne can adopt a non-spin-polarized rigid-band approach and

etain the basic DOS for the VEC = 18 situation (viz. the valenceaturated semiconductor case), but accept degrees of filling thatre insufficient (VEC = 16 or 17; Fig. 5(b), vertical dotted line) orxcessive (VEC = 19 or 20; Fig. 5(b), vertical dashed line). Theesulting phases will be non-magnetic and metallic, as demon-trated by the calculated DOS properties for FeTiSn (Fig. 6(a))nd CoTiSn (not shown). These phases will balance on the borderetween magnetic and non-magnetic variants, and depending on(EF) these phases will be highly influenced by any deviation

rom the ideal 1:1:1 stoichiometry or atomic disorder. FeTiSn isound experimentally to be non-magnetic [9] in accordanceith the calculation whereas CoTiSn is experimentally reported

o be weakly ferromagnetic with a magnetic moment of 0.35μB48] (more accurate magnetic data for this phase can be obtainedy full potential calculations). The latter phase may, therefore,rovide an example of the situation that arises when the fillingf DOS according to the rigid-band approach results in a largeumber of electrons at the Fermi-level. Such a situation shouldake a spin-polarized configuration preferred according to thetoner criterion [49,50]. For the phases with VEC lower than8, exchange interaction only results in minor energy shifts, asllustrated by, e.g., the calculated DOS for FeTiSb in Fig. 6(b),nd depicted schematically in Fig. 5(a). The spin polarizationf this phase gives rise to a calculated total magnetic momentf 0.96μB per formula unit and the local magnetic momentst the Fe(X) and Ti(Y) sites are 0.81 and 0.10μB, respectively.he VEC < 18 situation is accordingly different from that for

he VEC > 18 phases (see below) where magnetic moments areargely confined to the Y site. FeTiSb is experimentally reportedo be a metallic ferromagnet with a low magnetic moment [9].

For most phases with VEC > 18 the splitting of the Y-d

rbitals is complete (see Fig. 6(d–g)) giving rise to large localagnetic moments associated with the Y atom. For the minority-

pin electrons a gap opens up in the vicinity of EF and belowhis gap nine electrons will fill exactly one s, three p, and five d

4 ys an

baetnFtas

mpHbsmloVtaoAsebaepamios4lomtmvfdtsmst

HVmErmtc

lsp(MitahlaresocsCbatlws

mon(sTaI(tiacsobrv

dwaCNOsti

4 L. Offernes et al. / Journal of Allo

ands in the minority-spin channel. Corresponding orbitals arelso accommodated in the majority-spin channel, and additionallectrons can enter the majority Y-d band. As a rule of thumb,he orbital splitting of the Y-d states results in spin-only mag-etic moments corresponding to about 2ST = VEC − 18 [33,51].or the special case of VEC = 22, EF is located at, or within,

he minority-spin gap, and several of the VEC = 22 phases areccordingly to be classified as HMF materials (schematicallyhown in Fig. 5(c), solid vertical line).

Of the VEC = 22 phases, PtMnSb and NiMnSb are com-only recognized as HMF phases, while another interesting

hase, AuMnSn, has been obtained recently by the authors [20].owever, AuMnSn is not a proper HMF phase since both tight-inding and full potential calculations establish that EF is locatedlightly below the gap in the minority-spin channel. This phaseust accordingly be classified as a ferromagnetic (F) metal-

ic phase with poor conductivity although the overall featuresf its electronic structure are similar to the other phases withEC = 22. In the partial DOS profiles of AuMnSn the s elec-

rons lay low in energy (as for the NiTiSn phase, see above),nd are separated from the p and d electrons by a band gapf more than 1 eV. The weakly spin-polarized d electrons ofu dominate the region between −6.5 and −3.5 eV and the

trongly spin-polarized d electrons of Mn dominate the highernergy region. The majority-spin d states are found in the regionetween about −3.5 and −1 eV, while the minority-spin d statesre found above EF from about 0.4–3 eV. Both the Au- and Mn-dnergy levels are well localized. Compared to the elemental fcchase of gold the bulk of the Au-d states for AuMnSn falls innarrower energy range (∼−7 to −5 eV) and these electronsust thus be regarded as more localized than those in gold. This

s also evident from the peaked DOS features and the separationf the eg and t2g states by a pseudo-gap (Fig. 7(b)). For Mn thetriking feature is the exchange splitting of the d states by aroundeV. In relation to the band filling the majority-spin states are

ocated well below EF, whereas most of the minority-spin statesccur above EF and remain empty. This leads to a calculatedagnetic moment of 4.00μB for AuMnSn, in accordance with

he prediction of the (VEC − 18) rule. The calculated magneticoment complies accordingly quite well with the experimental

alue (3.8μB [3]) in particular when attention is called on theact that the measurements were done on samples with someeviation from the 1:1:1 stoichiometry assumed in the calcula-ions. The Sn-s states are lowered in energy compared to theituation in the element Sn (�-modification; fcc-type structure;etal), and the characteristic higher-lying peaks attributed to p

tates in the DOS of �-Sn are completely lost and spread overhe whole energy range in the DOS of AuMnSn.

Using rigid-band reasoning with the VEC = 22 spin-polarizedMF case as a starting point, it follows that the phases withEC = 19, 20, and 21 should have EF below the gap in theinority-spin channel while phases with VEC = 23 should haveF above the gap (Fig. 5(c), dotted and dashed vertical line,

espectively). These phases are accordingly to be classified asetals. The actual calculations for IrMnAl (VEC = 19) show that

here is no proper gap in the minority-spin channel, a feature thatan be traced back to a lesser degree of hybridization (due to the

etSl

d Compounds 439 (2007) 37–54

ower polarizability of Al relative to Sn or Sb) and the corre-ponding limited splitting of the Mn states. However, a ratherrominent pseudo-gap is found at EF (Fig. 8(d)). In IrMnSnVEC = 20) and IrMnSb (VEC = 21) the exchange splitting of the

n states are significantly larger, and these phases exhibits a gapn the minority-spin channel 0.4 and 0.2 eV above EF, respec-ively. According to the (VEC − 18) rule, IrMnAl should havemagnetic moment of 1μB, while IrMnSn and IrMnSb shouldave moments of 2 and 3μB, respectively. The actual, calcu-ated values are 1.15, 3.45, and 3.08μB, respectively, implying

marked discrepancy with the predictions of the (VEC − 18)ule for IrMnSn. However, this discrepancy is not confirmedxperimentally, where IrMnSn samples (indeed with some non-toichiometry [38]) have been reported with a magnetic momentf 2.25μB. In general, structural disorder or deviation in stoi-hiometry can have large effects on physical properties and evenmall changes in the position of EF can alter N(EF) drastically.hanges in N(EF) obviously influence electrical conductivity,ut in spin-polarized cases also the magnetic properties can beltered appreciable and certain phases can go from magnetico non-magnetic and vice versa. Half-Heusler phases, which areocated close to a magnetic–non-magnetic boundary (viz. phasesith VEC = 16, 17, 19, and 20) will be especially sensitive to

uch imperfections [9].In the case of AuMnSb with VEC = 23, the gap in the

inority-spin channel lies 0.4 eV below EF. This leads to fillingf minority Y-d states, and thus to a reduction of the mag-etic moment from the expected value of 5μB according to theVEC − 18) rule. The calculated magnetic moment of 4.56μB istill somewhat higher than the experimental value of 4.2μB [1].his may be associated with the non-stoichiometry of the phasend/or the non-account of orbital moment in our calculations.f one goes from phases with VEC = 23 to those with VEC = 24e.g., by substitution) the expected gain in energy achieved byhe large exchange splitting and thereupon enhanced magneticnteraction now becomes lost. In other words, the extra electrondded to the minority channel lowers the magnetic moment andonsequently overrules the relative stability gained by exchangeplitting and magnetic interaction. There is, in fact, no reportsn any half-Heusler phases with VEC = 24. A calculation of theand structure for the hypothetical phase AuMnTe (VEC = 24)esult in a magnetic moment of about 4μB and a high N(EF)alue indicating instability of such a phase.

There are exceptions, which do not follow the above-escribed systematics, and a few representative phases (Fig. 8)ith anomalous behaviour are briefly considered below (see

lso Sections 3.2.1 and 3.2.2). Some VEC = 18 phases (e.g.,uMgSb) are found to be semimetals, viz. metals with low(EF). The calculated DOS for CuMgSb is shown in Fig. 8(c).n going from a heavy to a lighter T element on the X site, the

pin–orbit coupling gradually decreases [52], and exchange ofhe T-3d constituent on the Y site with an alkaline-earth elementntroduces a marked distinction between the X and Y atoms. The

ffect of these chemical encroachments is a substantial reduc-ion in the covalent component of the bonding interaction (seeection 3.2.1). The less pronounced covalent hybridization thus

eads to a more “smeared” electronic profile, resulting in a lower

ys an

dilf(sfiabspiiEihta

3

atamfmi

Hrltf

3e

(aewotbb

wb(Fcil

FfaEc

L. Offernes et al. / Journal of Allo

egree of localization and a loss of the band gap at EF. With moreonic character (found for, e.g., the ZnLiN phase; Fig. 8(b)) lowying DOS peaks can appear as a consequence of electron trans-er from the electropositive (here Li) to the more electronegativehere N) constituent. The DOS of MgLiSb, however, shows someurprising features (Fig. 8(a)). This phase has VEC = 8 and ful-ls the traditional octet rule. From the nature of the alkali andlkaline-earth elements one might expect a highly ionic phase,ut the DOS indicates a large degree of covalent character, aseen by the almost identical profile of the partial DOS of thearticipating atomic constituents (see Section 3.2.1). Accord-ng to the DOS profile (Fig. 8(d)) the VEC = 19 phase IrMnAls metallic with a pseudo-gap in the minority-spin channel atF. The similarity of the partial DOSs of the Ir and Mn atoms

ndicate X–Z hybridization (from −5.0 to EF) as for the typicalalf-Heusler phases discussed above, but in the case of IrMnAlhe partial DOS also indicate covalent Ir–Al and Mn–Al inter-ctions (see Section 3.2.1).

.2. Chemical bonding

The bonding characteristics of intermetallic phases are usu-lly rather complex, and to investigate the bonding behaviour ofhe half-Heusler phases several computational tools have beenpplied. (Note that some of these analyses simply represent

athematical transpositions of one given dataset into another

orm in order to visualize different aspects of the data.) Toake the presentation as lucid as possible each “tool” will be

ntroduced accompanied by the results for two selected half-

bCto

ig. 9. Maps of (a) charge density; (b) ELF (electron-localization function); and (c)or AuMnSn. All plots refer to the (1 1 0)-plane and for easy comparison the same scctual scales and cut-off values are chosen for clarity. The charge density scale in parLF scale in parts b and e are from 0.20 to 0.50 with 8 levels in each map. In parts contour lines, respectively. The locations of the X, Y, and Z atoms are the same as in

d Compounds 439 (2007) 37–54 45

eusler phases, viz. NiTiSn and AuMnSn, which can serve asepresentatives for the semiconducting (VEC = 18) and metal-ic phases, respectively. Thereafter, a more comprehensive andrend-focusing discussion of the findings for the half-Heusleramily will be conducted.

.2.1. Charge density, charge transfer, andlectron-localization function

The charge density (CD) for NiTiSn in the (1 1 0) planeFig. 9(a)) unveils significant amounts of charge between Ni(X)nd Ti(Y) as well as between Ni(X) and Sn(Z). In general we seelectron accumulations between X and Y and between X and Zhereas that between the Y and Z is comparably smaller. Theutermost electrons of the soft Z atom are somewhat polarizedoward Y. In Fig. 10(a) the three-dimensional CD is representedy an isosurface, the value of which being chosen to visualizeonding regions.

From the CD alone it is difficult to draw safe conclusions as tohether an accumulated charge between two atoms stems fromonding electrons. One approach is to turn to charge-transferCT) maps (see Fig. 9(c) for CT in the (1 1 0) plane of NiTiSn).ig. 9(c) shows that there has occurred a significant transfer ofharge from the region of the atomic spheres of X and Y to thenterstitial region between them, indicating a directional, cova-ent character of the X–Y bond. The added charge in the region

etween the Sn(Z) atoms is (as evident from three-dimensionalT representations) actually the middle of the horse-shaped fea-

ure between Ni(X) and the two out-of-plane Ti(Y) atoms. Nother intermediary atomic constellations have gathered charge

charge transfer for NiTiSn; (d) charge density; (e) ELF; and (f) charge transferales and number of contour lines are used in corresponding plots. Note that thets a and d are from 0.020 to 0.10 e a.u.−3 with 10 levels in each map, while theand f regions with positive and negative values are given with solid and dashedFig. 2(b).

46 L. Offernes et al. / Journal of Alloys and Compounds 439 (2007) 37–54

F (b) Ev scs pS

at

pfaqEitemgrim(paastaEtsbt[nEt0[trim

tuc

eticiE2iEiaoahiweic(fNbriaTmy

ig. 10. Three-dimensional isosurface representation of (a) charge density andalue of the ELF isosurface is 0.44 and the enclosed attractors are visible as din(Z) atoms are the same as in Fig. 2(b).

nd neither of the atomic sites have gained net charge relative tohe constituents in the free atomic form.

The electron-localization function (ELF) is a ground-stateroperty that is useful to visualize and distinguish between dif-erent bonding interactions in solids [53,54]. ELF is limited tovalue between 0 and 1 and for simple valence compounds theualitative interpretation of the ELF is often straightforward. AnLF value of 0.5 indicates that the Pauli repulsion at the point

n question is the same as that of uniform electron gas withhe same density [55]. Such intermediate ELF values will thus,.g., be found for regions with metallic bonding in typical Drudeetals. ELF should be close to one in regions ruled by light main-

roup elements with paired electron configurations (low Pauliepulsion) such as covalent bonds or lone pairs. Paired electronsn covalent bonds are expected to manifest themselves in ELF

aps as enclosed valence-electron basins with high ELF valuescalled attractors), symmetrically centred along the bond axis. Inolar-covalent materials the attractor will be positioned closer to,nd bent toward, the more electronegative atom. Ionic-like inter-ctions can be recognized through electron basins with roughlypherical distribution around the atomic cores, positioned sohat no attractor is found on the direct line between two inter-cting atoms. Bonding s and p orbitals in simple materials givesLF values close to one. (Typical maximum values for attrac-

ors in covalent-bonded organic molecules are 0.8 [53].) Theame qualitative picture of ELF holds for intermetallic phases,ut since the ELF is derived from an expression that varies withhe quantum number l, the ELF values diminish as l increases56]. Therefore, as a rule, higher-angular-momentum quantum-umber orbitals, such as d orbitals, tend to give rather indistinctLF characteristics, both with respect to attractor volume and

he numerical size of ELF (maximum attractor values of only.4–0.5 are found for polar covalent bonds involving 3d orbitals57]). In addition, intermetallic phases show an intimate mix-

ure of a wide variety of bonding forms (see, e.g., Ref. [34]),esulting in considerably smaller differences between extremesn terms of ELF. For intermetallic phases containing both late

ain-group and transition-metal constituents, such as found for

tArs

LF for NiTiSn. The value of the charge density isosurface is 0.37 e a.u.−3. Theositioned between the Ni and Ti atoms. The blacking of the Ni(X), Ti(Y), and

he typical members of the Heusler family, the higher ELF val-es are likely to be found in the region of the late main-grouponstituent, regardless of the bonding characteristics.

The ELF of NiTiSn in the (1 1 0) plane (Fig. 9(b)) shows thexpected large ELF level around the Sn site. Turning the atten-ion to the more interesting interstitial regions, the main features the attractor located between Ni(X) and Ti(Y) confirming aertain covalent character for this bond. The attractor has a max-mum ELF value of 0.51 (see, e.g., Ref. [57] for a discussion onLF values in transition-metal compounds) and located some0% closer to Ni(X) than Ti(Y). This is in line with the find-ngs from DOS. In the three-dimensional representation of theLF for NiTiSn (Fig. 10(b)), the value of the isosurface, 0.37,

s chosen to enclose and emphasize the Ni–Ti attractors (seens discs in between the atomic spheres). Insight into the naturef the Ni–Sn (X–Z) bond is harder to extract from ELF, sinceny attractors close to the Sn(Z) atom would be blurred by theigh ELF values associated with the atom itself. To avoid thisnterference the CD and ELF were plotted for certain energyindows only (Fig. 11), thus visualizing the situation by differ-

ntiating between electrons with different energies. In NiTiSn its mainly electrons with energy between −2.25 eV and EF thatontribute to the covalent bonding between Ni(X) and Ti(Y)Fig. 11(a and b)). This is in accordance with the expectationsrom DOS since this energy range comprises the main part of thei-d states (see Fig. 7(a)). The main charge distribution foundetween Ni(X) and Sn(Z) (Fig. 11(c)) is from the lower energyange from −5.10 to −2.25 eV, and the ELF map (Fig. 11(d))ndicates some polar-covalent character of this bond since anttractor bent toward Sn(Z) is located between the two atoms.he s electrons found in the energy range −10 and −7 eV areostly confined to the Sn atoms as expected from the DOS anal-

sis (Fig. 11(e and f)).The CD distribution of AuMnSn (Fig. 9(d)) is similar to

hat of NiTiSn, demonstrating finite charge distribution betweenu(X) and Mn(Y) as well as between Au(X) and Sn(Z). The cor-

esponding CT to the interstitial regions of Fig. 9(f) also showsimilar features to (c), but the magnitude of charge transferred

L. Offernes et al. / Journal of Alloys an

Fig. 11. Charge density and ELF maps for NiTiSn in selected energy windows:(a) −2.25 eV to EF; (b) −5.10 to −2.25 eV; (c) −10 to −7 eV. The topologicalpfa

iawtaiof

Vwmwptf

tataaltS

foAnpsbptVciiwZTfiptvdbtblbhmaoTe

3

devipTbt(

artitioning of each map is conveniently chosen to present the most prominenteatures in the respective energy windows. The locations of the X, Y, and Ztoms are the same as in Fig. 2(b).

s significantly less in AuMnSn than NiTiSn. These similaritiesnd distinctions are also reflected in the ELF map (Fig. 9(e))here the attractor found between Au and Mn is located along

he connecting line between the atomic-sphere domains. Thettractor is also in this case positioned closer to the Au(X) siten analogy with the findings for NiTiSn, but the maximum valuef ELF for AuMnSn is lower (0.40) than the corresponding valueor NiTiSn.

The larger ELF value at the acceptor maximum for theEC = 18 phase NiTiSn could at first glance be associatedith the semiconducting behaviour of NiTiSn compared to theetallic behaviour of AuMnSn (see Section 3.2.2). However,

hen the maxima in the attractor values for all half-Heuslerhases subject to this study are compared, it becomes clearhat the semiconducting behaviour is not solely responsibleor higher ELF attractor values. As can be seen from Table 1,

opit

d Compounds 439 (2007) 37–54 47

he phases with VEC = 16 or 17 have in general higher ELF-ttractor values and those with VEC > 18 have lower valueshan the VEC = 18 phases. Several factors like the electroneg-tivity difference between X and Y, the selection of the Ztom, and the lowering of ELF for constituents with highvalues, seem to be of some importance for evaluation of

he strength and character of the X–Y interaction (see alsoection 3.2.2).

The above view of the half-Heusler phases has somewhat dif-erent perspective from that seen from DOS (Fig. 8) and bringsut different features in terms of CD and ELF characteristics.

few exceptions from the general trends indicated above willow be considered in terms of ELF (Fig. 12). The VEC = 18hase CuMgSb behaves like the other half-Heusler phases, butince this phase is metallic the degree of hybridization muste somewhat smaller than for the semiconducting VEC = 18hases. This is reflected in a maximum value of only 0.46 forhe ELF attractor (Fig. 12(a) and Table 1). The other deviatingEC = 18 phase in Table 1, ZnLiN, does not show any signifi-

ant attractors between the atomic spheres. The maximum valuen the ELF between Zn and Li is too small to be considered asndication of covalent bonding, another notable feature being aeak polarization of N toward Li. The largely ionic character fornLiN is thus confirmed by the ELF characteristics (Fig. 12(b)).he covalence of the MgLiSb phase with VEC = 8 is also con-rmed by the ELF map (Fig. 12(c)), the Sb atoms being largelyolarized toward Mg and the attractor is located close to Sb onhe connecting line between Mg and Li with a maximum ELFalue of 0.88. This bond should accordingly be classified as aistinctly polar-covalent bond. Another special case is providedy the VEC = 19 phase IrMnAl. For IrMnAl both the CD andhe ELF map (Fig. 12(d)) show a significant charge distributionetween all three kinds of atoms and the ELF map indicates aarge degree of hybridization for the Ir–Mn, Ir–Al, and Mn–Alonds. The attractor associated with the Ir–Mn bond is locatedalf way between the atomic spheres and exhibits a lower maxi-um value than the attractors on the Ir–Al and Mn–Al bonds that

re positioned close to, and bent toward, Al. These features leadne to characterize the Ir–Al and Ir–Al bonds as polar covalent.he shorter Ir–Al bond takes the highest ELF attractor value asxpected (Table 1).

.2.2. Crystal-orbital-Hamilton populationThe COHP, which is the Hamilton-population-weighted

ensity-of-states, is a partitioning scheme for the band-structurenergy in terms of orbital-pair contributions [58,59]. Negativealues for the COHP parameter indicate bonding, whereas pos-tive values indicate antibonding behaviour, and COHP thusrovides energy-resolved visualization of the chemical bonding.he bond strength between two interacting atoms in a solid cane investigated by looking at the complete COHP between them,aking into account all valence orbitals. The integrated COHPICOHP) up to EF is, therefore, used as a qualitative measure

f mainly covalent bond strength. The bond strength betweenairs of interacting X, Y, and Z atoms in the half-Heusler phasess thus investigated by calculating the COHP and comparinghe ICOHP values (data for selected, presumably representative

48 L. Offernes et al. / Journal of Alloys and Compounds 439 (2007) 37–54

F ts refev to 0.7o

pa

bdbabc−Nfttwn

ow(lNpec

pbiiNhetoaa

sbmItdtfm

ig. 12. ELF for: (a) CuMgSb; (b) ZnLiN; (c) MgLiSb; and (d) IrMnAl. All ploary (from 0.10 to 0.60, from 0.050 to 0.70, from 0.010 to 0.85, and from 0.0010f the X, Y, and Z atoms are the same as in Fig. 2(b).

hases are included in Table 1). The COHP profiles for NiMnSnnd AuMnSn are shown in Fig. 13.

The COHP for all bonds of NiTiSn (VEC = 18) confirmsonding electrons in the entire energy region below EF and pre-ominantly antibonding electrons above EF. In the energy regionelow −5.1 eV the s-orbital interactions predominate. The inter-ctions associated with Sn, viz. Ni–Sn and Ti–Sn, show strongonding interaction as expected from the more pronounced sharacter usually connected with Sn. The energy region from5.1 eV to the pseudo-gap at −2.3 eV, is also dominated by thei–Sn interactions, but here one also finds bonding contributions

rom Ni–Ti and Ti–Sn interactions. The main bonding interac-ion in the energy region from −2.3 eV to EF originates fromhe Ni–Ti bonds, especially around −2 eV (−2.3 to −1.6 eV)here COHP exhibits a large feature for this bond and little oro contribution from the other bonds.

The shape of the COHP curve for the Ni–Ti bond mirrors thatf the partial DOS of Ti (except for differences in magnitude),hich in return show large similarities to the partial DOS of Ni

see Fig. 7(a)). This behaviour of the COHP is typical for cova-ent interactions. The pronounced feature in the COHP for the

i–Ti interaction around −2 eV can be recognized in the orbitalrojected partial DOSs for the atoms with d character, both atomsxhibiting large peaks of d character in this region. The COHPurve for the Ni–Sn bond also shows distinct similarities to the

itwr

r to the (1 1 0) plane. Note that the scale and cut-off values on these illustrations5 in parts a, b, c, and d, respectively, with 10 levels in each map). The locations

artial DOSs for Ni and Sn. Such features are usually found toe correlated with covalent interaction, but the COHP bondingnteractions are most dominant in the low energy region, whichs typical for ionic interactions. These features suggest that thei–Sn bond has a certain polar character. The longer Ti–Sn bondas a COHP profile with low-to-intermediate values in the entirenergy region. Orbital-resolved COHP calculations show thathe bonding interaction between the Ni and Sn atoms is mainlyf sp3 character (the orbital-projected COHP-calculated inter-ction between the s orbital of Sn and the three p orbitals of Nire equivalent and vice versa).

The equal lengths of the Ni–Ti and Ni–Sn bonds indicateimilar bond strength, and this is indeed also brought abouty the ICOHP values (see Table 1, recalling that ICOHP pri-ary reflects strength of covalent interaction). According to the

COHP data the longer Ti–Sn bond should have almost halfhe strength of the Ni–Ti and Ni–Sn bonds. The bond strengthserived from the ICOHP values for NiTiSn are large comparedo those for the other half-Heusler phases. The ICOHP valuesor Ni–Ti and Ni–Sn are slightly smaller than (really of the sameagnitude as) the ICOHP value of 2.3 found for the Sn–Sn bond

n semiconducting �-Sn. The Ni–Sn bonding interaction appearso be strongest in the lower part of the energy range, coincidingith the low lying states of mainly Sn-s character in the energy

ange −9.8 to −7.1 eV, and partially the Sn-p and Ni-d energy

L. Offernes et al. / Journal of Alloys an

Fig. 13. Crystal-orbital-Hamilton population (COHP) represented: (a) schemat-ically; (b) as obtained computationally for NiTiSn; (c) as obtained computation-ally for AuMnSn. In part a three different positions of EF are shown with verticallines, in parts b and c the interactions between X–Y, X–Z, and Y–Z atomicpl

simdiatCra

s

AiApSAuistTfie

tttTpmttsbett

mataltstpariaies

LtFamtsc

airs are distinguished by solid, dashed, and dotted profiles, respectively, as alsoabelled on the illustrations.

tates in the range −5.0 to −2.3 eV. The Ni–Ti interaction dom-nates the region from −5.0 eV to EF and the COHP profile is

irroring the partial DOS profiles of Ni and Ti, indicating a largeegree of hybridization (see Fig. 7(a)). There is a large featuren the COHP profile for the Ni–Ti bond around −2 eV, whichppears to be correlated to a peak in the Ni-d states. This fea-ure is not recognized in the Ni–Sn and Ti–Sn interactions. TheOHP of the Ti–Sn interaction is fairly low in the whole energy

ange and the resulting ICOHP indicates lower bond strength inccordance with the inferences from the bond lengths.

The ICOHP values for AuMnSn (Table 1) indicate that thetrength of the Au–Sn bond is of the same magnitude as the

ltbs

d Compounds 439 (2007) 37–54 49

u–Mn bond and about twice the strength of the Mn–Sn bond,n full accordance with the inferences from the bond lengths. Theu–Sn bonding interaction (Fig. 13(c)) is strongest in the lowerart of the energy range, coinciding with the band of mainlyn-s character at −10.2 to −8.1 eV, and the largely localizedu-d states at −6.4 to −4.9 eV (see Fig. 7(b)). However, the sit-ation for the Au–Mn interaction is different, since the bondingnteractions are strong both in the region dominated by Au-dtates and in the higher energy range from −6.4 eV to EF wherehe partial DOSs of Au and Mn show appreciable hybridization.he COHP profile in the latter region does not mirror the pro-le of the localized majority-Mn-d states since these unpairedlectrons are magnetic and non-bonding.

The strengthening of the X–Y bond on going from AuMnSno NiTiSn seen in the ELF maps is supported by the findings fromhe COHP. The COHP of the Ni–Ti interaction show a large fea-ure around−2 eV, which is not recognized in the Au–Mn COHP.his distinction can be understood by looking at the orbital-rojected partial DOS of the atoms concerned. At −2 eV theain feature in the site-projected DOS of Ni is of d character and

his has a predominant d-orbital feature as counterpart in the par-ial DOS of Ti. This d–d interaction is not favoured in AuMnSnince the d electrons of Mn participate in magnetism rather thanonding. The strengthening of the X–Z bond is also reflected innhanced bonding contributions in the higher energy range, buthese are more evenly distributed over the energy range and can,herefore, not easily be assigned to a single interaction.

A comparison of the ICOHP data for NiTiSn and AuMnSnay be somewhat misleading. The semiconducting phases such

s CoTiSb and CoVSn also have ICOHP values comparable tohose of AuMnSn. A comparison (in this respect more appropri-te) between NiTiSn and NiTiSb shows that the bond strength isowered when the semiconducting state disappears. Apart fromhis, the differences in the ICOHP values seen in Table 1 seem totem from intrinsic differences inherent in the element combina-ions concerned rather than differences in VEC. For half-Heuslerhases containing light elements, e.g., Li, N, Mg, and Al, thetomic interactions will be necessarily different because of largeelative size differences that occurs when these constituents arenvolved and the lower polarizability of these atoms. In MgLiSbnd ZnLiN, which contain two light elements, only one of thenteraction pairs takes a significantly large COHP level, strength-ning the claim that these phases, bonding-wise, belong to aeparate branch of the half-Heusler family.

According to Rytz and Hoffman [60], Dronskowski [61], andandrum and Dronskowski [59] one can also obtain insight into

he stability of a phase by examining the COHP for all bonds.or a stable, semiconducting phase EF will separate bondingnd antibonding states and thus ensure optimized bonding andinimized energy. Such a state can be recognized and charac-

erized by the “optimized” COHP and this situation is depictedchematically in Fig. 13(a) as the horizontal part of the COHPurve in the region near EF (level indicated by a dashed vertical

ine). If the atomic interactions are antibonding at EF accordingo COHP (see the dotted vertical line in Fig. 13(a)), stability maye induced by removing or reducing the amount of antibondingtates by some kind of perturbation, viz. introducing changes

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n electronic structure, crystal structure or compositional stoi-hiometry. Conversely, if COHP is bonding at and above EF (seehe dotted-dashed line in Fig. 13(a)) a perturbation that maxi-izes the bonding will also stabilize the phase.The present calculations have been performed on phases,

hich are known to exist and with the experimentally estab-ished lattice parameter for a confirmed AlLiSi-type atomicrrangement. However, the strict 1:1:1 stoichiometry used in thealculations for all phases is certainly not generally correct sinceome of the more carefully investigated phases definitely haveeen shown [20] to exhibit deviations from the equiatomic stoi-hiometry. A closer examination of the COHP near EF for suchhases is likely to provide an estimate for how large the proba-le deviation in stoichiometry might be. Another utilization ofhe COHP concept could be to stipulate the effect of variationn VEC on substitution by examining the occurrence of bond-ng, non-bonding, and antibonding states around EF. Then uponntegration of DOS this situation may be converted into electronontent for a hypothetical substituted phase, which in turn maye subjected to experimental testing.

The basic empirical hypothesis that the numerous VEC = 18hases are more stable than those with VEC �= 18 is substantiatedy COHP. For the VEC = 18 phases EF lies between bonding andntibonding states as expected for semiconducting phases andEC = 18, therefore, serves as a point of reference. For exam-le, looking at the COHP for the electron-deficient phase FeTiSnVEC = 16) optimum VEC is found 0.41 eV above EF. A crudentegration of DOS from EF to 0.41 eV indicates that a supply ofpproximately 2 electrons will fill up the bonding states. Hence,n atomic substitution in FeTiSn that does not change the DOSrofile significantly (viz. assuming a proper rigid-band filling)s likely to provide a stable phase. The existing VEC = 17 phaseseTiSb and CoTiSn exhibit COHP and integrated DOS profileshich indicate stabilization upon adding one electron by substi-

ution, e.g., only bonding or non-bonding states are filled in suchubstitutions. The “two-electron-substituted” VEC = 18 phaseoTiSb exists and the COHP characteristics of this reflect semi-onductivity. However, exchanging all atoms in FeTiSn with theearest neighbour higher valent atom from the periodic tablelso gives an existing phase, namely CoVSb (VEC = 19). TheOHP for this phase indicates a certain destabilizing tendencyrought about by excess of electrons (the electrons close to EFre involved in antibonding interactions). Exchange of Sb for Sneads to removal of these states and yet another existing phase,oVSn (VEC = 18), is obtained. The VEC = 19 phase NiTiSblso comprizes antibonding states at EF and its “optimized”OHP occurs at 0.80 eV below EF. A “optimized” situation can

n this case be obtained by removing one electron by substitu-ion of one of the constituents by a lower valent element. All theesulting VEC = 18 combinations CoTiSb, NiScSb, and NiTiSnxist and exhibit semiconducting properties.

The VEC = 19 phase IrMnAl does not follow the pattern out-ined above. The DOS of IrMnAl show that a rigid-band-like

emoval of electrons to VEC = 18 will not give a semiconduct-ng phase. The light main-group-element Al gives rise to a muchmaller degree of hybridization than the heavier atoms gener-lly found in half-Heusler phases, and subsequently no gap is

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d Compounds 439 (2007) 37–54

ound in the DOS profile. According to the COHP there appearso be no gain in bonding interaction on substitution with antom with one electron less. On the other hand, a substitutionhich brings about an additional 1.5 electrons should provide a

table phase. There is no report on any other Al-containing half-eusler phases, but IrMnSn (VEC = 20) and IrMnSb (VEC = 21)o exist. The COHP and DOS data for these phases indicatehat stable phases can be obtained by addition of electrons.he phases PtMnSn (VEC = 21) and PtMnSb (VEC = 22) bothxist and the latter belong to the HMF category. It is worth not-ng that even though AuMnSn is a VEC = 22 phase, maximumonding interaction occurs above EF for the ideal 1:1:1 com-ound. This phase should, therefore, be stabilized by addinglectrons in the Au and Mn bands to maximize the bondingnteraction. The experimental solid-solubility range of AuMnSn20] does indeed indicate that a fraction of the Sn atoms iseplaced by Mn and/or Au. The VEC = 23 AuMnSb phase havets optimum COHP below EF. A removal of about 0.5 elec-rons would, therefore, bring about a more stable situation. Arocessing of AuMnSb according to rigid-band assumptionsould theoretically give a HMF phase. FLMTO-supercell calcu-

ations on the solid-solution series AuMnSn1−xSbx predict HMFehaviour within the solid-solubility range, more specifically at.50 < x < 0.75 [13]. CuMnSb is also a VEC = 23 phase, but inhis case rigid-band removal of one electron (viz. the conditionor optimum COHP) fails to predict a HMF phase, although theomplete exhange of Cu with Ni or Pt leads to known HMFhases.

An essential lesson from these considerations is accordinglyhat both VEC = 18 and VEC = 22 configurations lead to opti-

um bonding and stability conditions, which in turn implieshat stability of phases derived by simple substitutions can beredicted.

.3. Magnetic aspects

The magnetic electrons have not been considered specificallyn the discussion on bonding since they do not contribute to theonding properties. Many of the half-Heusler phases are inter-sting in relation to their magnetic properties as referred to inection 1. There are also a large group of half-Heusler phasesith more indistinct magnetic properties, e.g., only small mag-etic moments are found for most of the phases with VEC = 16,7, and 19 (as shown schematically in Fig. 5). The ferromag-etic Mn-based half-Heusler phases with VEC > 18, however,ave large magnetic moments and are commonly considered asrue local moments magnets. In the present study, accurate the-retical calculations have been performed for four such phases;amely AuMnSn, AuMnSb, PtMnSb, and NiMnSn. Calculatedotal magnetic moments for these phases are listed in Table 1.he spin-polarized calculations of the other phases (including a

arger number of phases not documented in Table 1) are madeo estimate magnetic moments only.

The non-spin-polarized band structures for the XMnZ half-eusler phases show a peak at EF (viz. a distinct N(EF)), which

ndicates that the non-magnetic phases have low stability andpontaneous magnetism [49] will prevail. As already established

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xperimentally [2,3], the characteristics of these magneticallynteresting phases is that they are basically ferromagneticallyrdered, with large magnetic moments confined to the Mn atomnd only small induced magnetic moments at the X and Z sites.he X atoms are ferromagnetically coupled to Mn and this istrongest among the possible magnetic interactions. The antifer-omagneticlly induced moment on the Z site could be indicativef a superexchange interaction between the Mn moments medi-ted by the intermediate Z atoms. However, the present authorselieve that the major contribution to the ferromagnetic cou-ling stems from a RKKY-like mechanism brought about by theonduction electrons.

To investigate the magnetic interactions further, spin den-ity were estimated for the magnetic phases. Spin-density mapsere obtained by subtracting the minority-spin electron density

rom the majority-spin electron density, and regions with higherrobability of up-spin electrons here appear with positive spinensity and conversely regions with higher probability of down-pin electrons appear with negative values. In the spin-densityap for AuMnSn (chosen as representative for the four phasesentioned above; Fig. 14(a)) a large up-spin density is seen

t the Mn(Y) site, as expected from considerations accordingo local magnetism as well as manifested by the experimen-al data [3]. Only small positive spin-density values are foundt the Au(X) site and small negative values at the Sn(Z) site.ore interestingly we note there are two major spin-density

eatures on interstitial locations between the atoms. The ferro-agnetically coupled magnetic moments of Au(X) and Mn(Y)

re apparently mediated by antiferromagnetically coupled spinn regions between them. This indicates a d–d exchange mecha-ism through the covalent bond between these atoms. Secondly,here are interconnected regions of up-spin electrons connect-ng the Mn(Y) atoms via the crystallographically empty 4d site,hich may be taken as manifestation of the suggested RKKY-

ike interaction via conduction electrons. To substantiate these

ndications calculations were also made for electrons in thenergy region close to EF only. Fig. 15(b and d) shows spinensities for AuMnSn for electrons in the range from −1.45 eVo EF and −0.050 eV to EF. To provide a pictorial impression

Pi(i

ig. 14. (a) Total spin density and (b) UNPAIR (see text) for AuMnSn. In part a, resolid contour lines), regions which is dominated by down-spin electrons are shown bolid line). The locations of the X, Y, and Z atoms are the same as in Fig. 2(b).

d Compounds 439 (2007) 37–54 51

f the contribution of a given atom in a certain energy rangehe corresponding charge-density plots are shown in Fig. 15(and c). The energy range −1.45 to 0 eV should reflect the ener-etically uppermost Mn-d(Y-d) electrons together with contri-utions from d electrons associated with the Au(X) atoms. Thepin-density distribution for this energy range (Fig. 15(b)) illus-rates how the up-spin electrons at the Mn(Y) site is orientedoward the Au(X) sites. (The four equivalent interaction pathsppear here as two quite pronounced maxima oriented toward then-plane Au(X) sites and two weaker maxima oriented towardut-of-plane Au(X) sites.) Corresponding up-spin regions at theu(X) site are oriented toward the Mn(Y) sites. As in the total

pin-density plot, these up-spin regions are connected to down-pin regions located on the lines connecting the atomic sites,hus confirming the findings from the total spin-density plot.he regions of up-spin electrons connecting the Mn(Y) atoms

hrough the empty 4d sites are easily recognized in Fig. 15(b),ut since this plot also includes electrons that do not contribute toonduction even at elevated temperatures, attention should rathere turned to the region closer to EF (Fig. 15(d)) for informationbout magnetic exchange interaction via conduction electrons.his plot shows that the conduction electrons are indeed spin-olarized with interconnecting regions dominated by up- andown-spin electrons. There is especially a pile up of down-spinlectrons around both the Mn(Y) and Au(X) sites (which to somextent also includes the empty 4d site) that suggest a broader flowf electrons than suggested by the total spin density. Our overallonclusion is accordingly that there are definite indications ofRKKY-like magnetic exchange mechanism being operative in

hese phases.Another way to illustrate possible accumulations of unpaired

lectrons is to multiply the ELF with the CD and for the inter-retation of this composite function make the (not too illogical)ssumption that high values reflect paired electrons (PAIR). Aext step could be to subtract the thus modified CD function

AIR from the plain CD function (valence-electron density) andnterpret high values as a coarse measure of unpaired electronUNPAIR) constellations. Such an UNPAIR plot for AuMnSns given in Fig. 14(b) and this manipulation of the potential con-

gions which are dominated by up-spin electrons are shown by positive valuesy negative values (dashed contour lines), and the border line is at zero (thicker,

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uction electrons once again supports the inference that there isn accumulation of unpaired electrons in the otherwise emptyegion between the Au(X) and Mn(Y) atoms.

.4. Concluding remarks

The trends in electronic structure with respect to valence-lectron content (VEC) are quite clear-cut for the majority ofhe half-Heusler XYZ phases which comprises X = heavy tran-ition metal, Y = light transition metal, and Z = late main-grouplement. A simple schematic representation of the electronicand structure with distinctly identifiable contributions from-d, Y-d, and Z-s and -p states gives a good qualitative pic-

ure of the electronic structure for a given half-Heusler phaseased on knowledge of the constituents that are involved. Thusrom the chemical composition only, one can predict density-of-tate (DOS) profile, chemical bonding features including degreef hybridization, and magnetic properties with reasonable esti-ates for the size of magnetic moments. However, it must be

mphasized that care must be exercised in considerations of

hases, which occur in border regions between co-operativend non-co-operative magnetism. Here, small deviations in lat-ice parameter and/or stoichiometry may alter the scenery quiteppreciably.

aVos

ig. 15. Maps of (a) charge density from −1.45 eV to EF; (b) spin density from −10.050 eV to EF, all for AuMnSn. In parts b and d, regions which are dominated byhich are dominated by down-spin electrons are shown by negative values (dashed c

he X, Y, and Z atoms are the same as in Fig. 2(b).

d Compounds 439 (2007) 37–54

The reason behind the close correlation in fundamental prop-rties of the mainstream half-Heusler phases is the common,ubic, positional-parameter-free AlLiSi-type structural arrange-ent. This leaves only a few degrees of freedom, like lattice

arameter and stoichiometry, and the structural arrangement ofhe atoms, therefore, really predetermines main bond characternd DOS profiles. In the idealized case, this leads to a rigid-and system where properties are governed by VEC rather thanhe combination of atoms involved. A quick look at clustersf phases with correspondingly simple cubic crystal structuresnd large number of representatives show that similar consid-rations may be successfully applied to suitable selections ofhases with, e.g., NaCl-, CsCl-, ZnS(zink blende)-, Cr3Si-, oru3Au-type structure.

For most of the half-Heusler intermetallic phases, the “opti-ized” COHP (crystal-orbital-Hamilton population) demon-

trates the simple chemical significance of VEC = 18 for theemiconducting representatives and VEC = 22 for the half-etallic ferromagnetic representatives. The former VEC value

pplies to the situation in which all bonding states are filled

nd all antibonding/non-bonding states are empty, and the latterEC value to the corresponding situation for the majority bandnly. The VEC = 18 configuration is easily rationalized as theituation with complete filling of one set of s, p, and d states. A

.45 eV to EF; (c) charge density −0.050 eV to EF; and (d) spin density fromup-spin electrons are shown by positive values (solid contour lines), regions

ontour lines), and border lines are at zero (thicker, solid line). The locations of

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imilar consideration for the VEC = 22 yields complete fillingf s, p, and d orbitals in the minority-spin channel and an addi-ional filling of four electrons in the 10 Y-d states. With full spinolarization and filling of half of the 10 Y-d states, however, theoptimized” VEC would occur at 23 rather than 22. This canationalize why the PtMnSb and NiMnSn phases has an “opti-ized” COHP at VEC = 22, while the corresponding gold phases

btains an “optimized” COHP for VEC > 22. For the rare-earthhases, the VEC varies between 25 and 32, the former valueorresponding to complete filling of s, p, and d, and half fillingf f states and the latter to situations with complete filling ofll s, p, d, and f states. In a quest for identifying new phases,he “optimized” COHP concept may be utilized both as a toolor prediction of substitutionally derived phases as well as fordentifying stable non-stoichiometry and solid-solution ranges.

The ELF (electron-localization function) and the COHP anal-ses have proved to be powerful tools to establish the nature ofonding interactions that is “typical” for half-Heusler phases,iz. the largely covalent bonding character. The X–Y and X–Zond strengths are of corresponding magnitude, but the latterond exhibits a pronounced polarity with the bonding electronsloser to Z than X. The variation in covalent bond strength variesith the degree of hybridization, which in turn depends on theEC as well as more empirical quantities like electronegativety,

ize difference, and polarizability of the atoms. The strict struc-ural constraints require that the constituents, especially the Ztoms, must be polarized to some extent to facilitate enhancedybridization. In the struggle to fulfil this demand the preferablehoice of constituents, X turns out to be a late transition metal,hile Z presents itself as a heavy main-group metal, viz. Sn orb as empirically established. The electron-deficient phases (asompared with the idealized semiconducting VEC = 18 phases)lso have traits of character of metallic bonding, while the extralectrons in the electron-excessive phases lead to metallic bond-ng as well as appearance of electrons that are classified ason-bonding and magnetic. However, the metallic half-Heuslerhases have poor electrical conductivity properties as evidentrom experimental measurements (unpublished work by somef the authors) and other physical properties, e.g., the gray, non-etallic luster and a brittle nature with little or no ductility. These

eatures and the similarity in bonding throughout the bulk ofhe half-Heusler phases indicate that the existence of a metalliconding component in the VEC �= 18 phases is of minor impor-ance in the overall bonding picture. The metallic contributiono the bonding is, however, important for the total energy of theystem and thus contributes to stability.

.5. Postscript

After the present contributions had been submitted, and in factccepted for publication, a paper by Kandpal et al. [62] came tour attention. This paper also deals with the bonding situationn the half-Heusler phases on the basis of first-principles DFT

alculations according to essentially the same computationalodes as used in this work and the methodological similarityxtends even further by encompassing the same visualizationools (notably CD, COHP, and ELF) to examine the findings.

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d Compounds 439 (2007) 37–54 53

t is therefore not surprising that some of the interpretationsre similar. However, as emphasized in the introduction therere known over a hundred half-Heusler phases and the specifichases subjected to the computational treatment in the two inde-endent studies are, with a few exceptions (CoVSn, CoTiSb, andiMnSb), not the same. This makes the studies complementary

ather than competitive.Kandpal et al. focus largely on the VEC = 8 phases and

onclude that their bonding situation can be described asonic interactions between cationic Y constituents and anionicZ sublattices with internal X–Z covalent bonding. This is

ssentially the same as found for the VEC = 8 phases in theresent study (see the above consideration on MgLiSb). For theEC ≥ 18 phases Kandpal et al. used three-dimensional isosur-

ace representations of CD and ELF to extract information onhe character of the chemical bonding. Again they draw muchhe same conclusions as established above. However, the exam-nation of ELF in the (1 1 0) plane allowed the present studyo detect covalent bonding within the XY sublattice (identifiedhrough the occurrence of attractors) when two transition met-ls are involved. This concurs both with the similarities in theartial DOS profiles for the X and Y constituents, and the largeCOHP values found for the X–Y bonds. In general Kandpal etl. appear to draw somewhat different conclusions from us inases where two transition metals (X and Y) are involved andhe value of VEC is high.

Our overall conclusion is that the Kandpal et al. and presenttudies together significantly contribute to a deeper understand-ng of the half-Heusler phases.

cknowledgements

LO and PR appreciate the financial support from the Researchouncil of Norway. Parts of the calculations were carried out on

he Norwegian supercomputer facilities.

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