prediction of composition for stable half-heusler...
TRANSCRIPT
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Journal of Alloys and Compounds 458 (2008) 47–60
Prediction of composition for stable half-Heusler phases fromelectronic-band-structure analyses
L. Offernes ∗, P. Ravindran, C.W. Seim, A. KjekshusDepartment of Chemistry, University of Oslo, P.O. Box 1033, Blindern, N-0315 Oslo, Norway
Received 14 March 2007; accepted 3 April 2007Available online 12 April 2007
bstract
This report describes a procedure to predict the frequently occurring non-stoichiometry of the half-Heusler XYZ alloys (viz. deviations fromhe equiatomic 1:1:1 composition and the usually accompanied narrow homogeneity regions) from ab initio calculated electronic-band-structureharacteristics. The essential feature of this approach is to utilize the valence electron content (VEC) and the calculated electronic band structureo expose factors that according to rigid-band considerations should determine the possible deviations from 1:1:1 stoichiometry and directionf the stable solid-solution regions. These means have been used to predict the direction of equilibrium solid-solution regions for a number ofernary phase diagrams that comprise half-Heusler phases and the predictions have been tested with experimental data from literature and presently
ynthesized and microprobe analysed samples of NiTiSn, PtTiSn, CoTiSb, PtMnSb, NiMnSb, and CoMnSb. The predictions are made based onaximum band filling of bonding states identified through the crystal-orbital-Hamilton population (COHP) analysis and density-of-states (DOS)ntegration.2007 Elsevier B.V. All rights reserved.
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eywords: Stoichiometry; Intermetallics; Half-Heusler phases; COHP; VEC
. Introduction
Our previous detailed examinations of the half-Heusler alloysuMnSn and AuMnSb [1–3] revealed that they do not take the
xact 1:1:1 composition at 400 ◦C. These phases rather showarrow composition ranges, indeed in the vicinity of the idealquiatomic composition, as displayed in the phase diagramsor the Au–Mn–Sn and Au–Mn–Sb systems [2,3] in Fig. 1.he formation of phases with compositions deviating from thequiatomic composition commonly associated with the half-eusler phases has intrigued and challenged us for some time
nd the aim of this report is to expose factors that can be respon-ible for the deviation from the simple 1:1:1 stoichiometry andstablish whether such deviations are likely to be common foralf-Heusler phases in general.
Another aspect of interest is the unusual complexity of suchhase diagrams and the impact the individual inherent featuresmpose on the preparation procedure and sample quality. In the
∗ Corresponding author. Tel.: +47 22857397; fax: +47 22855565.E-mail address: [email protected] (L. Offernes).
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sothermal sections of the Au–Mn–Sn and Au–Mn–Sb phaseiagrams in Fig. 1, single-phase regions are marked by blackreas, two-phase fields are represented by tie-lines or showns areas connecting two phases existing in equilibrium at theiven temperature, and three-phase fields are depicted by trian-les connecting the phases concerned. The relative amounts ofach phase in two- or three-phase fields follow from the leverule (see, e.g., Ref. [4]). If temperature is also considered as vari-ble, the phase diagrams become three-dimensional and aspectsuch as phase formation and stability become important. TheuMnSn and AuMnSb phases are, e.g., peritectically formed at470 and ∼575 ◦C, respectively, thus defining the upper sta-
ility limits for these phases. The different appearances (andomplexity) of these phase diagrams emphasize the importancef careful choice of nominal composition of samples for explo-ation of location and composition region for a genuine ternaryhase. For interpretation of phase-analytical data it may alsoe of vital importance to have a qualified opinion on the main
eatures of the phase diagram concerned.The family of half-Heusler phases includes well over 100hases and have been studied extensively in recent years (see,.g., Refs. [5–12]). Half-Heusler phases are known to form from
48 L. Offernes et al. / Journal of Alloys and Compounds 458 (2008) 47–60
F e-phal
agtevtwse[affVmtttsirsSmeMf
HhcsctcieeR
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ig. 1. Phase diagrams for (a) AuMnSn and (b) AuMnSb at 400 ◦C [2,3]. Singlines, and three-phase fields by triangles connecting coexisting phases.
wide variety of different elements and in a typical phase witheneral formula XYZ, X is a heavy transition metal, Y is a lightransition metal or a rare-earth metal, and Z is a late main-grouplement (most frequently Sb or Sn). These phases exhibit a greatariety of electronic states and physical properties. The proper-ies of the half-Heusler phases vary somewhat systematicallyith the valence-electron content (VEC; note, valences pre-
cribed by the periodic table) as manifested, e.g., by changes inlectronic conductivity and magnetic characteristics with VEC7,13]. Semiconducting features are associated with VEC = 8nd 18, which represent highly preferred electron configurationsulfilling the octet or expanded octet rule. So-called half-metallicerromagnetic (HMF) materials are found among phases withEC = 22 (see, e.g., Refs. [14–16]). These phases have largeagnetic moments and in the electronic structure of these phases
he majority-spin channel exhibits metallic characteristics, whilehe minority-spin channel exposes a semiconductor-like gap athe Fermi level (EF). This situation theoretically results in 100%pin-polarized materials which are technologically importantn the field of spintronics. HMF materials are, e.g., incorpo-ated in magnetic multilayers which, due to the spin-dependentcattering of electrons, exhibit giant magnetoresistance (GMR).ome of the HMF half-Heusler phases also exhibit interestingagneto-optical properties, e.g., the large magneto-optical Kerr
ffect (MOKE) found for PtMnSb [17]. Materials with highOKE are used in the erasable data-storage technology [18]
or read/write applications.Both theoretical and experimental investigations of half-
eusler phases have been extensive, but most studies of thealf-Heusler phases seem to simply postulate an equiatomicomposition. The focus of these studies has not been on thetoichiometry of the phases, but rather concentrated on spe-ific properties [12,19–21] or trends in properties throughouthe family [6,7,22,23]. Only a few of the ternary systems whichontain half-Heusler phases have been systematically mapped
n the form of phase diagrams [2,3,17], making it difficult tovince a qualified opinion on composition issues. There have,.g., been published over 300 articles about the (anticipated, seeefs. [16,24,25]) HMF phases PtMnSb and NiMnSb, but whilepArt
se fields are shown as black areas, tie-lines are represented by solid and dashed
ome reports [26,27] account for phase purity of the samplessed, only few have documented accurate data on composi-ion. In the theoretical treatment of these and other half-Heuslerhases the equiatomic composition has almost invariably beenostulated, but also experimental founded papers appear toake the equiatomic composition for granted. The authors haveuestioned this assumption and wondered whether this is onlyalse for a few Mn-based phases with large VEC or whetherolid-solution regions and deviations from the equiatomic com-osition are the rule, rather than the exceptions for a largerortion of half-Heusler phases.
To understand non-stoichiometry and its origin it is impor-ant to consider the crystal structure and possible mechanismsor disorder and composition alterations. The cubic crystal struc-ure of the half-Heusler phases is of the AlLiSi type (Fig. 2(a);pace group F43m; see Ref. [1]). The AlLiSi-type structure cane regarded as an ordered version of the CaF2 type. Anotheray to look at the structure is to start with the Cu2MnAl-type
tructure of the X2YZ full-Heusler phases (Fig. 2(b)). The lat-er structure consists of four interpenetrating fcc-lattices, two ofhich consist of X atoms, one of Y atoms, and one of Z atoms. Ifne of the X atoms is removed from the Cu2MnAl-type structureccording to an ordered pattern, the resulting structure (viz. theu2MnAl type with one empty site) has become of the AlLiSi
ype exhibited by the half-Heusler phases. Referring to the gen-ral formula XYZ, X takes a coordination number of 8 (four–Y bonds in tetrahedral configuration and another four X–Zonds in identical configuration). The Y and Z sites are crys-allographically identical, both have coordination numbers of0 with capped (by four X) octahedral geometry, amounting toour Y–X and six Y–Z bonds and four Z–X and six Z–Y bonds,espectively. The structure has no variable positional parame-ers, leaving the fully ordered, stoichiometric XYZ phases withhe cubic lattice parameter (a) as the only structural variable.
As established for AuMnSn and AuMnSb, half-Heusler
hases are not doomed to be stoichiometric and ordered [1–3].t first glance it is natural to assume that the solid-solubilityegions of AuMnSn and AuMnSb are brought about by (i) addi-ion of Au, Mn, and/or Sn/Sb atoms to the just mentioned empty
L. Offernes et al. / Journal of Alloys and Compounds 458 (2008) 47–60 49
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Fig. 2. Crystal structures of (a) half-Heusler (AlLiSi type, general formu
rystallographic site, (ii) partial removal (subtraction) of oneonstituent or (iii) substitution of one of the constituent bynother. Other more complex non-stoichiometry variants arisen combination of the pure-cultivated mechanisms (i)–(iii), e.g.,n the form of imbalanced distribution between and subsequentisorder on the mutually equivalent Mn and Sn sites as suggestedor PtMnSn [28,29]. The solid-solution fields of AuMnSn anduMnSb lie on opposite sides of the equiatomic composition
see Fig. 1) and this show that also in these cases no single ofhe mechanisms (i)–(iii) can satisfactory explain the deviationsrom 1:1:1 stoichiometry.
The rest of this paper is organized as follows. Section 2ives a brief account of the computational and experimentalethods used to collect the data foundation for the delibera-
ions. The findings are presented and discussed in Section 3hich concludes with an overview of stoichiometry predictions
ccording to rigid-band considerations [30,31] based on the cal-ulated electronic-band-structure data. The predictions are thenonfronted with experimental findings in Section 4 and overallonclusions are finally summarized in Section 5.
. Data collection
Comparatively detailed descriptions of the computational andxperimental methods used in this work can be found elsewhere32–34], but a brief account is also included here for the con-enience of the readers. A considerable part of the theoreticalackground is in fact already published in an earlier commu-ication [32], but all computational findings considered in thisaper will nevertheless be presented as if they represented freshesults.
.1. Computational methods
All calculations are performed within the framework of theeneralized-gradient approximation (GGA), with exchange cor-elation according to Perdew et al. [35], and density-functionalheory (DFT) in the local-density approximation (LDA).
fops
Z) and (b) full-Heusler (Cu2MnAl type, general formula X2YZ) phases.
First-principles, self-consistent, tight-binding linear-muffin-in-orbital calculations within the atomic-sphere approximationTB-LMTO-ASA) [36] were performed to obtain the density-f-states (DOS) characteristics for all phases considered in thisaper. The calculations are semi-relativistic (i.e., without spin-rbit coupling, but with all other relativistic effects included)aking also into account combined correction terms. The basisets consisted of appropriate s, p, and d orbitals for the elements.he integration over the Brillouin zone (BZ) was made by the
etrahedron method, sampling a grid of 245 k points in the irre-ucible part of BZ (4096 in the full BZ). The crystal structureas divided into space filling, slightly overlapping, spheres cen-
red on each of the atomic sites. An empty sphere is included athe empty crystallographic 4d site of the AlLiSi-type structure.he Wigner–Seitz-sphere radii used were scaled so that the totalolume of all the spheres equalled the volume of the unit cell.xperimental lattice parameters were used in the calculations
listed in Table 1) for all the phases considered here.Full potential linear muffin-tin orbital (FLMTO) calculations
37] have also been used to obtain the DOS for some of the phasessee Refs. [33,34] for details).
Crystal-orbital-Hamiltonian population (COHP) plots werealculated according to the TB-LMTO code as implementedn the TBLMTO-47 package [38]. The COHP, which ishe Hamiltonian-population-weighted DOS, is a partitioningcheme for the band-structure energy in terms of orbital-pairontributions [39,40]. A negative value for the COHP indicatesonding states, whereas positive COHP values indicate anti-onding states, which provides an energy-resolved visualizationf the chemical bonding.
.2. Experimental methods
Samples were made by melting (heating at 1100–1300 ◦C
or about 1 min under vigorous shaking) weighed amountsf the elements in sealed, evacuated, silica-glass tubes. Highurity elements were used as starting materials for the synthe-es (Ti: Mackay 99.99%; Mn: Aldrich 99.98%; Co: Koch-Light
50 L. Offernes et al. / Journal of Alloys and Compounds 458 (2008) 47–60
Table 1Experimental lattice parameters from the present study or literature data quoted from Refs. [1,3,7,13,21,28], the condition of optimum bonding (relative to EF)according to COHP, and the composition or dominant composition ranges for the experimentally investigated phases
VECeq veceq Phase XYZ(X2YZ)
a (A)a “Optimum”COHP (eV)
DOS estimate(no. of e−)
VECst vecst Composition (x,y,z)
Composition range (x,y,z)–(x,y,z)
16 5.33 FeTiSn 6.056 +0.38 +0.2 ∼16.2 5.4
17 5.67 FeTiSb 5.997 +0.03 +0.2 ∼17.2 5.73CoTiSn 5.957 +0.17 +1 ∼18 6.0
18 6.0 NiTiSn 5.941, 5.926 0 0 18 6.0 (0.33,0.33,0.33)– – (Ni2TiSn) 6.06 – – – –
PtTiSn 6.168, 6.163 0 0 18 6.0 (0.33,0.33,0.33)CoVSn 5.98 0 0 18 6.0CoTiSb 5.832, 5.875 0 0 18 6.0 (0.33,0.33,0.33)
19 6.33 NiTiSb 5.872 −0.80 −1 ∼18 6.0CoVSb 5.766 −0.25 −1 ∼18 6.0
20 6.67 IrMnSn 6.182 +0.45 +1 ∼21 7.0RhMnSn 5.947b +0.30 +0.8 ∼20.8 6.93
21 7.0 IrMnSb 6.164 +0.15 +0.2 ∼21.2 7.07RhMnSb 6.145 +0.32 +0.5 ∼21.5 7.17CoMnSb 5.875, 5.865–5.878 0 0 21 7.0 (0.27,0.40,0.33)–(0.37,0.32,0.31)
– – (Co2MnSb) 5.923, 5.919 – – – –PtMnSn 6.264 +0.45 +0.5 ∼21.5 7.17
22 7.33 NiMnSb 5.909, 5.914–5.981 0 0 22 7.33 (0.30,0.34,0.36)–(0.41, 0.29, 0.30)PtMnSb 6.201, 6.194–6.225 0 0 22 7.33 (0.30, 0.37, 0.33)–(0.33, 0.32, 0.35)AuMnSn 6.323 +0.01 +0.2 ∼22.2 7.4
23 7.67 AuMnSb 6.379 −0.47 −0.5 ∼22.5 7.5
ective
9G9ttw
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a Values given in roman and italics are from literature and present study, respb From volume optimized TB-LMTO calculations.
9.998%; Ni: Goodfellow 99.98%; Sn: Merck 99.95%; Sb:oodfellow 99.999%; Pt: Rasmussen 99.99%; Au: Rasmussen9.95%). The initial heat treatment was concluded by quenchinghe samples from the molten state into water. The samples werehen annealed for 30 days at 400 ◦C and finally quenched intoater.Powder X-ray diffraction (PXD) was used to confirm the pres-
nce of the desired half-Heusler phase with the characteristiclLiSi-type structure. The data was also utilized to derive the
attice parameter (a) for the half-Heusler phases (and when rele-ant also for the full-Heusler phases) and to establish the possibleresence of other phases.
Composition analysis was performed using an automaticavelength-dispersive CAMECA SX 100 electron microprobetted with an energy-dispersive system. Acceleration voltagef 20 keV, sample current of 15–20 nA, and counting timef 10–20 s were used. Pure metals or oxides were used astandards and a thin layer of carbon was evaporated on the met-llographic specimens used for electron microprobe analyses.atrix corrections were performed by the CAMECA software
nd the achieved analytical precision (2σ; evaluated on theasis of repeated analyses of individual grains) is better than1%. Unless otherwise stated, the composition for a given
alf-Heusler phase represents a mean value of at least three inde-endent measurements. Back-scattered-electron (BSE) scansere used for imaging the samples. In BSE images the con-
rast is given by the average atomic weight of the different
esc
ly.
hases, e.g., a phase mainly containing heavy atoms will appears light colored. Optical microscopy and BSE were used to checkhe homogeneity of the half-Heusler phases considered and tostablish whether a sample had reached equilibrium or not.
A well-chosen selection of a limited number of samples withominal composition meant to establish the solid-solution regionf a phase really requires prior information about the phase dia-ram. As can be seen from the diagrams in Fig. 1, the idealocations for such surveying samples would be at points withinhe three-phase fields surrounding the phase in question. Pro-ided equilibrium is reached, such samples would then containhe ternary phase with different terminal compositions, outlin-ng the solid-solution region at the temperature concerned. Sincehe location of such phase fields usually is as unknown as thehase diagrams themselves, our selection of nominal compo-itions was made according to a common pattern around (notoo close or too far from) the equiatomic composition. In addi-ion, one sample with nominal composition corresponding tohe equiatomic composition was prepared for each system. Theominal composition of a sample is given with the general for-ula XxYyZz, where
+ y + z = 1 (1)
.g., using fractions rather than percentages. For the systemsubjected to experimental examination the following nominalompositions (x,y,z) was used: (0.33, 0.33, 0.33), (0.20, 0.50,
ys an
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L. Offernes et al. / Journal of Allo
.30), (0.25, 0.35, 0.40), (0.40, 0.35, 0.25), and (0.45, 0.20,
.35) for the Ni–Ti–Sn, Pt–Ti–Sn, Co–Ti–Sb, Ni–Mn–Sb, ando–Mn–Sb systems and (0.33, 0.33, 0.33), (0.20, 0.50, 0.30),
0.20, 0.35, 0.45), (0.25, 0.20, 0.55), (0.30, 0.50, 0.20), (0.38,.20, 0.42), (0.40, 0.30, 0.30), and (0.40, 0.40, 0.20) for thet–Mn–Sb system.
In the analyses of the annealed samples, we have recognizednumber of factors which have influenced the interpretation
f the findings. (i) Several of the samples turned out to be inocal equilibrium, but were hampered by macro segregatione.g., low-density crystals floating on the top of a higher-densityelt or vice versa). This phenomenon (which is easily observed
n vertically mounted metallographic cross sections) causeshanges in composition throughout the sample (and possiblyhase exclusion in bulk XRD analyses). (ii) Characteristic lamel-ar microstructures are formed in eutectic solidification of a
elt. In such solidified melts the crystals of some of the phasesnvolved can be too small for proper microprobe analyses. (iii)wo of the systems investigated experimentally (Ni–Ti–Sn ando–Mn–Sb) contained both full- and half-Heusler phases. The
imilarity in structure and lattice parameter made it difficulto distinguish these by XRD alone. (iv) Some of the inves-igated phases (notably NiMnSb and CoMnSb) showed clearigns of distinct variation in homogeneity region with temper-ture. When a sample which comprises a phase with a largeolid-solubility region at high temperatures and a narrower com-osition region at lower temperatures is cooled, parts of the phaseeld become unstable and the associated excess of one or twof the components will either be trapped (thus causing tensionnd/or composition inhomogeneities) or precipitate crystals ofnother phase inside the crystal domain of the reformed phase.v) Peritectic phases are formed on crystals of another phasen a reaction between these crystals and a melt. As the newhase forms, the reaction rate gradually slows down consequentn increased solid-diffusion paths. The result is often seen as aore of the original phase inside the peritectically formed phase.ote that cases (iv) and (v) refer to samples that have not reached
ocal equilibrium (for general background on non-equilibriumituations see Ref. [4]).
Specific cases in relation to half-Heusler phases are consid-red in Section 4.
. Theoretical considerations on non-stoichiometry ofalf-Heusler phases
The hitherto performed examinations of the bonding prop-rties of half-Heusler phases are based on calculations for aostulated “ideal” 1:1:1 composition [32]. However, accordingo information from COHP, neither of our introductory examplesuMnSn and AuMnSb obtains an optimized bonding situation
or the equiatomic composition. The condition for “optimized”onding according to COHP occurs when all bonding statesre filled and all antibonding states are left empty, viz. a phase
hich experiences optimized bonding should have all nega-ive COHP values within the energy region of the occupiedtates, and all positive values are associated with the unoccu-ied states, i.e., above EF [41]. For phases where bonding is not
erae
d Compounds 458 (2008) 47–60 51
ptimized, COHP is used to establish the difference in energyin eV) between the actually filled states and states with bond-ng character (corresponding stability evaluations have earliereen derived from DOS [42]). Under the supposition that thelectronic band structure remains virtually unchanged upon rel-tively minor subtraction, addition or substitution of atoms, onean achieve this by removing or adding electrons (so-calledigid-band filling). This rigid-band approach usually works veryell for substitutional half-Heusler phases [32]. The forma-
ion of non-stoichiometric half-Heusler phases is probably quiteommon considering the fact that theoretic calculations giveon-optimized bonding for most of the phases in the equiatomiconfiguration. Our predictions presuppose that the compositionf the phase under consideration remains reasonably close tohe equiatomic composition, but will be shifted toward the VEChich maximize the bonding interactions. The direction of the
omposition shift is toward the line in the phase diagram corre-ponding to this VEC (see below) and a possible solid-solutionegion is predicted to lie along the same line. To predict theomposition shift and solid-solution region for a given phase,he COHP plots and integrated DOS profiles originate fromalculations for the ideal equiatomic 1:1:1 phase were consulted.
Non-stoichiometric ternary half-Heusler phases are conve-iently represented by the formula XxYyZz. When the phasender consideration is entered into the phase diagram, x, y, andbecomes variable between 0 and 1, viz. fractional variables are
n this case more convenient than the commonly used percent-ges. An equiatomic stoichiometric ternary half-Heusler phaseill then have the composition X1/3Y1/3Z1/3. Trivially, the com-osition variables will be connected by Eq. (1) and the VECariable will thus be modified to:
ec = VEC
3(2)
here vec represent the valence-electron content per formulatom (viz. vec refers to X1/3Y1/3Z1/3). The valence-electron con-ribution from each constituent of a given half-Heusler phase ispecified as u electrons from X, v electrons from Y, and w elec-rons from Z (where u, v, and w are fixed when X, Y, and Z areefined). This in turn specifies vec as:
ec = ux + vy + wz (3)
r combined with Eq. (1) as
ec(x, y) = (u − w)x + (v − w)y + w, (4)
r correspondingly with the equivalent relation expressed withhe other variables. Eq. (4) represents a straight line trough thesothermal X–Y–Z phase diagram. For VEC values correspond-ng to the case when the stoichiometric equiatomic compositions satisfied, the vec(x,y) line will go through the equiatomicomposition and these particular VEC and vec(x,y) values areherefore denoted VECeq and veceq(x,y). In cases where the
lectronic-structure characteristics indicate that the phase wouldather prefer a modified composition for stability reasons, VECnd vec(x,y) will be denoted VECst and vecst(x,y). It should bemphasized that Eq. (2) does not deal with defect situations that
52 L. Offernes et al. / Journal of Alloys and Compounds 458 (2008) 47–60
Fig. 3. (a) Schematic illustration of the relationship between vec (x,y) (solid line) and vec (x,y) (dashed line) for a hypothetic half-Heusler phase X Y Z (see text).T orresi :1 cog
ct(sttcectt
gpF
c
3
Fa
eq
he equiatomic 1:1:1 composition is indicated by a small black dot in this and cs well within the 50%X–50%Y–50%Z cut. The location of the equiatomic 1:1rey portion in part a corresponds to the section reproduced in part b.
omprise added or subtracted atoms from the half-Heusler struc-ural unit cell. However, it may still be justified to choose Eq.2) for simplicity, when the deviations from stoichiometry aremall and the predictions are intended to be of a more qualita-ive rather than quantitative nature. VEC is then averaged overhree atoms regardless of the true number of atoms in the unitell.When u, v, and w are specified one can represent the lin-
ar relationships for veceq(x,y) and vecst(x,y) in an isothermalross-section of the X–Y–Z phase diagram (see Fig. 3). Sincehe actual phase-composition problem in this case is related tohe concentration region around the centre of gravity of the dia-afi
ig. 4. DOS for (a) AuMnSn and (c) AuMnSb and COHP for (b) AuMnSn and (d) AuMmong X–Z, Y–Z, and X–Y, atomic pairs are distinguished by solid, dashed, and dot
st x y z
ponding diagrams. (b) The field of interest for most of the investigated systemsmposition is marked by a open circle in this and corresponding diagrams. The
ram (1/3, 1/3, 1/3) only the portion of the diagram around thisoint is reproduced in most of the following representations (seeig. 3(b)).
Specific values of veceq(x,y) and vecst(x,y) for all phasesonsidered in some detail in this report are given in Table 1.
.1. Preliminary testing with AuMnSn and AuMnSb
According to analyses of the band filling of bonding states,system will gain extra stability when all bonding states arelled and all antibonding states are empty [31,43,44]. Gener-
nSb. EF is marked by vertical dashed lines, and in parts b and d the interactionsted profiles, respectively.
ys an
aimcbumEpgA∼Hr
Tibras2
A
Fpc
L. Offernes et al. / Journal of Allo
lly, the bonding and antibonding states are separated by a gapn semiconductors, a gap in the minority-spin channel in half-etals, and a so-called pseudogap (a deep valley in the DOS
urve in the vicinity of EF) in intermetallic compounds. Theonding and antibonding states in solids can easily be locatedsing COHP. The COHP for AuMnSn (Fig. 4(b)) shows thataximum filling of the bonding states occurs at 0.01 eV aboveF and on conferring this finding with the corresponding DOSrofile (Fig. 4(a)) this amounts to addition of electrons up to theap in the minority-spin channel, which in turn would convert
uMnSn into a HMF phase. Integration of DOS indicates that0.2 added electrons can be accommodated in bonding states.ence, the predicted VECst of AuMnSn is not 22 electrons, butather ∼22.2 electrons.
rsec
ig. 5. Predicted equilibrium compositions or homogeneity regions for the (stable) (ahases, are marked by a grey disk in each diagram. The veceq(x,y) and vecst(x,y) reomposition by a black dot.
d Compounds 458 (2008) 47–60 53
AuMnSb with the equiatomic composition has VECeq = 23.he COHP for this phase (Fig. 4(d)) shows that optimum bond-
ng (viz. maximum filling of bonding states) occurs at 0.47 eVelow EF, which according to integration of DOS (Fig. 4(c)) cor-esponds to removal of about 0.5 electrons. Again one ends upt a gap in the minority-spin channel and the creation of a HMFtate. The predicted value for VECst of AuMnSb is accordingly2.5 electrons.
Fig. 5(a, b) gives the lines for veceq(x,y) and vecst(x,y) foruMnSn and AuMnSb together with the predicted homogeneity
egions [2] for the two phases. The actually observed compo-ition regions of these phases are found in the vicinity of thequiatomic composition and between or on the lines. It is espe-ially gratifying to note that the composition of the AuxMnySnz
) AuMnSn, (b) AuMnSb, (c) NiTiSn, (d) CoTiSb, (e) PtMnSb, and (f) NiMnSblations are given by solid and dashed lines, respectively, and the equiatomic
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4 L. Offernes et al. / Journal of Allo
hase is predicted to lie below the veceq(x,y) line, while the com-osition of the AuxMnySbz phase is predicted to lie above theeceq(x,y) line, as established experimentally (see also Fig. 1).
.2. The semiconducting and HMF phases
Phases with VEC = 18 are predicted to be semiconductors.emiconductors have filled bonding states and empty anti-onding states separated by a gap, and accordingly optimizedonding. This trait of character must also be reflected in COHP.he HMF phases mimic the semiconducting case in the sense
hat they have a gap at EF in the minority-spin channel and thus alled-band/empty-band configuration for this channel. A closer
ook at COHP shows that this configuration also optimizes bond-ng. As a result, both semiconducting and HMF phases satisfyhe condition that veceq(x,y) equals vecst(x,y). These phases arexpected to take the exact equiatomic 1:1:1 composition or existver a composition range along the veceq(x,y) line.
DOS and COHP profiles for the VEC = 18 phase NiTiSn areiven in Fig. 6, which manifest the “semiconductor gap” and theoptimized” COHP. The predicted location for the VEC = 18hases NiTiSn and CoTiSb are given in Fig. 5(c, d) and theorresponding phase-diagrams for the VEC = 22 phases PtMnSbnd NiMnSb are given in Fig. 5(e, f).
.3. Predictions
For a phase which is neither semiconductor nor HMF, COHPlots should be able to demonstrate that optimum bonding cantill be achieved by addition or subtraction of electrons (pro-ided, of course, that the rigid-band assumption holds [30,32],hich is often the case for alloys and intermetallic compounds).s for the gold phases (considered in Section 3.1), the pre-icted location of such half-Heusler phases tend toward the
ecst(x,y) line. Fig. 7 shows a selection of ternary phase diagramsisualizing the predicted location of its half-Heusler inhabitantnumerical data are given in Table 1). The composition and pos-ible homogeneity region are for all these phases predicted toig. 6. (a) DOS and (b) COHP for NiTiSn. EF is shown by vertical dashedines. In part b the interactions among Ni–Sn, Ti–Sn, and Ni–Ti atomic pairs areistinguished by solid, dashed, and dotted profiles, respectively.
ocePNe
etdaaabwaHaodRsM
d Compounds 458 (2008) 47–60
e found in the vicinity of the equiatomic composition, locatedlong and between the veceq(x,y) and vecst(x,y) lines.
For the phases with VECeq < 18 we find that a certain amountf added electrons will lead to maximum filling of bondingtates, thus increasing the stability of the phase which by thiseans come closer to the semiconducting VEC = 18 situation.or FeTiSn (16), FeTiSb (17), and CoTiSn (17) the predictedomposition or composition region lie on the electron-rich sidef the equiatomic composition (Fig. 7(a–c)). For phases withEC = 19 loss of electrons will have the same effect, and theredicted homogeneity region and/or composition for NiTiSbnd CoVSb is located above the veceq(x,y) line in Fig. 7(d, e).hMnSn (20) is reported to exist [45], but experimental data onomposition, unit cell dimension, and magnetic moment are notvailable. This phase is nevertheless included in these consider-tions since it provide an example of a phase with VEC equallyistant from the semiconducting and HMF situations. As cane seen from Fig. 7(f) the predicted stabilization occurs with anddition of electrons, viz. a preference for the HMF situation, inhich magnetically non-bonding electrons on Mn are utilized
or splitting of states and lowering of energy.For several of the above considered phases the predicted devi-
tion from VECeq is large (addition or removal of some 0.5–1lectrons per formula unit) which implies that the postulatedpplicability of the rigid-band approximation strictly speakingo longer should be valid since for any real phase such largehanges in VEC are likely to lead to significant alterations inlectronic structure. In these cases more moderate composi-ion changes are anticipated, may be associated with structuralhanges (as, e.g., reported for CoMnSb [26,28,46]).
.4. Phase-analytical data from literature
As a first test of the validity of the way of thinking devel-ped from the ternary gold-containing half-Heusler phases, weompare our predictions with composition information from lit-rature. Data for IrMnSn (20), IrMnSb (21), RhMnSb (21), andtMnSn (21) are presented in Fig. 8 and data for PtMnSb andiMnSb are presented and discussed in Section 4 since fresh
xperimental data have been collected for these phases.Masumoto et al. [47] has reported deviation from the
quiatomic composition for IrMnSn (20) and as seen in Fig. 8(a),he actually observed composition is shifted in the direction pre-icted by the vecst(x,y) line, but the amount of added electronsre not as extensive as anticipated according to the rigid-bandpproach. Masumoto et al. reported only one composition valuend conclusions on the homogeneity region can accordingly note drawn. Turning to the corresponding Sb phase IrMnSb (21)here the anticipated deviation in composition is not as large
s first expected owing to the fact that this phase is close to theMF situation with VEC = 22 (Fig. 8(b)). A report [21] on devi-
tion from the equiatomic composition is found, but this sourcenly shifts the composition along the veceq(x,y) line, not in the
irection predicted for added electrons. For the isoelectric phasehMnSb we also predict that addition of electrons is needed totabilize the phase (Fig. 8(c)). Both van Engelen et al. [48] andasumoto and Watanabe [47] report compositions other than
L. Offernes et al. / Journal of Alloys and Compounds 458 (2008) 47–60 55
F iSn,m re giva
1otocCPt[ot[s
au
4
6ta
ig. 7. Predicted compositions or homogeneity regions for the (stable) (a) FeTarked by a grey disk in each diagram. The veceq(x,y) and vecst(x,y) relations ablack dot.
:1:1, but while the former authors confirm a rather large amountf added electrons the latter authors reports a composition alonghe veceq(x,y) line. Without further experimental investigationsf the Rh–Mn–Sb system it is difficult to judge whether theseompositions coexist as parts of a larger homogeneity region.ompared to other half-Heusler phases treated in the literaturetMnSn (21) is rather well surveyed with regard to composi-
ion and the findings are appropriately documented in literature29,49]. Our predictions again assume stabilization by addition
f electrons, but the experimental compositions instead lie beau-ifully along the veceq(x,y) line. Structural disorder is reported29] for the PtMnSn phase and the present authors speculate thatuch encroachment itself may cause stabilization of an atomicprin
(b) FeTiSb, (c) CoTiSn, (d) NiTiSb, (e) CoVSb, and (f) RhMnSn phases, areen by solid and dashed lines, respectively, and the equiatomic composition by
rrangement which makes electronically induced adjustmentsnnecessary.
. Experimental findings
To substantiate the predictions, we prepared 31 samples fromdifferent systems which contain half-Heusler phases. Quali-
ative investigation of the samples made by optical microscopynd BSE showed that quite a few of the samples had not reached
roper equilibrium and some were hampered by macro seg-egation. Taken the low number of samples for each of thenvestigated phases and the indicative nature of our investigation,one of the samples where disregarded so that all information
56 L. Offernes et al. / Journal of Alloys and Compounds 458 (2008) 47–60
F the ([ ositior
gap
mcrNNgtaraltA
4
ttsce
eNtfp(fmrtslpslcpr(
4
ig. 8. Predicted location of composition compared with data from literature for29,49] phases. The equiatomic composition is marked by open circle, and compelations are given by solid and dashed lines, respectively.
athered were used in some manner to shed light on the phasend system in question. The compositional findings for thesehases are presented in Fig. 9.
To investigate whether a semiconducting half-Heusler phaseore or less by definition is likely to take the exact equiatomic
omposition and relinquish the possibility for homogeneityegions, samples were made for the ternary systems Co–Ti–Sb,i–Ti–Sn, and Pt–Ti–Sn. The HMF phases PtMnSb andiMnSb are well known, and although there exist a phase dia-ram for the Pt–Mn–Sb system [50] we took special interest inhese phases. The CoMnSb (21) phase was chosen to representdeviation from the HMF situation. The latter phase has been
eported [26,28] to take both the AlLiSi- and BiF3-type structurend although these structures are very similar the distinctions areikely to be important for stability reasons. However, we haveaken the liberty to disregard such small deviations from thelLiSi-type structure in the consideration of our samples.
.1. NiTiSn
The NiTiSn (18) phase is predicted to be a semiconduc-or and experimentally measured NiTiSn crystals were found
o be close to the 1:1:1 composition in all our samples. Thishould make comparison between data from computational cal-ulations and experimental findings rather straightforward. Thexperimental samples from the Ni–Ti–Sn system are, how-ppl
stable) (a) IrMnSn [47], (b) IrMnSb [21], (c) RhMnSb [47,48], and (d) PtMnSnn data from literature are shown by filled symbols. The veceq(x,y) and vecst(x,y)
ver, “hampered” with the presence of the full-Heusler phase,i2TiSn, from which the half-Heusler phase NiTiSn forms peri-
ectically. In a couple of samples, the half-Heusler phase hasormed on crystals of the full-Heusler phase leaving the sam-les in a far-from-equilibrium state after 4 weeks of annealingsee Fig. 10(a) and the accompanying figure caption). Theull- and half-Heusler phases have similar atomic arrange-ents (see Fig. 2) and nearly equal lattice parameters which
esult in almost indistinguishable XRD patterns. In fact, fromhe inspection of XRD diagrams alone we concluded that aample with the nominal composition Ni45Ti20Sn30 containarge amounts of the half-Heusler phase. However, micro-robe analyses showed that this was not the case at all. Theample contains large amounts of the full-Heusler phase andittle of the half-Heusler phase. Two out of four samples werelearly not at equilibrium. The composition of the half-Heuslerhase in these samples matched that of the samples that hadeached equilibrium, viz. exhibit the equiatomic compositionFig. 9(a)).
.2. PtTiSn
Microprobe analysis of PtTiSn (18) also indicates that thishase takes the 1:1:1 composition (Fig. 9(b)). The present sam-les showed some internal macro segregation, but had reachedocal equilibrium, making the analysis quite uncomplicated. Too
L. Offernes et al. / Journal of Alloys and Compounds 458 (2008) 47–60 57
Fig. 9. Predicted location of composition compared with experimental data for the (stable) (a) NiTiSn, (b) PtTiSn, (c) CoTiSb, (d) PtMnSb, (e) NiMnSb, and (f)C expel re gived
fpTs
4
sptup
cpeCCacwt
oMnSb phases. The equiatomic composition is marked by open circle, presentiterature data by filled squares [27,47]. The veceq(x,y) and vecst(x,y) relations aata points in parts e and f see text for more details.
ew samples were measured to justify a qualified opinion on theossible existence of a homogeneity region for the PtTiSn phase.here is, however, nothing from our investigation that indicatesuch a region.
.3. CoTiSb
Microprobe analyses of the CoTiSb phase yielded a compo-ition close to the stoichiometric formula and showed that the
hase has little or no homogeneity region at 400 ◦C (Fig. 9(c);he spread in the analytical data turned out to be below thencertainty of the microprobe technique). Also these sam-les showed some degree of macro segregation. We haveCCai
rimental data by filled circles (and open triangles for more uncertain data), andn by solid and dashed lines, respectively. About the scatter in the experimental
onfidence in the equiatomic composition estimate for thishase since our five samples are from at least three differ-nt phase fields. In a sample with the nominal compositiono0.33Ti0.33Sb0.33 with some internal macro segregation, theoTiSb phase seems to be in local equilibrium with smallmounts of two Sb-rich Ti–Sb phases. Another sample (nominalomposition Co0.40Ti0.35Sb0.25) showed CoTiSb in equilibriumith a Ti-rich Ti–Sb phase and a Co-rich Co–Ti phase. A third
hree-phase field was found to exist between CoTiSb and two
o-rich Co–Sb phases in a sample with nominal compositiono0.45Ti0.20Sb0.35. From these findings one can even producetentative phase-diagram sketch, that can be used for furthernvestigations of the Co–Ti–Sb system.
58 L. Offernes et al. / Journal of Alloys and Compounds 458 (2008) 47–60
Fig. 10. BSE scan of sample with nominal composition (a) Ni Ti Sn and (b) Ni Ti Sn . The sample in part a have not reached equilibrium, the coreo ains th( ast). Te (whic
4
cal0tasePtpa
4
ctt
cbttSNdtrptXpdsatc
FadlAss
0.45 0.20 0.30
f the grey crystals consists of the full-Heusler phase while the porous rim contvisible by optical microscopy, but not seen in the BSE images due to poor contrquilibrium with the light colored crystals of a Ti–Sn phase and a Sn-rich melt
.4. PtMnSb
The homogeneity region indicated for PtMnSb in Fig. 9(d)losely matches the region previously reported by Masumotond Watanabe [50]. The present measurements generally yieldedess Pt than the earlier study, but the deviation (approximately–1 at.%) can not properly be regarded as significant sincehe discrepancy lies within the uncertainty for the microprobenalyses. There were few problems in the analyses of theseamples which proved to be homogenous and to have reachedquilibrium. It is, however, important to emphasize that thetMnSb phase exhibits a definite solid-solution region and that
he implied non-stoichiometry has consequences for both sam-le preparation and the comparisons with the results derivedccording to the idealized computational model.
.5. NiMnSb
The accurate microprobe analyses by Hanssen et al. [27] con-luded with a composition close to the equiatomic formula forhe NiMnSb phase. Our findings show a more scattered pat-ern (see Fig. 9(e)), but the findings enclose the equiatomic
ocop
ig. 11. BSE scans of sample with nominal composition (a) Ni0.45Mn0.20Sb0.35 andre seen dispersed in a solidified eutectic melt. According to microprobe analysis thesired ternary phase, XRD showed reflections from two other phases, which accordiight and dark colored in the BSE scans, respectively. (b) Heterogeneous dendritic crccording to microprobe analysis the ternary phase has a varying composition from C
howed reflections from two other phases, which according to microprobe analysis arcans, respectively.
0.25 0.35 0.40
e half-Heusler phase. The lighter colored matrix consists of two Ni–Sn phaseshe grey crystals in part b have the exact 1:1:1 composition and have establishedh has solidified into an eutectic of (Sn) and a Ti–Sn phase).
omposition. The homogeneity region for NiMnSb appears toe relatively large, at least at high temperatures. In most ofhe present samples NiMnSb turned out to be inhomogeneous,he composition variation being clearly visible in BSE images.uch inhomogeneities are also reported by Otto et al. [26]. TheiMnSb crystals have been formed at higher temperatures anduring the cooling to 400 ◦C or the subsequent annealing at thisemperature, the phase has apparently become unstable withegard to composition leading to either composition variation orrecipitation of another phase (see Fig. 11(a)). It is not obvioushat such inhomogeneities or precipitates will be discovered byRD analyses alone. First, because small changes in the latticearameter of the half-Heusler phase cause broadening of theiffraction peaks rather than a set of new reflections. Second,mall amounts of the involved phases and large grain bound-ry to bulk ratios will result in a low signal-to-noise ratio inhe probing of such samples by any analytical technique. Suchomplications may certainly have large impact on interpretation
f experimental results since such inhomogeneities and/or pre-ipitates are likely to have a large effect on the properties. Onlyne of our five samples turned out to be homogenous. This sam-le had the nominal composition Ni25Mn35Sb40 and contained(b) Co0.33Mn0.33Sb0.33. In part a, heterogeneous dendritic crystals of NiMnSbe ternary phase has the composition Ni0.410Mn0.285Sb0.305. In addition to the
ng to microprobe analysis are a Ni–Sb phase and a Mn-rich phase, appearing asystals of the CoMnSb phase surround small regions of solidified eutectic melt.o0.33Mn0.33Sb0.34 to Co0.38Mn0.30Sb0.32. In addition to the ternary phase, XRDe of a Co–Sb and a Mn–Sb phase, appearing light and dark colored in the BSE
ys an
tfif
4
spspo(ia4CHpdtXdarbhttfiaf
5
ihsaprValhtbaolehirei
nab
hrrrsnttecttVegittscp
pfctccpbtsctoretsb
A
Dwfic
L. Offernes et al. / Journal of Allo
he half-Heusler phase with a relatively low Ni content, but thisnding shows that, with care, the phase can satisfactory be maderom simple melting of the elements.
.6. CoMnSb
The composition results for CoMnSb (Fig. 9(f)) are ascattered as those for the NiMnSb case and the measured com-ositions again lie on both sides of the veceq(x,y) line. Theamples of the CoMnSb phase are also hampered by anotherroblem also experienced for NiMnSb, namely precipitationf another phase inside the domain of the half-Heusler phaseFig. 11(b)). This is not seen in all samples, but the findingsndicate a more extended homogeneity region at high temper-tures. It is difficult to estimate the solid-solution region at00 ◦C without a more detailed investigation of the system. Theo–Mn–Sb system also contains both a full-Heusler and a half-eusler phase, but unlike the findings for the Ni–Ti–Sn system,reparation of the half-Heusler phase of the Co–Mn–Sb systemoes not invoke problems. First, the lattice parameters of thewo phases are different enough to be distinguished by simpleRD analyses (Table 1) and, second, the half-Heusler phaseoes not appear to have formed peritectically. The indication ofrelatively large homogeneity range at elevated temperatures
aises the question of a partly overlapping solid-solubility fieldetween the full- and half-Heusler phases. The direction of theomogeneity region of CoMnSb does not follow the direction ofhe vec(x,y) lines, but rather extends toward the composition ofhe full-Heusler phase (see open triangles in Fig. 9(f)). To con-rm whether such a range exists the temperature dependencend stability limit of the involved phases must be investigatedurther.
. Discussion and conclusions
We have shown that the valence-electron content (VEC)s an important parameter for composition considerations onalf-Heusler phases. The optimum electronic-band-structure-tabilized VECst found by combining information from DOS-nd COHP-calculated profiles for the equiatomic half-Heuslerhase is used to predict the composition or possible homogeneityegion for a stable half-Heusler phase. Generally, for phases withECeq close to the electron content in semiconducting phases,
dding or removing electrons so that VEC gets closer to 18ead to increased stability of the phase. When VECeq becomesigher than 19 the desired alteration in VEC will be towardhe half-metallic ferromagnetic (HMF) situation characterizedy VEC = 22. This means that, without experimentally knowingnything else about a half-Heusler phase than its constituents,ne can predict whether the composition of the stable phase isikely to be shifted away from the equiatomic 1:1:1 stoichiom-try or not and also the position and direction of a possibleomogeneity region. The electron content is not the only count-
ng factor for stability, so in some cases, half-Heusler phasesather undergo a systematic structural change (as indicated for,.g., PtMnSn [29] and CoMnSb [26,46]). We believe that thenvolved structural perturbation implies maximizing of bondingR
d Compounds 458 (2008) 47–60 59
ot only by band filling but also by hybridization and other inter-ctions and that the consequent modification of the electronicand structure acts as a substitute for a change in VEC.
The simple probing of a few ternary systems which contain aalf-Heusler phase revealed several aspects which are useful toemember. First, even simple surveying of a ternary phase mayequire an appreciable number of samples if the homogeneityegion is to be mapped with reasonable accuracy. Second,imple melting of appropriate amounts of the constituents mayot yield single-phase samples. This problem may be difficulto detect by common bulk analytical methods, especially whenhere are only small amounts of other phases present. Third,ven when single-phase samples are obtained, the question oforrect composition still remains. The present work indicateshat although some of the semiconducting half-Heusler phasesake the equiatomic composition and also phases with otherEC values exhibit well-defined composition regions, more
xperimental data are needed before these inferences can beeneralized. Our experimental investigation was originallyntended only as a test for theoretically founded predictions, buthe findings also serve as a remainder of the various processeshat can hamper preparation of a single-phase half-Heuslerample. However, most of these challenges can be handled byareful sample preparation based on knowledge of the ternaryhase diagram and its temperature dependence.
From our experimental data we found that the theoreticalredictions based on band filling of bonding states seem validor half-Heusler phases. Both literature data and the presentlyollected experimental test data seem to be consistent withhe prenotion that electron content to some extent governs theomposition of a phase. The phases calculated to be stable semi-onductors all take the equiatomic composition, while the otherhases show deviation from this composition and they are sta-le over homogeneity regions more or less in accordance withhe predicted change in VEC. However, the amount of conclu-ive experimental data is fare too small for a definite generalonclusion. We leave it to interested readers to verify whetherhe predictions holds for, e.g., the phases presented in Fig. 7r any other half-Heusler phase. However, such verificationsequire detailed mapping of phase diagrams. It would be inter-sting to extend this kind of predictions to other systems knowno exhibit non-stoichiometry. Especially families of phases withimple crystal structures and mixed bonding situations shoulde interesting candidates for exploring.
cknowledgements
The authors are grateful to Dr. Muriel Erambert (at theepartment of Geosciences, University of Oslo) for assistanceith the microprobe analysis. L.O. and P.R. appreciates thenancial and supercomputing support from the Research Coun-il of Norway.
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