Journal of Alloys and Compounds 439 (2007) 37–54
Electronic structure and chemical bonding in half-Heusler phases
Laila Offernes ∗ , P. Ravindran, A. Kjekshus
Department of Chemistry, University of Oslo, PO Box 1033, Blindern, N-0315 Oslo, Norway
Received 16 August 2006; accepted 30 August 2006
Available online 23 October 2006
Abstract
The electronic structure and chemical bonding in half-Heusler phases have been systematically investigated using first-principles, self-consistent
tight-binding linear-muffin-tin-orbital calculations within the atomic-sphere approximation (TB-LMTO-ASA). The density-of-states profiles for
the mainstream half-Heusler phases exhibit very similar features originating from the cubic, positional-parameter-free AlLiSi-type structural
arrangement and resemblances in composition. The electronic structures of these half-Heusler phases are accordingly very suitable for rigidband considerations and predictions made on this basis are tested against actually calculated data. The nature of the chemical bonding has
been systematically explored for the large transition-metal branch of the half-Heusler family using density-of-states, charge-density, chargetransfer, electron-localization-function, and crystal-orbital-Hamilton-population plots. The study has laid stress on the remarkable consistency in
bonding behaviour among the considered intermetallic phases even though the properties range from non-magnetic metals and semiconductors via
ferromagnetic metals to half-metallic ferromagnetic metals, brought about by large spread in element combinations and valence-electron content.
The typical half-Heusler phase exhibits an appreciable covalent contribution to the bonding independent of the electronic state at the Fermi-level.
The 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.
Keywords: Half-Heusler phases; Intermetallics; Electronic band structure; ELF; COHP
1. Introduction
During earlier studies [1–3] of the half-Heusler phases
AuMnSb and AuMnSn, a number of interesting features came to
our attention. In this and a couple of forthcoming papers, we will
address some of these aspects, the present contribution focusing
on electronic structure and chemical bonding in the half-Heusler
family. AuMnSb and AuMnSn are metallic, soft ferromagnets
with large magnetic moments and poor electric conductivity.
Structurally these phases belong to the large Heusler family,
which comprises the so-called full-Heusler (X2 YZ) phases in
addition to the half-Heusler (XYZ) variants. There is no single set of properties that characterizes the entire Heusler family.
However, from a structure-chemical point of view the Heusler
family are described by only two variables, viz. composition
(element 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: [email protected] (L. Offernes),
[email protected] (A. Kjekshus).
0925-8388/$ – see front matter © 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.jallcom.2006.08.316
cubic lattice parameter. The Heusler family trademark is a simple structural framework that can accommodate a vast amount of
different element combinations, resulting in phases with a large
diversity in physical properties. In the half-Heusler phases, X is
typically a heavy transition metal (T), Y is a light transition metal
or a rare-earth metal (R; this branch of the family will, however,
only be briefly considered here), while Z is a late main-group element (M), most frequently Sb or Sn. The resulting total number
of valence electrons per formula unit, referred to as the valenceelectron content (VEC), varies to a great extent. Fig. 1 illustrates
this variation and gives an impression of the multitude of known
half-Heusler phases. Gold-containing phases only constitute a
small branch of the large half-Heusler family so the scope of the
investigation was extended to most of the half-Heusler family.
The crystal structure of the half-Heusler phases is of the
AlLiSi type (Fig. 2(a); space group F 43m; see Ref. [2]), which
can be regarded as an ordered ternary version of the CaF2 -type
structure. Referring to the general formulae XYZ, X takes a
coordination number of 8 (four X–Y bonds in tetrahedral configuration and another four X–Z bonds in identical configuration).
The coordination numbers of Y and Z are 10, both with capped
38
L. Offernes et al. / Journal of Alloys and Compounds 439 (2007) 37–54
Fig. 1. Histogram for reported half-Heusler phases as a function of valenceelectron content (VEC). The total number of phases included in this representation is 102.
(by 4 X) octahedral geometry, amounting to 4 Y–X and 6 Y–Z
bonds and 4 Z–X and 6 Z–Y bonds, respectively. The structure
has as mentioned no variable positional parameters, leaving the
fully ordered, stoichiometric XYZ phases with the lattice parameter, a, as the only structural variable.
The other family branch (viz. the X2 YZ full-Heusler phases,
named after German chemist Fritz Heusler) is probably still best
known for the appearance of ferromagnetism in phases of traditionally non-magnetic elements [4] and localized magnetism in
metallic phases [5]. However, full-Heusler phases are nowadays
studied with great interest owing to potential utilization in giant
magnetoresistance (GMR) [6] and magnetic shape memory [7]
devices. The full-Heusler alloys take the Cu2 MnAl-type structure (Fig. 2(c)) from which the AlLiSi type formally derives
by leaving half of the X sub-lattice empty in an ordered pattern. The, thus, created voids in the structure give rise to less
orbital overlap and consequently to a larger degree of localization and presence of gaps in the density-of-states (DOS).
These distinctions give the half-Heusler phases electronic and
magnetic properties different from the full-Heusler phases. The
electronic structure of the half-Heusler phases varies somewhat
systematic as manifested by changes from metallic to semiconducting, through metallic to half-metallic (see below), and back
to metallic behaviour as VEC increases [8,9]. The semiconducting characteristics are associated with VEC = 18, and as seen
from Fig. 1, VEC = 18 is a highly preferred configuration, over
a third of the half-Heusler phases belonging to this category.
In general the phases with VEC = 18 are either narrow-gap
semiconductors or semimetals (defined as materials with low
number of electrons at the Fermi-level, N(EF )). Phases with
VEC = 18 have been investigated for potential use in thermoelectric devices [10] and some of the R-containing phases are
found to display GMR features [11]. So-called half-metallic ferromagnetic (HMF) materials are found among the phases with
VEC = 22 [12]. In the HMF phases, the majority-spin channel
exhibits metallic characteristics, while the minority-spin channel
exposes a semiconductor-like gap at the Fermi-level (EF ). This
should theoretically result in 100% spin-polarized electronic
conduction [13–15]. Highly spin-polarized materials, like the
HMFs, are technological important in the growing field of spintronics since on adding the spin degree of freedom the data processing speed is imagined increased by orders of magnitude, the
electric power consumption correspondingly decreased, and the
overall size of electronic devices can maintain the continuously
shrinking trend [14,16,17]. HMF materials are, e.g., incorporated in magnetic multilayers, which, due to the spin-dependent
scattering of electrons, exhibit GMR. These multilayers are considered important in the development of magnetic recording
technology. Some of the HMF half-Heusler phases also show
interesting magneto-optical properties, like the large magnetooptical Kerr effect (MOKE) found for PtMnSb [18]. Materials
with high MOKE are used in read/write applications in datastorage technology [19].
Several of the ferromagnetic half-Heusler phases with VEC
around 22 are not half-metallic, but exhibit a gap in the minorityspin channel in the vicinity of EF . Some of these phases also have
a narrow solid-solution range [1,20–22], which does not necessarily include the exact 1:1:1 stoichiometry as, e.g., established
Fig. 2. (a) One unit cell of the AlLiSi-type structure adopted by the half-Heusler phases. (b) Extract of the (1 1 0) plane in the AlLiSi-type structure. The dashed cut
represents the section for which computational results are presented in following illustrations. (c) The Cu2 MnAl-type structure of the full-Heusler phases. Atoms are
labelled according to the general designations X, Y, and Z used in the text.
L. Offernes et al. / Journal of Alloys and Compounds 439 (2007) 37–54
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 appear
twice in this representation. The total number of unique phases included in the
illustration is 74.
for AuMnSn and AuMnSb [1,20]. However, since only a few of
the ternary systems which contain half-Heusler phases has been
systematically mapped in the form of phase diagrams, a further
discussion of trends in non-stoichiometry is postponed to one of
the forthcoming papers [23].
The uncovering of interesting magnetic properties has put
the focus on half-Heusler phases, which contain 3d elements.
Fig. 3 shows how the present knowledge distributes such phases
along the 3d series. More than half of them contain either Mn or
Ni. Especially, the Mn-based full- and half-Heusler phases are
considered to be true local moment phases with the magnetic
moment essentially confined to the Mn sites of the structure
[24–26]. The occurrence of the unoccupied sites in the structure
of the half-Heusler phases relative to that of the full-Heusler
phases gives rise to narrower bands and enhances the localized character of the magnetic moments. In fact, the magnetic
moments in these Mn-based ferromagnetic half-Heusler phases
are approximately integers, specified by the excess amount of
electrons (VEC − 18) compared with the 18-valence-electron
semiconductors. In view of the large separation of the magnetic
Mn(Y) atoms (Mn–Mn distances exceeding 4 Å), the ferromagnetism is thought to be established through indirect exchange
interactions, rather than originating from direct d–d overlap
[25] between Mn atoms. It has been suggested that the magnetic coupling in the full- and half-Heusler alloys is dominated by Ruderman–Kittel–Kasuya–Yoshida-like (RKKY [27])
exchange interaction between localized moments and itinerant
electrons [28], in which the magnetic information is transferred
through local spin polarization of the conduction electrons. The
resulting co-operative magnetic state should then exhibit ferromagnetic or antiferromagnetic alignment of the moments largely
depending on the interatomic distances; a consequence of oscillating variations in the conduction-electron density. Other competing exchange interactions may, e.g., involve superexchange
through the s and p electrons of the Z atoms, RKKY indirect s–d
39
exchange, and d–d exchange through polarization and interband
mixing (hybridization) [29–32].
The bonding situation in the Heusler family has certainly
been discussed in the literature on earlier occasions (see, e.g.,
Refs. [6,12,29,33]), but the focus has mainly been targeted on the
band gaps in semiconducting and HMF phases. The occurrence
of semiconducting phases with VEC = 18 has been explained by
the “18-electron rule” using an ionic electron counting procedure [33]. The Z atoms constitute the most electronegative part
of the half-Heusler phases, followed by the X atoms, leaving
the Y element as the electron donating constituent. The formal
electron accounting scheme leads to the hypothetic (ionic) configurations X−a (d10 ), Y+b (d0 ), and Z−c (s2 p6 ), where trivially
a + c = b. However, this and similar approaches do not convey
any information about the bonding situation in the material, the
electron counts gathered are merely useful for accounting purposes. All experimental and “theoretical” findings suggest that
the half-Heusler phases are largely covalent bonded. The gap
in the minority-spin channel in the HMF phases are, e.g., commonly accepted as an effect of covalent hybridization, which
leads to bonding and antibonding minority-spin states separated
by a gap [29].
The bonding situation in intermetallic materials is messy
and the field is in need of new impulses [34]. The traditional
empirical-based approach has made use of concepts like electronegativity and atomic size. The elements, which enter a
typical half-Heusler phase belong to a fairly narrow range in
the electronegativity scale where the tabulated values moreover
are burdened with various uncertainties. Strict electronegativity
considerations must accordingly be considered unproductive in
this case. The simple rigid-sphere packing model is not applicable for the half-Heusler phases either. As evident from Fig. 2,
the (position-parameter-free) AlLiSi-type structure requires that
the X–Y and X–Z bond distances are equal, which in turn would
imply that rigid-sphere Y and Z objects should have identical
size. Such a strict size criterion can obviously not be fulfilled for
all element combinations, which occur within the half-Heusler
family. At least one of the atomic constituents is, therefore,
required to exhibit a certain diffuse and polarizable character,
which will allow the atom in question to enter the structural
framework as a non-spherical object. The Z constituent (late
main-group element) of the half-Heusler phases will probably
be the softer partner in this case. Although meaningful bondlength data cannot be extracted and evaluated for individual
half-Heusler phases, it is still possible to obtain qualitative information from trends in the structure data.
Fig. 4 depicts the relationship between experimentally established interatomic X–Y (= X–Z) distances and the Z constitutent
for series of half-Heusler phases with Z = Sn and Sb. Note that
all phases with available and relevant data are treated and that
we have taken the liberty to include a few phases with Z = Ga
to indicate the continuation of the relationships to Z from group
III of the periodic table (no half-Heusler phases with Z = In are
reported). Fig. 4 shows that the phases with VEC = 18 or 22
(viz. representatives with filled or half-filled orbital configuration) exhibit a relatively shorter X–Y bond length than those
with VEC = 18 or 22. Comparing the four series in the lower
40
L. Offernes et al. / Journal of Alloys and Compounds 439 (2007) 37–54
VEC is attempted, with emphasis on differences and similarities,
which have potential impact on bonding.
2. Computational details
Fig. 4. The X–Y = X–Z bond length (and lattice parameter a) as function of Z
component for series of half-Heusler phases. Legends to the symbols used to
distinguish the different series are shown on the illustration. The main purpose
of the illustration is to visualize that phases with VEC = 18 or 22 (emphasized
by thicker solid lines) has a relatively shorter X–Y bond length than the trend
established for phases with VEC = 18 or 22.
part of the illustration it is seen that there occurs a relative shortening of the X–Y bond on going from Sn to Sb in the NbCoZ
and NiTiZ series (with VEC = 18 at Z = Sn) and the CoTiZ and
FeVZ series (with VEC = 18 at Z = Sb). This is as expected from
the shrinkage in size introduced by shell contraction on electron
filling. However, the fact that the shrinkage is much smaller in
the two former series (than in the two latter series) suggest a
decrease in bond strength on going from the Sn phases to the
corresponding Sb phases. In the AuMnZ series (see the upper
part of Fig. 4) this behaviour is manifested even clearer since
the X–Y bond length increases from AuMnSn (VEC = 22) to its
Sb counterpart. In line with the above reasoning this behaviour
is not seen for the PtMnZ and PdMnZ phases where the Sb
phases (VEC = 22) has the shorter bond length. Likewise, the
Sn phases of the IrMnZ, TiRhZ, CuMgZ, and PtMnZ series
exhibit similar bond lengths (VEC = 18 or 22 for all phases),
whereas the Sb phases of the three latter series (VEC = 18
for TiRhSb and CuMgSb; VEC = 22 for PtMnSb) show a
larger contraction in bond length than found for the IrMnZ
series.
In the rest of this paper, we will explore connections between
physical properties and bonding by means of density-functionaltheory (DFT) calculations for “typical” XYZ phases with varying VEC. This paper is sectioned so that, after a brief outline of
the computational details, results and discussion are presented
together. First, a general overview of the DOS are given for all
phases. Then, we turn to the findings for a typical VEC = 18
representative where the different examination tools are also
briefly described. In order to evaluate the bonding interactions,
valence-charge-density analyses have been preformed for different crystal planes as well as for the three-dimensional structural
arrangement. Since the (1 1 0) plane of the AlLiSi-type structure conveniently cuts through the centres of the X, Y, and Z
constituents (Fig. 2(b)) most analyses will be presented for this
plane only. Finally, a full comparison of phases with different
The density-functional-theory (DFT) calculations of the electronic structure are performed within the framework of the
generalized-gradient approximation (GGA; with exchange correlation according to Perdew et al. [35]) and the local-density
approximation (LDA).
First-principles, self-consistent, tight-binding linear-muffintin-orbital calculations within the atomic sphere approximation
(TB-LMTO-ASA) [36] were performed for all phases subjected
to this study. These calculations are semi-relativistic (i.e., without spin–orbit coupling, but all other relativistic effects included)
taking also into account combined correction terms. The basis
sets consisted of 6s, 6p, and 5d orbitals for 5d elements such as
Au and Pt, 4s, 4p, and 3d orbitals for 3d elements such as Mn
and Ni, and 5s, 5p, and 5d orbitals for Sb and Sn. The integration over the Brillouin zone (BZ) was made by the tetrahedron
method, sampling a grid of 245 k points in the irreducible part of
BZ (4096 in the whole zone). The crystal lattice is divided into
space-filling, slightly overlapping spheres centred on each of the
occupied atomic sites. An empty sphere is included at the site
formally vacated on going from the Cu2 MnAl- to the AlLiSitype structure (Fig. 2(a and c)). The Wigner–Seitz-sphere radii
used are scaled so that the sum of the volume of all the spheres
equals the volume of the unit cell. The experimental lattice
parameters listed in Table 1 are used in the calculations. Volume
optimization was performed for some of the main phases, but
the outcome of these calculations did not indicate any significant
deviation (actual deviations ranging between 0.8 and 1.2%) from
the experimental values. Full potential linear-muffin-tin-orbital
(FLMTO) band-structure calculations [44,45] were performed
for selected phases. The FLMTO method includes spin–orbit
coupling and is evidently more accurate than the TB-LMTO
method, but although there are certain numeric differences and
small shifts in energy, the DOS obtained by the former method
gave no additional, qualitative information which justifies a
detailed account in this report. All band structures documented
in the succeeding section refer to the TB-LMTO calculations. A
few results from the FLMTO calculations are reported in Ref.
[13].
Crystal-orbital-Hamilton population (COHP) and electronlocalization-function (ELF) plots are calculated according to the
TB-LMTO code as implemented in the TBLMTO-47 package
[46]. Charge-density (CD) analyses have been performed for
the three-dimensional unit cell as well as for different crystal
planes. The analyses include charge transfer (CT), which is
the self-consistent valence-electron density of the phase under
consideration in a particular region, minus the valence-electron
density which the corresponding free atom would have
exhibited in the same region. This enables visualization of
the electron redistribution in the crystal lattice (compared to
the electron distribution of the involved free atoms placed on
the same locations) caused by the bonding interactions. The
present analyses also included the spin-resolved valence-charge
L. Offernes et al. / Journal of Alloys and Compounds 439 (2007) 37–54
41
Table 1
Experimental lattice parameter (quoted from Refs. [1,2,8,9,37–39]) electronic classification according to calculationsa , and calculated and experimental (quoted from
Refs. [1,2,8,9,38,40–43]) magnetic moments
VEC
8
Phase XYZ
MgLiSb
a (Å)
6.62
Moment (μB , f.u.−1 )
ICOHP
Calculated
Experimental
X–Y
X–Z
Y–Z
SC
0
–
0.22
1.70
0.18
Classification
Optimum COHP
Location (eV)
0
ELF
VEC
mismatch
X–Y
–
[X–Z]
[0.88]
16
FeTiSn
6.056
MET
∼10−4
–
1.00
0.92
0.43
+0.38
∼2
0.56
17
FeTiSb
CoTiSn
5.997
5.957
F
MET
0.96
∼10−4
–
–
1.04
1.00
1.01
0.90
0.48
0.44
+0.03
+0.17
∼0.2
∼1
0.58
0.54
18
NiTiSn
CoVSn
CoTiSb
CuMgSb
ZnLiN
5.941
5.98
5.832
6.164
4.877
SC
SC
SC
M
SC
0
0
0
0
0
0
0
0
0
0
1.68
0.95
1.07
0.39
0.05
1.72
0.86
1.03
0.86
0.06
0.95
0.46
0.51
0.35
1.19
0
0
0
0
0
–
–
–
–
–
0.51
–
0.58
0.46
0.27
19
NiTiSb
CoVSb
IrMnAl
5.872
5.766
5.992
MET
MET
F
∼10−6
∼10−6
1.15
–
0.18
0.4
0.97
1.16
1.28
0.87
1.01
0.96
0.52
0.54
0.29
−0.80
−0.25
+0.40
∼1
∼1
∼1.5
0.49
–
0.48
20
IrMnSn
6.182
F
3.44
2.25
1.16
1.04
0.35
+0.45
∼1
0.45
21
IrMnSb
RhMnSb
PtMnSn
6.164
6.145
6.264
F
F
F
3.08
3.27
3.53
3.1
3.35
3.37
1.21
1.01
1.04
1.10
0.95
0.97
0.40
0.47
0.39
+0.15
+0.32
+0.45
∼0.2
∼0.5
∼0.5
0.49
–
–
22
NiMnSb
PtMnSb
AuMnSn
5.909
6.201
6.323
HMF
HMF
F
3.91b
4.00
4.00
3.65
4.14
3.8
0.86
1.05
0.86
0.96
1.01
0.88
0.53
0.41
0.39
0
0
+0.01
–
–
∼0.2
0.45
0.45
0.40
23
AuMnSb
CuMnSb
6.379
6.088
F
AFc
4.56b
–
4.2
0 (AF)
0.78
0.56
0.85
0.79
0.38
0.52
−0.47
−0.59
∼0.5
∼1
0.41
–
[0.70]
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 the
appropriate attractor.
a Abbreviated as SC, semiconductor; MET, metal; HMF, half-metallic ferromagnet; F, ferromagnet; AF, antiferromagnet.
b From FLMTO calculations.
c From exp. only [39].
density. All these tools have been used to explore the bonding
properties.
3. Results and discussion
3.1. Electronic structure—density-of-states
Spin-polarized DOS were calculated for about 30 halfHeusler phases with VEC between 16 and 23, the number of
phases for each VEC reflecting the shape of the histogram in
Fig. 1. To facilitate comparisons and detection of trends we
chose phases with similar element combinations for the DOS
analyses (Table 1). Since phases with the same VEC turned
out to give similar DOS profiles, the following phases are presented as representative for our findings (VEC in parenthesis):
FeTiSn (16), FeTiSb (17), CoTiSn (17), CoTiSb (18), NiTiSn
(18), NiTiSb (19), IrMnSn (20), IrMnSb (21), PtMnSn (21),
PtMnSb (22), AuMnSn (22), and AuMnSb (23). Fig. 5 shows
schematic DOSs for VEC values ranging from 16 to 23, while
Fig. 6 shows the actual TB-LMTO-calculated total DOS for
some of the just mentioned phases. Total and partial DOSs for
NiTiSn (18) and AuMnSn (22) are presented in Fig. 7. A few
phases that should be considered somewhat different, namely
MgLiSb (8), ZnLiN (18), CuMgSb (18), and IrMnAl (19), have
also been investigated and the finding for these will be discussed
separately (Fig. 8).
Generally, the DOSs in Figs. 6 and 7 show very similar features as are to be expected because of the identical (AlLiSi-type;
see above) structural arrangement and the resemblances in chemical composition (please note that the phases presented here as
“typical”, make up only one branch of the half-Heusler family). The DOS profiles have peaked features, with valleys and
pseudo-gaps. The s states are mainly originating from Z and lie
low in energy (down to about −12 eV for the Sb phases and down
to about −10 eV for the Sn phases). The s states are separated
from the p and d states by an energy gap of 1–4 eV. The thereafter following band is essentially composed of X-d states with
some admixture of Z-p and Y-d states. The Y-d states are usually (depending on VEC) found above EF . For phases with only
small or no spin polarization, the Y-d states are mainly unoccupied, while for the electron-rich, highly spin-polarized phases
the majority Y-d states are found below EF , well separated from
higher-lying minority Y-d states.
The semiconducting VEC = 18 configuration is highly preferred (see Fig. 1). This is to be expected since filling of the
bands 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 indicated
on 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,
18, and 19, EF being indicated by dotted, solid, and dashed vertical line, respectively. (c) Strongly spin-polarized DOSs for VEC = 20 or 21, 22, and 23, EF being
indicated by dotted, solid, and dashed vertical line, respectively.
(Eg ) corresponds to a closed-shell configuration with exactly
2 electrons in the s band, 6 in the p bands and 10 in the d
bands. Filling of all bonding states while leaving all antibonding
states empty, separates bonding and anti-bonding states, lowers
the energy and adds stability. On inspection of the DOS for a
phase with VEC = 18, e.g., NiTiSn (Fig. 7(a)), some features
become evident. This phase is in a non-spin-polarized semiconducting state. The s states, with principally Sn character, lay
low in energy (down to about −10 eV) and are separated from
the p and d states by a gap of more than 2 eV. The delocalized electrons of the p states also have a large degree of Sn
character and extend up to about −5 eV. The more localized
d states have mainly Ni character and are also found in this
range, the eg and t2g states are separated by a pseudo-gap at
around −2 eV. These states are separated from the unoccupied
Ti-d states by a band gap Eg ≈ 0.5 eV (recalling that TB-LMTOcalculated band gaps are likely to be underestimated compared to
the actual Eg value [47]). The partial DOSs in Fig. 7(a) unveil a
definite degree of d–d hybridized interaction between Ni and
Ti, suggesting covalent-like bonding along the 1 1 1 directions of the structure. The extent of this hybridization varies
according to the energy separation between the d states of the
atoms in question, e.g., in the CoTiSb phase (and even more
so in the CoVSn phase) the d states of the two 3d elements
comes energetically closer together than in the NiTiSn phase
and, in turn, this gives rise to a larger degree of d–d mixing
in CoTiSb than NiTiSn (see also Section 3.2). As mentioned
earlier the VEC = 18 configuration can formally be imagined as
arising on complete filling of the d orbitals of X and s and p
orbitals of Z and this situation is schematically illustrated with
an idealized DOS in Fig. 5(b). In this simple ionic-motivated
picture, the Y component poses as cation, donating electrons
to the more electronegative X and Z constituents. In the halfHeusler phases, the bonding is certainly not pure ionic, but an
L. Offernes et al. / Journal of Alloys and Compounds 439 (2007) 37–54
Fig. 6. Spin-polarized total density-of-states (DOS) for selected phases with
VEC ranging between 16 and 23 (VEC value in parenthesis). Schematic versions 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 ,
respectively, profiles e/f, g, and h to Fig. 5(c) with dotted, solid, and dashed marking of EF , respectively. Locations of EF according to the theoretical calculations
are indicated by dashed vertical lines.
ionic description can still be used as a simple electron counting
tool.
The half-Heusler structure does allow deviations from the
VEC = 18 configuration. For the phases where X and Y both are
T elements the VEC mainly varies between 16 and 23. According
to the simple ionic description VEC = 23 corresponds to filling of
half of the d orbitals of the Y constituent, giving rise to electron
configurations which exhibit co-operative magnetic properties.
For the phases where X is a T element, while Y is a R element,
VEC mainly varies between 25 and 32, which corresponds to
a variation between half-filled and filled f orbitals for the R
element. Concentrating on the phases where X and Y both are T
Fig. 7. Total and partial spin-polarized density-of-states (DOS) for (a) NiTiSn
and (b) AuMnSb. Total DOS profiles are shown by solid lines, while X, Y,
and Z partial DOS profiles are marked by dashed, dot-dashed, and dotted lines,
respectively. EF is marked by dashed vertical lines.
43
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 DOS
profiles are marked by solid lines, while X, Y, and Z partial DOS profiles are
marked by dashed, dot-dashed, and dotted lines, respectively. EF is marked by
dashed vertical lines. Note that the scale on the DOS axis varies between the
illustrations.
elements, two possibilities should be considered for VEC = 18.
One can adopt a non-spin-polarized rigid-band approach and
retain the basic DOS for the VEC = 18 situation (viz. the valence
saturated semiconductor case), but accept degrees of filling that
are insufficient (VEC = 16 or 17; Fig. 5(b), vertical dotted line) or
excessive (VEC = 19 or 20; Fig. 5(b), vertical dashed line). The
resulting phases will be non-magnetic and metallic, as demonstrated by the calculated DOS properties for FeTiSn (Fig. 6(a))
and CoTiSn (not shown). These phases will balance on the border
between magnetic and non-magnetic variants, and depending on
N(EF ) these phases will be highly influenced by any deviation
from the ideal 1:1:1 stoichiometry or atomic disorder. FeTiSn is
found experimentally to be non-magnetic [9] in accordance
with the calculation whereas CoTiSn is experimentally reported
to be weakly ferromagnetic with a magnetic moment of 0.35μB
[48] (more accurate magnetic data for this phase can be obtained
by full potential calculations). The latter phase may, therefore,
provide an example of the situation that arises when the filling
of DOS according to the rigid-band approach results in a large
number of electrons at the Fermi-level. Such a situation should
make a spin-polarized configuration preferred according to the
Stoner criterion [49,50]. For the phases with VEC lower than
18, exchange interaction only results in minor energy shifts, as
illustrated by, e.g., the calculated DOS for FeTiSb in Fig. 6(b),
and depicted schematically in Fig. 5(a). The spin polarization
of this phase gives rise to a calculated total magnetic moment
of 0.96μB per formula unit and the local magnetic moments
at the Fe(X) and Ti(Y) sites are 0.81 and 0.10μB , respectively.
The VEC < 18 situation is accordingly different from that for
the VEC > 18 phases (see below) where magnetic moments are
largely confined to the Y site. FeTiSb is experimentally reported
to be a metallic ferromagnet with a low magnetic moment [9].
For most phases with VEC > 18 the splitting of the Y-d
orbitals is complete (see Fig. 6(d–g)) giving rise to large local
magnetic moments associated with the Y atom. For the minorityspin electrons a gap opens up in the vicinity of EF and below
this gap nine electrons will fill exactly one s, three p, and five d
44
L. Offernes et al. / Journal of Alloys and Compounds 439 (2007) 37–54
bands in the minority-spin channel. Corresponding orbitals are
also accommodated in the majority-spin channel, and additional
electrons can enter the majority Y-d band. As a rule of thumb,
the orbital splitting of the Y-d states results in spin-only magnetic moments corresponding to about 2ST = VEC − 18 [33,51].
For the special case of VEC = 22, EF is located at, or within,
the minority-spin gap, and several of the VEC = 22 phases are
accordingly to be classified as HMF materials (schematically
shown in Fig. 5(c), solid vertical line).
Of the VEC = 22 phases, PtMnSb and NiMnSb are commonly recognized as HMF phases, while another interesting
phase, AuMnSn, has been obtained recently by the authors [20].
However, AuMnSn is not a proper HMF phase since both tightbinding and full potential calculations establish that EF is located
slightly below the gap in the minority-spin channel. This phase
must accordingly be classified as a ferromagnetic (F) metallic phase with poor conductivity although the overall features
of its electronic structure are similar to the other phases with
VEC = 22. In the partial DOS profiles of AuMnSn the s electrons lay low in energy (as for the NiTiSn phase, see above),
and are separated from the p and d electrons by a band gap
of more than 1 eV. The weakly spin-polarized d electrons of
Au dominate the region between −6.5 and −3.5 eV and the
strongly spin-polarized d electrons of Mn dominate the higher
energy region. The majority-spin d states are found in the region
between about −3.5 and −1 eV, while the minority-spin d states
are found above EF from about 0.4–3 eV. Both the Au- and Mn-d
energy levels are well localized. Compared to the elemental fcc
phase of gold the bulk of the Au-d states for AuMnSn falls in
a narrower energy range (∼−7 to −5 eV) and these electrons
must thus be regarded as more localized than those in gold. This
is also evident from the peaked DOS features and the separation
of the eg and t2g states by a pseudo-gap (Fig. 7(b)). For Mn the
striking feature is the exchange splitting of the d states by around
4 eV. In relation to the band filling the majority-spin states are
located well below EF , whereas most of the minority-spin states
occur above EF and remain empty. This leads to a calculated
magnetic moment of 4.00μB for AuMnSn, in accordance with
the prediction of the (VEC − 18) rule. The calculated magnetic
moment complies accordingly quite well with the experimental
value (3.8μB [3]) in particular when attention is called on the
fact that the measurements were done on samples with some
deviation from the 1:1:1 stoichiometry assumed in the calculations. The Sn-s states are lowered in energy compared to the
situation in the element Sn (␤-modification; fcc-type structure;
metal), and the characteristic higher-lying peaks attributed to p
states in the DOS of ␤-Sn are completely lost and spread over
the whole energy range in the DOS of AuMnSn.
Using rigid-band reasoning with the VEC = 22 spin-polarized
HMF case as a starting point, it follows that the phases with
VEC = 19, 20, and 21 should have EF below the gap in the
minority-spin channel while phases with VEC = 23 should have
EF above the gap (Fig. 5(c), dotted and dashed vertical line,
respectively). These phases are accordingly to be classified as
metals. The actual calculations for IrMnAl (VEC = 19) show that
there is no proper gap in the minority-spin channel, a feature that
can be traced back to a lesser degree of hybridization (due to the
lower polarizability of Al relative to Sn or Sb) and the corresponding limited splitting of the Mn states. However, a rather
prominent pseudo-gap is found at EF (Fig. 8(d)). In IrMnSn
(VEC = 20) and IrMnSb (VEC = 21) the exchange splitting of the
Mn states are significantly larger, and these phases exhibits a gap
in the minority-spin channel 0.4 and 0.2 eV above EF , respectively. According to the (VEC − 18) rule, IrMnAl should have
a magnetic moment of 1μB , while IrMnSn and IrMnSb should
have moments of 2 and 3μB , respectively. The actual, calculated values are 1.15, 3.45, and 3.08μB , respectively, implying
a marked discrepancy with the predictions of the (VEC − 18)
rule for IrMnSn. However, this discrepancy is not confirmed
experimentally, where IrMnSn samples (indeed with some nonstoichiometry [38]) have been reported with a magnetic moment
of 2.25μB . In general, structural disorder or deviation in stoichiometry can have large effects on physical properties and even
small changes in the position of EF can alter N(EF ) drastically.
Changes in N(EF ) obviously influence electrical conductivity,
but in spin-polarized cases also the magnetic properties can be
altered appreciable and certain phases can go from magnetic
to non-magnetic and vice versa. Half-Heusler phases, which are
located close to a magnetic–non-magnetic boundary (viz. phases
with VEC = 16, 17, 19, and 20) will be especially sensitive to
such imperfections [9].
In the case of AuMnSb with VEC = 23, the gap in the
minority-spin channel lies 0.4 eV below EF . This leads to filling
of minority Y-d states, and thus to a reduction of the magnetic moment from the expected value of 5μB according to the
(VEC − 18) rule. The calculated magnetic moment of 4.56μB is
still somewhat higher than the experimental value of 4.2μB [1].
This may be associated with the non-stoichiometry of the phase
and/or the non-account of orbital moment in our calculations.
If one goes from phases with VEC = 23 to those with VEC = 24
(e.g., by substitution) the expected gain in energy achieved by
the large exchange splitting and thereupon enhanced magnetic
interaction now becomes lost. In other words, the extra electron
added to the minority channel lowers the magnetic moment and
consequently overrules the relative stability gained by exchange
splitting and magnetic interaction. There is, in fact, no reports
on any half-Heusler phases with VEC = 24. A calculation of the
band structure for the hypothetical phase AuMnTe (VEC = 24)
result in a magnetic moment of about 4μB and a high N(EF )
value indicating instability of such a phase.
There are exceptions, which do not follow the abovedescribed systematics, and a few representative phases (Fig. 8)
with anomalous behaviour are briefly considered below (see
also Sections 3.2.1 and 3.2.2). Some VEC = 18 phases (e.g.,
CuMgSb) are found to be semimetals, viz. metals with low
N(EF ). The calculated DOS for CuMgSb is shown in Fig. 8(c).
On going from a heavy to a lighter T element on the X site, the
spin–orbit coupling gradually decreases [52], and exchange of
the T-3d constituent on the Y site with an alkaline-earth element
introduces a marked distinction between the X and Y atoms. The
effect of these chemical encroachments is a substantial reduction in the covalent component of the bonding interaction (see
Section 3.2.1). The less pronounced covalent hybridization thus
leads to a more “smeared” electronic profile, resulting in a lower
L. Offernes et al. / Journal of Alloys and Compounds 439 (2007) 37–54
degree of localization and a loss of the band gap at EF . With more
ionic character (found for, e.g., the ZnLiN phase; Fig. 8(b)) low
lying DOS peaks can appear as a consequence of electron transfer from the electropositive (here Li) to the more electronegative
(here N) constituent. The DOS of MgLiSb, however, shows some
surprising features (Fig. 8(a)). This phase has VEC = 8 and fulfils the traditional octet rule. From the nature of the alkali and
alkaline-earth elements one might expect a highly ionic phase,
but the DOS indicates a large degree of covalent character, as
seen by the almost identical profile of the partial DOS of the
participating atomic constituents (see Section 3.2.1). According to the DOS profile (Fig. 8(d)) the VEC = 19 phase IrMnAl
is metallic with a pseudo-gap in the minority-spin channel at
EF . The similarity of the partial DOSs of the Ir and Mn atoms
indicate X–Z hybridization (from −5.0 to EF ) as for the typical
half-Heusler phases discussed above, but in the case of IrMnAl
the partial DOS also indicate covalent Ir–Al and Mn–Al interactions (see Section 3.2.1).
3.2. Chemical bonding
The bonding characteristics of intermetallic phases are usually rather complex, and to investigate the bonding behaviour of
the half-Heusler phases several computational tools have been
applied. (Note that some of these analyses simply represent
mathematical transpositions of one given dataset into another
form in order to visualize different aspects of the data.) To
make the presentation as lucid as possible each “tool” will be
introduced accompanied by the results for two selected half-
45
Heusler phases, viz. NiTiSn and AuMnSn, which can serve as
representatives for the semiconducting (VEC = 18) and metallic phases, respectively. Thereafter, a more comprehensive and
trend-focusing discussion of the findings for the half-Heusler
family will be conducted.
3.2.1. Charge density, charge transfer, and
electron-localization function
The charge density (CD) for NiTiSn in the (1 1 0) plane
(Fig. 9(a)) unveils significant amounts of charge between Ni(X)
and Ti(Y) as well as between Ni(X) and Sn(Z). In general we see
electron accumulations between X and Y and between X and Z
whereas that between the Y and Z is comparably smaller. The
outermost electrons of the soft Z atom are somewhat polarized
toward Y. In Fig. 10(a) the three-dimensional CD is represented
by an isosurface, the value of which being chosen to visualize
bonding regions.
From the CD alone it is difficult to draw safe conclusions as to
whether an accumulated charge between two atoms stems from
bonding electrons. One approach is to turn to charge-transfer
(CT) maps (see Fig. 9(c) for CT in the (1 1 0) plane of NiTiSn).
Fig. 9(c) shows that there has occurred a significant transfer of
charge from the region of the atomic spheres of X and Y to the
interstitial region between them, indicating a directional, covalent character of the X–Y bond. The added charge in the region
between the Sn(Z) atoms is (as evident from three-dimensional
CT representations) actually the middle of the horse-shaped feature between Ni(X) and the two out-of-plane Ti(Y) atoms. No
other intermediary atomic constellations have gathered charge
Fig. 9. Maps of (a) charge density; (b) ELF (electron-localization function); and (c) charge transfer for NiTiSn; (d) charge density; (e) ELF; and (f) charge transfer
for AuMnSn. All plots refer to the (1 1 0)-plane and for easy comparison the same scales and number of contour lines are used in corresponding plots. Note that the
actual scales and cut-off values are chosen for clarity. The charge density scale in parts a and d are from 0.020 to 0.10 e a.u.−3 with 10 levels in each map, while the
ELF scale in parts b and e are from 0.20 to 0.50 with 8 levels in each map. In parts c and f regions with positive and negative values are given with solid and dashed
contour lines, respectively. The locations of the X, Y, and Z atoms are the same as in Fig. 2(b).
46
L. Offernes et al. / Journal of Alloys and Compounds 439 (2007) 37–54
Fig. 10. Three-dimensional isosurface representation of (a) charge density and (b) ELF for NiTiSn. The value of the charge density isosurface is 0.37 e a.u.−3 . The
value of the ELF isosurface is 0.44 and the enclosed attractors are visible as discs positioned between the Ni and Ti atoms. The blacking of the Ni(X), Ti(Y), and
Sn(Z) atoms are the same as in Fig. 2(b).
and neither of the atomic sites have gained net charge relative to
the constituents in the free atomic form.
The electron-localization function (ELF) is a ground-state
property that is useful to visualize and distinguish between different bonding interactions in solids [53,54]. ELF is limited to
a value between 0 and 1 and for simple valence compounds the
qualitative interpretation of the ELF is often straightforward. An
ELF value of 0.5 indicates that the Pauli repulsion at the point
in question is the same as that of uniform electron gas with
the same density [55]. Such intermediate ELF values will thus,
e.g., be found for regions with metallic bonding in typical Drude
metals. ELF should be close to one in regions ruled by light maingroup elements with paired electron configurations (low Pauli
repulsion) such as covalent bonds or lone pairs. Paired electrons
in covalent bonds are expected to manifest themselves in ELF
maps as enclosed valence-electron basins with high ELF values
(called attractors), symmetrically centred along the bond axis. In
polar-covalent materials the attractor will be positioned closer to,
and bent toward, the more electronegative atom. Ionic-like interactions can be recognized through electron basins with roughly
spherical distribution around the atomic cores, positioned so
that no attractor is found on the direct line between two interacting atoms. Bonding s and p orbitals in simple materials gives
ELF values close to one. (Typical maximum values for attractors in covalent-bonded organic molecules are 0.8 [53].) The
same qualitative picture of ELF holds for intermetallic phases,
but since the ELF is derived from an expression that varies with
the quantum number l, the ELF values diminish as l increases
[56]. Therefore, as a rule, higher-angular-momentum quantumnumber orbitals, such as d orbitals, tend to give rather indistinct
ELF characteristics, both with respect to attractor volume and
the numerical size of ELF (maximum attractor values of only
0.4–0.5 are found for polar covalent bonds involving 3d orbitals
[57]). In addition, intermetallic phases show an intimate mixture of a wide variety of bonding forms (see, e.g., Ref. [34]),
resulting in considerably smaller differences between extremes
in terms of ELF. For intermetallic phases containing both late
main-group and transition-metal constituents, such as found for
the typical members of the Heusler family, the higher ELF values are likely to be found in the region of the late main-group
constituent, regardless of the bonding characteristics.
The ELF of NiTiSn in the (1 1 0) plane (Fig. 9(b)) shows the
expected large ELF level around the Sn site. Turning the attention to the more interesting interstitial regions, the main feature
is the attractor located between Ni(X) and Ti(Y) confirming a
certain covalent character for this bond. The attractor has a maximum ELF value of 0.51 (see, e.g., Ref. [57] for a discussion on
ELF values in transition-metal compounds) and located some
20% closer to Ni(X) than Ti(Y). This is in line with the findings from DOS. In the three-dimensional representation of the
ELF for NiTiSn (Fig. 10(b)), the value of the isosurface, 0.37,
is chosen to enclose and emphasize the Ni–Ti attractors (seen
as discs in between the atomic spheres). Insight into the nature
of the Ni–Sn (X–Z) bond is harder to extract from ELF, since
any attractors close to the Sn(Z) atom would be blurred by the
high ELF values associated with the atom itself. To avoid this
interference the CD and ELF were plotted for certain energy
windows only (Fig. 11), thus visualizing the situation by differentiating between electrons with different energies. In NiTiSn it
is mainly electrons with energy between −2.25 eV and EF that
contribute to the covalent bonding between Ni(X) and Ti(Y)
(Fig. 11(a and b)). This is in accordance with the expectations
from DOS since this energy range comprises the main part of the
Ni-d states (see Fig. 7(a)). The main charge distribution found
between Ni(X) and Sn(Z) (Fig. 11(c)) is from the lower energy
range from −5.10 to −2.25 eV, and the ELF map (Fig. 11(d))
indicates some polar-covalent character of this bond since an
attractor bent toward Sn(Z) is located between the two atoms.
The s electrons found in the energy range −10 and −7 eV are
mostly confined to the Sn atoms as expected from the DOS analysis (Fig. 11(e and f)).
The CD distribution of AuMnSn (Fig. 9(d)) is similar to
that of NiTiSn, demonstrating finite charge distribution between
Au(X) and Mn(Y) as well as between Au(X) and Sn(Z). The corresponding CT to the interstitial regions of Fig. 9(f) also shows
similar features to (c), but the magnitude of charge transferred
L. Offernes et al. / Journal of Alloys and Compounds 439 (2007) 37–54
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 topological
partitioning of each map is conveniently chosen to present the most prominent
features in the respective energy windows. The locations of the X, Y, and Z
atoms are the same as in Fig. 2(b).
is significantly less in AuMnSn than NiTiSn. These similarities
and distinctions are also reflected in the ELF map (Fig. 9(e))
where the attractor found between Au and Mn is located along
the connecting line between the atomic-sphere domains. The
attractor is also in this case positioned closer to the Au(X) site
in analogy with the findings for NiTiSn, but the maximum value
of ELF for AuMnSn is lower (0.40) than the corresponding value
for NiTiSn.
The larger ELF value at the acceptor maximum for the
VEC = 18 phase NiTiSn could at first glance be associated
with the semiconducting behaviour of NiTiSn compared to the
metallic behaviour of AuMnSn (see Section 3.2.2). However,
when the maxima in the attractor values for all half-Heusler
phases subject to this study are compared, it becomes clear
that the semiconducting behaviour is not solely responsible
for higher ELF attractor values. As can be seen from Table 1,
47
the phases with VEC = 16 or 17 have in general higher ELFattractor values and those with VEC > 18 have lower values
than the VEC = 18 phases. Several factors like the electronegativity difference between X and Y, the selection of the Z
atom, and the lowering of ELF for constituents with high
l values, seem to be of some importance for evaluation of
the strength and character of the X–Y interaction (see also
Section 3.2.2).
The above view of the half-Heusler phases has somewhat different perspective from that seen from DOS (Fig. 8) and brings
out different features in terms of CD and ELF characteristics.
A few exceptions from the general trends indicated above will
now be considered in terms of ELF (Fig. 12). The VEC = 18
phase CuMgSb behaves like the other half-Heusler phases, but
since this phase is metallic the degree of hybridization must
be somewhat smaller than for the semiconducting VEC = 18
phases. This is reflected in a maximum value of only 0.46 for
the ELF attractor (Fig. 12(a) and Table 1). The other deviating
VEC = 18 phase in Table 1, ZnLiN, does not show any significant attractors between the atomic spheres. The maximum value
in the ELF between Zn and Li is too small to be considered as
indication of covalent bonding, another notable feature being a
weak polarization of N toward Li. The largely ionic character for
ZnLiN is thus confirmed by the ELF characteristics (Fig. 12(b)).
The covalence of the MgLiSb phase with VEC = 8 is also confirmed by the ELF map (Fig. 12(c)), the Sb atoms being largely
polarized toward Mg and the attractor is located close to Sb on
the connecting line between Mg and Li with a maximum ELF
value of 0.88. This bond should accordingly be classified as a
distinctly polar-covalent bond. Another special case is provided
by the VEC = 19 phase IrMnAl. For IrMnAl both the CD and
the ELF map (Fig. 12(d)) show a significant charge distribution
between all three kinds of atoms and the ELF map indicates a
large degree of hybridization for the Ir–Mn, Ir–Al, and Mn–Al
bonds. The attractor associated with the Ir–Mn bond is located
half way between the atomic spheres and exhibits a lower maximum value than the attractors on the Ir–Al and Mn–Al bonds that
are positioned close to, and bent toward, Al. These features lead
one to characterize the Ir–Al and Ir–Al bonds as polar covalent.
The shorter Ir–Al bond takes the highest ELF attractor value as
expected (Table 1).
3.2.2. Crystal-orbital-Hamilton population
The COHP, which is the Hamilton-population-weighted
density-of-states, is a partitioning scheme for the band-structure
energy in terms of orbital-pair contributions [58,59]. Negative
values for the COHP parameter indicate bonding, whereas positive values indicate antibonding behaviour, and COHP thus
provides energy-resolved visualization of the chemical bonding.
The bond strength between two interacting atoms in a solid can
be investigated by looking at the complete COHP between them,
taking into account all valence orbitals. The integrated COHP
(ICOHP) up to EF is, therefore, used as a qualitative measure
of mainly covalent bond strength. The bond strength between
pairs of interacting X, Y, and Z atoms in the half-Heusler phases
is thus investigated by calculating the COHP and comparing
the ICOHP values (data for selected, presumably representative
48
L. Offernes et al. / Journal of Alloys and Compounds 439 (2007) 37–54
Fig. 12. ELF for: (a) CuMgSb; (b) ZnLiN; (c) MgLiSb; and (d) IrMnAl. All plots refer to the (1 1 0) plane. Note that the scale and cut-off values on these illustrations
vary (from 0.10 to 0.60, from 0.050 to 0.70, from 0.010 to 0.85, and from 0.0010 to 0.75 in parts a, b, c, and d, respectively, with 10 levels in each map). The locations
of the X, Y, and Z atoms are the same as in Fig. 2(b).
phases are included in Table 1). The COHP profiles for NiMnSn
and AuMnSn are shown in Fig. 13.
The COHP for all bonds of NiTiSn (VEC = 18) confirms
bonding electrons in the entire energy region below EF and predominantly antibonding electrons above EF . In the energy region
below −5.1 eV the s-orbital interactions predominate. The interactions associated with Sn, viz. Ni–Sn and Ti–Sn, show strong
bonding interaction as expected from the more pronounced s
character usually connected with Sn. The energy region from
−5.1 eV to the pseudo-gap at −2.3 eV, is also dominated by the
Ni–Sn interactions, but here one also finds bonding contributions
from Ni–Ti and Ti–Sn interactions. The main bonding interaction in the energy region from −2.3 eV to EF originates from
the Ni–Ti bonds, especially around −2 eV (−2.3 to −1.6 eV)
where COHP exhibits a large feature for this bond and little or
no contribution from the other bonds.
The shape of the COHP curve for the Ni–Ti bond mirrors that
of the partial DOS of Ti (except for differences in magnitude),
which in return show large similarities to the partial DOS of Ni
(see Fig. 7(a)). This behaviour of the COHP is typical for covalent interactions. The pronounced feature in the COHP for the
Ni–Ti interaction around −2 eV can be recognized in the orbital
projected partial DOSs for the atoms with d character, both atoms
exhibiting large peaks of d character in this region. The COHP
curve for the Ni–Sn bond also shows distinct similarities to the
partial DOSs for Ni and Sn. Such features are usually found to
be correlated with covalent interaction, but the COHP bonding
interactions are most dominant in the low energy region, which
is typical for ionic interactions. These features suggest that the
Ni–Sn bond has a certain polar character. The longer Ti–Sn bond
has a COHP profile with low-to-intermediate values in the entire
energy region. Orbital-resolved COHP calculations show that
the bonding interaction between the Ni and Sn atoms is mainly
of sp3 character (the orbital-projected COHP-calculated interaction between the s orbital of Sn and the three p orbitals of Ni
are equivalent and vice versa).
The equal lengths of the Ni–Ti and Ni–Sn bonds indicate
similar bond strength, and this is indeed also brought about
by the ICOHP values (see Table 1, recalling that ICOHP primary reflects strength of covalent interaction). According to the
ICOHP data the longer Ti–Sn bond should have almost half
the strength of the Ni–Ti and Ni–Sn bonds. The bond strengths
derived from the ICOHP values for NiTiSn are large compared
to those for the other half-Heusler phases. The ICOHP values
for Ni–Ti and Ni–Sn are slightly smaller than (really of the same
magnitude as) the ICOHP value of 2.3 found for the Sn–Sn bond
in semiconducting ␣-Sn. The Ni–Sn bonding interaction appears
to be strongest in the lower part of the energy range, coinciding
with the low lying states of mainly Sn-s character in the energy
range −9.8 to −7.1 eV, and partially the Sn-p and Ni-d energy
L. Offernes et al. / Journal of Alloys and Compounds 439 (2007) 37–54
Fig. 13. Crystal-orbital-Hamilton population (COHP) represented: (a) schematically; (b) as obtained computationally for NiTiSn; (c) as obtained computationally for AuMnSn. In part a three different positions of EF are shown with vertical
lines, in parts b and c the interactions between X–Y, X–Z, and Y–Z atomic
pairs are distinguished by solid, dashed, and dotted profiles, respectively, as also
labelled on the illustrations.
states in the range −5.0 to −2.3 eV. The Ni–Ti interaction dominates the region from −5.0 eV to EF and the COHP profile is
mirroring the partial DOS profiles of Ni and Ti, indicating a large
degree of hybridization (see Fig. 7(a)). There is a large feature
in the COHP profile for the Ni–Ti bond around −2 eV, which
appears to be correlated to a peak in the Ni-d states. This feature is not recognized in the Ni–Sn and Ti–Sn interactions. The
COHP of the Ti–Sn interaction is fairly low in the whole energy
range and the resulting ICOHP indicates lower bond strength in
accordance with the inferences from the bond lengths.
The ICOHP values for AuMnSn (Table 1) indicate that the
strength of the Au–Sn bond is of the same magnitude as the
49
Au–Mn bond and about twice the strength of the Mn–Sn bond,
in full accordance with the inferences from the bond lengths. The
Au–Sn bonding interaction (Fig. 13(c)) is strongest in the lower
part of the energy range, coinciding with the band of mainly
Sn-s character at −10.2 to −8.1 eV, and the largely localized
Au-d states at −6.4 to −4.9 eV (see Fig. 7(b)). However, the situation for the Au–Mn interaction is different, since the bonding
interactions are strong both in the region dominated by Au-d
states and in the higher energy range from −6.4 eV to EF where
the partial DOSs of Au and Mn show appreciable hybridization.
The COHP profile in the latter region does not mirror the profile of the localized majority-Mn-d states since these unpaired
electrons are magnetic and non-bonding.
The strengthening of the X–Y bond on going from AuMnSn
to NiTiSn seen in the ELF maps is supported by the findings from
the COHP. The COHP of the Ni–Ti interaction show a large feature around −2 eV, which is not recognized in the Au–Mn COHP.
This distinction can be understood by looking at the orbitalprojected partial DOS of the atoms concerned. At −2 eV the
main feature in the site-projected DOS of Ni is of d character and
this has a predominant d-orbital feature as counterpart in the partial DOS of Ti. This d–d interaction is not favoured in AuMnSn
since the d electrons of Mn participate in magnetism rather than
bonding. The strengthening of the X–Z bond is also reflected in
enhanced bonding contributions in the higher energy range, but
these are more evenly distributed over the energy range and can,
therefore, not easily be assigned to a single interaction.
A comparison of the ICOHP data for NiTiSn and AuMnSn
may be somewhat misleading. The semiconducting phases such
as CoTiSb and CoVSn also have ICOHP values comparable to
those of AuMnSn. A comparison (in this respect more appropriate) between NiTiSn and NiTiSb shows that the bond strength is
lowered when the semiconducting state disappears. Apart from
this, the differences in the ICOHP values seen in Table 1 seem to
stem from intrinsic differences inherent in the element combinations concerned rather than differences in VEC. For half-Heusler
phases containing light elements, e.g., Li, N, Mg, and Al, the
atomic interactions will be necessarily different because of large
relative size differences that occurs when these constituents are
involved and the lower polarizability of these atoms. In MgLiSb
and ZnLiN, which contain two light elements, only one of the
interaction pairs takes a significantly large COHP level, strengthening the claim that these phases, bonding-wise, belong to a
separate branch of the half-Heusler family.
According to Rytz and Hoffman [60], Dronskowski [61], and
Landrum and Dronskowski [59] one can also obtain insight into
the stability of a phase by examining the COHP for all bonds.
For a stable, semiconducting phase EF will separate bonding
and antibonding states and thus ensure optimized bonding and
minimized energy. Such a state can be recognized and characterized by the “optimized” COHP and this situation is depicted
schematically in Fig. 13(a) as the horizontal part of the COHP
curve in the region near EF (level indicated by a dashed vertical
line). If the atomic interactions are antibonding at EF according
to COHP (see the dotted vertical line in Fig. 13(a)), stability may
be induced by removing or reducing the amount of antibonding
states by some kind of perturbation, viz. introducing changes
50
L. Offernes et al. / Journal of Alloys and Compounds 439 (2007) 37–54
in electronic structure, crystal structure or compositional stoichiometry. Conversely, if COHP is bonding at and above EF (see
the dotted-dashed line in Fig. 13(a)) a perturbation that maximizes the bonding will also stabilize the phase.
The present calculations have been performed on phases,
which are known to exist and with the experimentally established lattice parameter for a confirmed AlLiSi-type atomic
arrangement. However, the strict 1:1:1 stoichiometry used in the
calculations for all phases is certainly not generally correct since
some of the more carefully investigated phases definitely have
been shown [20] to exhibit deviations from the equiatomic stoichiometry. A closer examination of the COHP near EF for such
phases is likely to provide an estimate for how large the probable deviation in stoichiometry might be. Another utilization of
the COHP concept could be to stipulate the effect of variation
in VEC on substitution by examining the occurrence of bonding, non-bonding, and antibonding states around EF . Then upon
integration of DOS this situation may be converted into electron
content for a hypothetical substituted phase, which in turn may
be subjected to experimental testing.
The basic empirical hypothesis that the numerous VEC = 18
phases are more stable than those with VEC = 18 is substantiated
by COHP. For the VEC = 18 phases EF lies between bonding and
antibonding states as expected for semiconducting phases and
VEC = 18, therefore, serves as a point of reference. For example, looking at the COHP for the electron-deficient phase FeTiSn
(VEC = 16) optimum VEC is found 0.41 eV above EF . A crude
integration of DOS from EF to 0.41 eV indicates that a supply of
approximately 2 electrons will fill up the bonding states. Hence,
an atomic substitution in FeTiSn that does not change the DOS
profile significantly (viz. assuming a proper rigid-band filling)
is likely to provide a stable phase. The existing VEC = 17 phases
FeTiSb and CoTiSn exhibit COHP and integrated DOS profiles
which indicate stabilization upon adding one electron by substitution, e.g., only bonding or non-bonding states are filled in such
substitutions. The “two-electron-substituted” VEC = 18 phase
CoTiSb exists and the COHP characteristics of this reflect semiconductivity. However, exchanging all atoms in FeTiSn with the
nearest neighbour higher valent atom from the periodic table
also gives an existing phase, namely CoVSb (VEC = 19). The
COHP for this phase indicates a certain destabilizing tendency
brought about by excess of electrons (the electrons close to EF
are involved in antibonding interactions). Exchange of Sb for Sn
leads to removal of these states and yet another existing phase,
CoVSn (VEC = 18), is obtained. The VEC = 19 phase NiTiSb
also comprizes antibonding states at EF and its “optimized”
COHP occurs at 0.80 eV below EF . A “optimized” situation can
in this case be obtained by removing one electron by substitution of one of the constituents by a lower valent element. All the
resulting VEC = 18 combinations CoTiSb, NiScSb, and NiTiSn
exist and exhibit semiconducting properties.
The VEC = 19 phase IrMnAl does not follow the pattern outlined above. The DOS of IrMnAl show that a rigid-band-like
removal of electrons to VEC = 18 will not give a semiconducting phase. The light main-group-element Al gives rise to a much
smaller degree of hybridization than the heavier atoms generally found in half-Heusler phases, and subsequently no gap is
found in the DOS profile. According to the COHP there appears
to be no gain in bonding interaction on substitution with an
atom with one electron less. On the other hand, a substitution
which brings about an additional 1.5 electrons should provide a
stable phase. There is no report on any other Al-containing halfHeusler phases, but IrMnSn (VEC = 20) and IrMnSb (VEC = 21)
do exist. The COHP and DOS data for these phases indicate
that stable phases can be obtained by addition of electrons.
The phases PtMnSn (VEC = 21) and PtMnSb (VEC = 22) both
exist and the latter belong to the HMF category. It is worth noting that even though AuMnSn is a VEC = 22 phase, maximum
bonding interaction occurs above EF for the ideal 1:1:1 compound. This phase should, therefore, be stabilized by adding
electrons in the Au and Mn bands to maximize the bonding
interaction. The experimental solid-solubility range of AuMnSn
[20] does indeed indicate that a fraction of the Sn atoms is
replaced by Mn and/or Au. The VEC = 23 AuMnSb phase have
its optimum COHP below EF . A removal of about 0.5 electrons would, therefore, bring about a more stable situation. A
processing of AuMnSb according to rigid-band assumptions
would theoretically give a HMF phase. FLMTO-supercell calculations on the solid-solution series AuMnSn1−x Sbx predict HMF
behaviour within the solid-solubility range, more specifically at
0.50 < x < 0.75 [13]. CuMnSb is also a VEC = 23 phase, but in
this case rigid-band removal of one electron (viz. the condition
for optimum COHP) fails to predict a HMF phase, although the
complete exhange of Cu with Ni or Pt leads to known HMF
phases.
An essential lesson from these considerations is accordingly
that both VEC = 18 and VEC = 22 configurations lead to optimum bonding and stability conditions, which in turn implies
that stability of phases derived by simple substitutions can be
predicted.
3.3. Magnetic aspects
The magnetic electrons have not been considered specifically
in the discussion on bonding since they do not contribute to the
bonding properties. Many of the half-Heusler phases are interesting in relation to their magnetic properties as referred to in
Section 1. There are also a large group of half-Heusler phases
with more indistinct magnetic properties, e.g., only small magnetic moments are found for most of the phases with VEC = 16,
17, and 19 (as shown schematically in Fig. 5). The ferromagnetic Mn-based half-Heusler phases with VEC > 18, however,
have large magnetic moments and are commonly considered as
true local moments magnets. In the present study, accurate theoretical calculations have been performed for four such phases;
namely AuMnSn, AuMnSb, PtMnSb, and NiMnSn. Calculated
total magnetic moments for these phases are listed in Table 1.
The spin-polarized calculations of the other phases (including a
larger number of phases not documented in Table 1) are made
to estimate magnetic moments only.
The non-spin-polarized band structures for the XMnZ halfHeusler phases show a peak at EF (viz. a distinct N(EF )), which
indicates that the non-magnetic phases have low stability and
spontaneous magnetism [49] will prevail. As already established
L. Offernes et al. / Journal of Alloys and Compounds 439 (2007) 37–54
experimentally [2,3], the characteristics of these magnetically
interesting phases is that they are basically ferromagnetically
ordered, with large magnetic moments confined to the Mn atom
and only small induced magnetic moments at the X and Z sites.
The X atoms are ferromagnetically coupled to Mn and this is
strongest among the possible magnetic interactions. The antiferromagneticlly induced moment on the Z site could be indicative
of a superexchange interaction between the Mn moments mediated by the intermediate Z atoms. However, the present authors
believe that the major contribution to the ferromagnetic coupling stems from a RKKY-like mechanism brought about by the
conduction electrons.
To investigate the magnetic interactions further, spin density were estimated for the magnetic phases. Spin-density maps
were obtained by subtracting the minority-spin electron density
from the majority-spin electron density, and regions with higher
probability of up-spin electrons here appear with positive spin
density and conversely regions with higher probability of downspin electrons appear with negative values. In the spin-density
map for AuMnSn (chosen as representative for the four phases
mentioned above; Fig. 14(a)) a large up-spin density is seen
at the Mn(Y) site, as expected from considerations according
to local magnetism as well as manifested by the experimental data [3]. Only small positive spin-density values are found
at the Au(X) site and small negative values at the Sn(Z) site.
More interestingly we note there are two major spin-density
features on interstitial locations between the atoms. The ferromagnetically coupled magnetic moments of Au(X) and Mn(Y)
are apparently mediated by antiferromagnetically coupled spin
in regions between them. This indicates a d–d exchange mechanism through the covalent bond between these atoms. Secondly,
there are interconnected regions of up-spin electrons connecting the Mn(Y) atoms via the crystallographically empty 4d site,
which may be taken as manifestation of the suggested RKKYlike interaction via conduction electrons. To substantiate these
indications calculations were also made for electrons in the
energy region close to EF only. Fig. 15(b and d) shows spin
densities for AuMnSn for electrons in the range from −1.45 eV
to EF and −0.050 eV to EF . To provide a pictorial impression
51
of the contribution of a given atom in a certain energy range
the corresponding charge-density plots are shown in Fig. 15(a
and c). The energy range −1.45 to 0 eV should reflect the energetically uppermost Mn-d(Y-d) electrons together with contributions from d electrons associated with the Au(X) atoms. The
spin-density distribution for this energy range (Fig. 15(b)) illustrates how the up-spin electrons at the Mn(Y) site is oriented
toward the Au(X) sites. (The four equivalent interaction paths
appear here as two quite pronounced maxima oriented toward the
in-plane Au(X) sites and two weaker maxima oriented toward
out-of-plane Au(X) sites.) Corresponding up-spin regions at the
Au(X) site are oriented toward the Mn(Y) sites. As in the total
spin-density plot, these up-spin regions are connected to downspin regions located on the lines connecting the atomic sites,
thus confirming the findings from the total spin-density plot.
The regions of up-spin electrons connecting the Mn(Y) atoms
through the empty 4d sites are easily recognized in Fig. 15(b),
but since this plot also includes electrons that do not contribute to
conduction even at elevated temperatures, attention should rather
be turned to the region closer to EF (Fig. 15(d)) for information
about magnetic exchange interaction via conduction electrons.
This plot shows that the conduction electrons are indeed spinpolarized with interconnecting regions dominated by up- and
down-spin electrons. There is especially a pile up of down-spin
electrons around both the Mn(Y) and Au(X) sites (which to some
extent also includes the empty 4d site) that suggest a broader flow
of electrons than suggested by the total spin density. Our overall
conclusion is accordingly that there are definite indications of
a RKKY-like magnetic exchange mechanism being operative in
these phases.
Another way to illustrate possible accumulations of unpaired
electrons is to multiply the ELF with the CD and for the interpretation of this composite function make the (not too illogical)
assumption that high values reflect paired electrons (PAIR). A
next step could be to subtract the thus modified CD function
PAIR from the plain CD function (valence-electron density) and
interpret high values as a coarse measure of unpaired electron
(UNPAIR) constellations. Such an UNPAIR plot for AuMnSn
is given in Fig. 14(b) and this manipulation of the potential con-
Fig. 14. (a) Total spin density and (b) UNPAIR (see text) for AuMnSn. In part a, regions which are dominated by up-spin electrons are shown by positive values
(solid contour lines), regions which is dominated by down-spin electrons are shown by negative values (dashed contour lines), and the border line is at zero (thicker,
solid line). The locations of the X, Y, and Z atoms are the same as in Fig. 2(b).
52
L. Offernes et al. / Journal of Alloys and Compounds 439 (2007) 37–54
duction electrons once again supports the inference that there is
an accumulation of unpaired electrons in the otherwise empty
region between the Au(X) and Mn(Y) atoms.
3.4. Concluding remarks
The trends in electronic structure with respect to valenceelectron content (VEC) are quite clear-cut for the majority of
the half-Heusler XYZ phases which comprises X = heavy transition metal, Y = light transition metal, and Z = late main-group
element. A simple schematic representation of the electronic
band structure with distinctly identifiable contributions from
X-d, Y-d, and Z-s and -p states gives a good qualitative picture of the electronic structure for a given half-Heusler phase
based on knowledge of the constituents that are involved. Thus
from the chemical composition only, one can predict density-ofstate (DOS) profile, chemical bonding features including degree
of hybridization, and magnetic properties with reasonable estimates for the size of magnetic moments. However, it must be
emphasized that care must be exercised in considerations of
phases, which occur in border regions between co-operative
and non-co-operative magnetism. Here, small deviations in lattice parameter and/or stoichiometry may alter the scenery quite
appreciably.
The reason behind the close correlation in fundamental properties of the mainstream half-Heusler phases is the common,
cubic, positional-parameter-free AlLiSi-type structural arrangement. This leaves only a few degrees of freedom, like lattice
parameter and stoichiometry, and the structural arrangement of
the atoms, therefore, really predetermines main bond character
and DOS profiles. In the idealized case, this leads to a rigidband system where properties are governed by VEC rather than
the combination of atoms involved. A quick look at clusters
of phases with correspondingly simple cubic crystal structures
and large number of representatives show that similar considerations may be successfully applied to suitable selections of
phases with, e.g., NaCl-, CsCl-, ZnS(zink blende)-, Cr3 Si-, or
Cu3 Au-type structure.
For most of the half-Heusler intermetallic phases, the “optimized” COHP (crystal-orbital-Hamilton population) demonstrates the simple chemical significance of VEC = 18 for the
semiconducting representatives and VEC = 22 for the halfmetallic ferromagnetic representatives. The former VEC value
applies to the situation in which all bonding states are filled
and all antibonding/non-bonding states are empty, and the latter
VEC value to the corresponding situation for the majority band
only. The VEC = 18 configuration is easily rationalized as the
situation with complete filling of one set of s, p, and d states. A
Fig. 15. Maps of (a) charge density from −1.45 eV to EF ; (b) spin density from −1.45 eV to EF ; (c) charge density −0.050 eV to EF ; and (d) spin density from
−0.050 eV to EF , all for AuMnSn. In parts b and d, regions which are dominated by up-spin electrons are shown by positive values (solid contour lines), regions
which are dominated by down-spin electrons are shown by negative values (dashed contour lines), and border lines are at zero (thicker, solid line). The locations of
the X, Y, and Z atoms are the same as in Fig. 2(b).
L. Offernes et al. / Journal of Alloys and Compounds 439 (2007) 37–54
similar consideration for the VEC = 22 yields complete filling
of s, p, and d orbitals in the minority-spin channel and an additional filling of four electrons in the 10 Y-d states. With full spin
polarization and filling of half of the 10 Y-d states, however, the
“optimized” VEC would occur at 23 rather than 22. This can
rationalize why the PtMnSb and NiMnSn phases has an “optimized” COHP at VEC = 22, while the corresponding gold phases
obtains an “optimized” COHP for VEC > 22. For the rare-earth
phases, the VEC varies between 25 and 32, the former value
corresponding to complete filling of s, p, and d, and half filling
of f states and the latter to situations with complete filling of
all s, p, d, and f states. In a quest for identifying new phases,
the “optimized” COHP concept may be utilized both as a tool
for prediction of substitutionally derived phases as well as for
identifying stable non-stoichiometry and solid-solution ranges.
The ELF (electron-localization function) and the COHP analyses have proved to be powerful tools to establish the nature of
bonding interactions that is “typical” for half-Heusler phases,
viz. the largely covalent bonding character. The X–Y and X–Z
bond strengths are of corresponding magnitude, but the latter
bond exhibits a pronounced polarity with the bonding electrons
closer to Z than X. The variation in covalent bond strength varies
with the degree of hybridization, which in turn depends on the
VEC as well as more empirical quantities like electronegativety,
size difference, and polarizability of the atoms. The strict structural constraints require that the constituents, especially the Z
atoms, must be polarized to some extent to facilitate enhanced
hybridization. In the struggle to fulfil this demand the preferable
choice of constituents, X turns out to be a late transition metal,
while Z presents itself as a heavy main-group metal, viz. Sn or
Sb as empirically established. The electron-deficient phases (as
compared with the idealized semiconducting VEC = 18 phases)
also have traits of character of metallic bonding, while the extra
electrons in the electron-excessive phases lead to metallic bonding as well as appearance of electrons that are classified as
non-bonding and magnetic. However, the metallic half-Heusler
phases have poor electrical conductivity properties as evident
from experimental measurements (unpublished work by some
of the authors) and other physical properties, e.g., the gray, nonmetallic luster and a brittle nature with little or no ductility. These
features and the similarity in bonding throughout the bulk of
the half-Heusler phases indicate that the existence of a metallic
bonding component in the VEC = 18 phases is of minor importance in the overall bonding picture. The metallic contribution
to the bonding is, however, important for the total energy of the
system and thus contributes to stability.
3.5. Postscript
After the present contributions had been submitted, and in fact
accepted for publication, a paper by Kandpal et al. [62] came to
our attention. This paper also deals with the bonding situation
in the half-Heusler phases on the basis of first-principles DFT
calculations according to essentially the same computational
codes as used in this work and the methodological similarity
extends even further by encompassing the same visualization
tools (notably CD, COHP, and ELF) to examine the findings.
53
It is therefore not surprising that some of the interpretations
are similar. However, as emphasized in the introduction there
are known over a hundred half-Heusler phases and the specific
phases subjected to the computational treatment in the two independent studies are, with a few exceptions (CoVSn, CoTiSb, and
NiMnSb), not the same. This makes the studies complementary
rather than competitive.
Kandpal et al. focus largely on the VEC = 8 phases and
conclude that their bonding situation can be described as
ionic interactions between cationic Y constituents and anionic
XZ sublattices with internal X–Z covalent bonding. This is
essentially the same as found for the VEC = 8 phases in the
present study (see the above consideration on MgLiSb). For the
VEC ≥ 18 phases Kandpal et al. used three-dimensional isosurface representations of CD and ELF to extract information on
the character of the chemical bonding. Again they draw much
the same conclusions as established above. However, the examination of ELF in the (1 1 0) plane allowed the present study
to detect covalent bonding within the XY sublattice (identified
through the occurrence of attractors) when two transition metals are involved. This concurs both with the similarities in the
partial DOS profiles for the X and Y constituents, and the large
ICOHP values found for the X–Y bonds. In general Kandpal et
al. appear to draw somewhat different conclusions from us in
cases where two transition metals (X and Y) are involved and
the value of VEC is high.
Our overall conclusion is that the Kandpal et al. and present
studies together significantly contribute to a deeper understanding of the half-Heusler phases.
Acknowledgements
LO and PR appreciate the financial support from the Research
Council of Norway. Parts of the calculations were carried out on
the Norwegian supercomputer facilities.
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