Elastic properties of 5d transition-metal carbides: An ab initio study
We have systematically studied the mechanical stability of group V transition metal carbides TMC₂ (TM=Hf, Ta, W, Re, Os, Ir, Pt, and Au) in the pyrite and fluorite phase, by calculating their elastic constants within the density functional theory scheme. It was found that all but ReC₂ and OsC₂ are s...
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Mex, L. Aguayo, A. Murrieta, G. 2019-06-16T15:15:08Z 2019-06-16T15:15:08Z 2015 Elastic properties of 5d transition-metal carbides: An ab initio study / L. Mex, A. Aguayo, G. Murrieta // Condensed Matter Physics. — 2015. — Т. 18, № 3. — С. 33801: 1–7. — Бібліогр.: 24 назв. — англ. 1607-324X DOI:10.5488/CMP.18.33801 arXiv:1510.06917 PACS: 81.05.Je, 81.05.Zx, 71.15.Mb, 71.20.Be https://nasplib.isofts.kiev.ua/handle/123456789/155269 We have systematically studied the mechanical stability of group V transition metal carbides TMC₂ (TM=Hf, Ta, W, Re, Os, Ir, Pt, and Au) in the pyrite and fluorite phase, by calculating their elastic constants within the density functional theory scheme. It was found that all but ReC₂ and OsC₂ are stable in pyrite phase. On the other hand, all metal carbides studied were unstable in the fluorite phase. Проведено систематичнi дослiдження менханiчної стiйкостi карбiдiв перехiдних металiв п’ятої групи TMC₂ (TM = Hf, Ta, W, Re, Os, Ir, Pt, i Au) у пiритовiй i флюоритовiй фазах шляхом обчислення їх еластичних сталих в рамках теорiї функцiоналу густини. Встановлено, що всi карбiди металiв, за винятком ReC₂ i OsC₂, є стiйкими у пiритовiй фазi. З iншого боку, всi метали карбiдiв, що вивчалися, виявилися нестiйкими у флюоритовiй фазi. The authors would like to thank Carlos Brito for helpful comments. This work was supported by Facultad de Matámaticas-UADY under Grant no. FMAT–2012–0007 and CONACyT under Grant no. 025794. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Elastic properties of 5d transition-metal carbides: An ab initio study Еластичнi властивостi карбiдiв 5d перехiдних металiв: ab initio дослiдження Article published earlier |
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Elastic properties of 5d transition-metal carbides: An ab initio study |
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Elastic properties of 5d transition-metal carbides: An ab initio study Mex, L. Aguayo, A. Murrieta, G. |
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Elastic properties of 5d transition-metal carbides: An ab initio study |
| title_full |
Elastic properties of 5d transition-metal carbides: An ab initio study |
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Elastic properties of 5d transition-metal carbides: An ab initio study |
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Elastic properties of 5d transition-metal carbides: An ab initio study |
| title_sort |
elastic properties of 5d transition-metal carbides: an ab initio study |
| author |
Mex, L. Aguayo, A. Murrieta, G. |
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Mex, L. Aguayo, A. Murrieta, G. |
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2015 |
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Condensed Matter Physics |
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Інститут фізики конденсованих систем НАН України |
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Article |
| title_alt |
Еластичнi властивостi карбiдiв 5d перехiдних металiв: ab initio дослiдження |
| description |
We have systematically studied the mechanical stability of group V transition metal carbides TMC₂ (TM=Hf, Ta, W, Re, Os, Ir, Pt, and Au) in the pyrite and fluorite phase, by calculating their elastic constants within the density functional theory scheme. It was found that all but ReC₂ and OsC₂ are stable in pyrite phase. On the other hand, all metal carbides studied were unstable in the fluorite phase.
Проведено систематичнi дослiдження менханiчної стiйкостi карбiдiв перехiдних металiв п’ятої групи
TMC₂ (TM = Hf, Ta, W, Re, Os, Ir, Pt, i Au) у пiритовiй i флюоритовiй фазах шляхом обчислення їх еластичних сталих в рамках теорiї функцiоналу густини. Встановлено, що всi карбiди металiв, за винятком
ReC₂ i OsC₂, є стiйкими у пiритовiй фазi. З iншого боку, всi метали карбiдiв, що вивчалися, виявилися
нестiйкими у флюоритовiй фазi.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/155269 |
| citation_txt |
Elastic properties of 5d transition-metal carbides: An ab initio study / L. Mex, A. Aguayo, G. Murrieta // Condensed Matter Physics. — 2015. — Т. 18, № 3. — С. 33801: 1–7. — Бібліогр.: 24 назв. — англ. |
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| first_indexed |
2025-11-25T14:38:54Z |
| last_indexed |
2025-11-25T14:38:54Z |
| _version_ |
1850514941424959488 |
| fulltext |
Condensed Matter Physics, 2015, Vol. 18, No 3, 33801: 1–7
DOI: 10.5488/CMP.18.33801
http://www.icmp.lviv.ua/journal
Elastic properties of 5d transition-metal carbides:
An ab initio study
L. Mex1, A. Aguayo2, G. Murrieta2
1 Facultad de Ingeniería, Universidad Autynoma de Yucatán. Av. Industrias no Contaminantes por Periférico
Norte Apdo. Postal 150 Cordemex Mérida, Yucatán, México
2 Facultad de Matemáticas, Universidad Autynoma de Yucatán. Periférico Norte, Tablaje 13615, C. P. 97110,
Mérida, Yucatán, México
Received December 9, 2014, in final form April 14, 2015
We have systematically studied the mechanical stability of group V transition metal carbides TMC2 (TM =Hf, Ta,
W, Re, Os, Ir, Pt, and Au) in the pyrite and fluorite phase, by calculating their elastic constants within the density
functional theory scheme. It was found that all but ReC2 and OsC2 are stable in pyrite phase. On the other hand,
all metal carbides studied were unstable in the fluorite phase.
Key words: first-principles calculation, elastic constants, hard material, transition metals
PACS: 81.05.Je, 81.05.Zx, 71.15.Mb, 71.20.Be
1. Introduction
The elastic stability criterion has been used to establish the possible stability or metastability of a
crystallographic phase in the solid state. It has been shown that through the first-principles calculation it
is possible to obtain the elastic properties and to describe the strength of the bond between neighboring
atoms. Since the adequacy of the theoretical results made it possible to adequately predict the formation
of new crystal structures, it became possible to synthesize some of the predicted structures.
Intensive theoretical and experimental efforts have been focused on the possibility of finding new
low compressibility materials with hardness comparable with diamond [1]. Superhard materials are of
primary importance in modern science and technology due to their numerous applications, starting from
cutting and polishing tools up to wear resistant coatings. In this search, special interest has been taken
in the metal carbides and nitrides. The introduction of smaller atoms such as nitrogen or carbon into
interstitial sites in closely packed transition metal lattices changes their chemical and physical properties
with respect to themetal. Transitionmetal carbides and nitrides have been considerably investigated due
to their unique chemical and physical properties, such as high thermal conductivity, high melting point,
chemical inertness, high stiffness, high hardness, and metallic electrical conductivity [2–7]. Transition
metal carbides or nitrides had been little studied due to the difficulty of obtaining the crystalline sam-
ples. However, a series of works such as PtN, PtN2, IrN2, and PtC [8] triggered a rapid advance in the pro-
duction of new carbide and nitride materials. They can have different morphologies, atomic structures
and substantially differ from the corresponding crystalline phase relative to their physicochemical prop-
erties. To date, a wide group of nanocarbides of the d metals such as molecular clusters, nanocrystals,
nanospheres, nanowires, nanotubes, etc., have been synthesized. Several of these new systems formed
by transition metals and nitrogen or carbon have found no consensus on their crystal structure. Among
all these materials, the group of platinoid nitrides and carbides attracted a particular interest due to their
technological potential. Crowhurst et al. [9] in order to explain the high bulk modulus of platinum ni-
trides studied the Pt–N system in the stoichiometry PtN2. They analyzed the composite in two different
structures, fluorite and pyrite, and found that the most stable phase is the pyrite phase, with an internal
parameter u = 0.415.
© L. Mex, A. Aguayo, G. Murrieta, 2015 33801-1
http://dx.doi.org/10.5488/CMP.18.33801
http://www.icmp.lviv.ua/journal
L. Mex, A. Aguayo, G. Murrieta
To explore the possible existence of 5d transition-metal carbides with C/M = 2 stoichiometry in the
pyrite or fluorite phases, using density functional theory, we calculated the elastic constants in both cubic
phases and analyzed the performance of elastic stability criteria.
2. Methods
The first-principles calculations were performed based on the density functional theory (DFT). The
Kohn-Sham total energies were self-consistently calculated using the linearized augmented plane wave
method (FP-LAPW) with local orbital extensions [10], as implemented in theWIEN2k [11, 12] code, where
the core states are treated fully relativistically, and the semicore and valence states are computed in a
scalar relativistic approximation. The exchange-correlation terms were considered in the Perdew-Burke-
Ernzerhof form of the generalized gradient approximation (GGA) [13]. We have chosen the muffin-tin
radii (RMT) of 2.0 a.u. for the transition metals and 1.2 a.u. for the carbon atoms. The self-consistent
calculations were done with an LAPW basis set defined by the cutoff RMTKMAX=8.0. Inside the atomic
spheres, the potential and charge densities are expanded in crystal harmonics up to L = 10. Convergence
was assumed when the energy difference between the input and output charge densities was less than
1× 10−5 Ry. The calculations were carried out with a sufficiently large number of k points in the first
Brillouin zone (BZ). We used a 13×13×13 k-point mesh, yielding a different number of k points in the
irreducible wedge of the BZ depending on the structure: 256 for the fluorite, and 176 for the pyrite phase.
We evaluated the structure of TMC2 (TM=Hf, Ta, W, Re, Os, Ir, Pt, and Au) in the pyrite and fluorite
phase. Pyrite (FeS2 structure type) has a cubic crystal structure with space group Pa3 (205), the transition
metal atoms occupying Wyckoff site 4a (0,0,0), and the carbon atoms are grouped as dimers around the
fcc octahedral interstitial sites oriented in the 〈111〉 directions at 8c (u,u,u). The transition metal in the
pyrite structure is fixed by symmetry but the carbon atoms have one free parameter (u) [14]. Fluorite is
a particular high symmetry phase of pyrite. When the free parameter u = 0.25 at the 8c site, we obtain
the fluorite (CaF2 structure type) structure with space group cF12 (225).
The calculated total energy as a function of volume was fitted to the Birch–Murnaghan equation of
state [15]. From this process, the equilibrium lattice constant (a) and the bulk modulos (B) were obtained.
All crystals in a cubic structure have only three independent elastic constants, namely C11, C12, and C44.
One can use a small strain and calculate the change of energy or stress to obtain elastic constants Ci j .
In the crystal structures analyzed in this work, an external strain δ from −0.08 to + 0.08 was applied in
the directions as explained by Güemez et al. [16], associated with deformations: isotropic, tetragonal and
orthorhombic.
ǫiso =
(1+δ)1/3 0 0
0 (1+δ)1/3 0
0 0 (1+δ)1/3
,
ǫtet =
(1+δ)−
1
3 0 0
0 (1+δ)−
1
3 0
0 0 (1+δ)
2
3
, (1)
ǫort =
1 δ 0
δ 1 0
0 0 1+δ
2
,
to distort the lattice vectors, R′ = (1+ ǫ)R. The resulting changes of energy are associated with elastic
constants,
∆Eiso =
V0
2
(C11 +2C12)δ
2
=
2V0
3
Bδ
2
,
∆Etet =
V0
3
(C11 −C12)δ
2 and
33801-2
Elastic properties of 5d transition-metal carbides: An ab initio study
∆Eort = 2V0C44δ
2
.
In order to be mechanically stable, the elastic stiffness constants of a given crystal should satisfy
the generalized elastic stability criteria [17]. The elastic stability criteria for a cubic crystal at ambient
conditions are,
C11 +2C12 > 0, C11 −C12 > 0, and C44 > 0, (2)
i.e., all the bulk moduli (B), shear (C44), and tetragonal shear [C ′ = (C11 −C12)/2] moduli are positive. We
also calculated the Young’s modulus (E ), which provides a measure of stiffness and stability of the solids.
Another interesting elastic property for any applications, particularly for their anisotropy, is the Zener
factor A. These quantities are calculated in terms of the computed Ci j using the following relations
E =
9BG
3B +G
, (3)
A =
C44
C ′
=
2C44
C11 −C12
, (4)
where G is the isotropic shear modulus. For a cubic material with its two shear constants, C44 and C ′, the
value of G should be between these two constants. Therefore, G has a unique value if C44 = C ′. This hap-
pens if the material is isotropic, that is, the Zener factor is equal to one, as in the case of W. By assuming
a homogeneous strain on the compound, Voigth [18] established the upper limit of G as
GV=
1
5
C ′
(2+3A). (5)
On the other hand, assuming a homogeneous stress, as the lower bound Reuss [19] proposes
GR= 5C ′ A
3+2A
. (6)
In this work, we take the arithmetic average as proposed by Hill [20]
G =
1
2
(GV+GR). (7)
3. Results and discussion
In figure 1 we show the calculated total energy of pyrite HfC2 and AuC2 (pyrite-face) for seven values
of the cell volume (open circles). The calculated total energy as a function of volume was fitted to the
Birch-Murnaghan equation of state[15]. The fit is presented in figure 1 (solid line), were the energy is
given with respect to the minimum energy of the pyrite structure.
In order to find the value of the free parameter u in the pyrite phase, the DFT total energy and the
forces acting on the atoms were optimized. Figure 2 shows the DFT total energy as a function of the
free parameter u for HfC2. In this curve, we have got two local minima, one of them being observed at
u = 0.25, which correspond to a possible metastable fluorite phase. At u = 0.43, there is another minimum
at much lower energy, which corresponds to pyrite phase. In all the 5d transition metal carbides studied
in this work, we have got the same value for the free parameter u. The difference in the energy between
the two minimum phase suggests that pyrite phase should be a more stable phase at zero pressure.
From the analysis of elastic stability, applying a standard set of deformations [equation (1)], we find
that all compounds in fluorite phase are unstable, particularly under the tetragonal deformation, that is,
the relation C11 −C12 turned out to be negative.
The table 1 presents the results of our calculations for the pyrite phase. Our results demonstrate that
HfC2, TaC2, WC2, IrC2, PtC2 and AuC2 aremechanically stable, while ReC2 and OsC2 violate themechanical
stability conditions. In both of them, the shear modulus (C44) was negative, so the shear modulus C44 is
the main constraint on stability in those compounds in the pyrite phase. As can be seen, in general it
holds for pyrite phase that B >C ′ >C44 > 0.
33801-3
L. Mex, A. Aguayo, G. Murrieta
Figure 1. Calculated total energies for HfC2(top) and AuC2(bottom) under isotropic deformation (circles).
The energy of the pyrite phase at the equilibrium volume is the reference level, and is set to zero. The line
corresponds to a fit of the Birch-Murnaghan equation of state to the calculated energy values (see text).
Figure 2. Energy of HfC2 as a function of the position of C atoms (u). When u = 1/4, HfC2(pyrite) reduces
to HfC2(fluorite). For all the transition metal carbides studied in this work, the second minimum was in
u = 0.43
The Zener anisotropy factor A is a measure of the degree of elastic anisotropy in solids. A will take
the value of 1 for a completely isotropic material. A value of A smaller or greater than unity shows the
degree of elastic anisotropy. The calculated Zener anisotropy for pyrite structure (see table 1) implies
that all compounds are elastically anisotropic (A <1). In order to better visualize the anisotropy of these
33801-4
Elastic properties of 5d transition-metal carbides: An ab initio study
Table 1. DFT lattice constant a, zero pressure elastic constants ci j (GPa), bulk modulus B0 (GPa), shear
modulus G (GPa), Zener factor A, and Young’s modulus (GPa), calculated in the present work for pyrite
phase of period VI transition metal carbides.
a(Å) C11 C12 C44 B0 G A E
HfC2 5.34 275 127 53 176 61 0.72 158
TaC2 5.12 380 168 71 238 83 0.67 217
WC2 5.03 432 183 30 266 55 0.24 152
ReC2 4.98 Unstable Unstable Unstable 282 – – –
OsC2 4.96 Unstable Unstable Unstable 286 – – –
IrC2 4.95 569 133 69 279 112 0.32 289
PtC2 4.98 603 101 88 268 136 0.35 342
AuC2 5.06 445 102 116 217 135 0.68 326
compounds, we show a three-dimensional (3D) representation of Young’s modulus. For cubic crystals, the
directional dependence of the Young’s modulus in 3D representation can be given by
1
E
= S11 −2
(
S11 −S12 −
1
4
S44
)
(
l 2
1 l 2
2 + l 2
2 l 2
3 + l 2
3 l 2
1
)
, (8)
where Si j are the elastic compliance constants, and l1, l2 and l3 are the directional cosines to the x−, y−
and z−axes, respectively. In the 〈100〉 directions, the second term is zero, and for C44/C ′ < 1, a maximum
in 〈100〉 directions. In all the compounds, the Zener’s coefficient was less than one, so that 〈111〉 directions
are soft and 〈100〉 are hard. Three-dimensional representation of Young’s modulus is shown in figure 3.
It can be seen that the elastic anisotropy increases in the direction 〈100〉 as Zener’s coefficient decreases.
(a) (b)
Figure 3. 3D directional dependence of the Young’s modulus for (a) AuC2 (A = 0.68) and (b) WC2 (A = 0.24).
The bulk moduli follow the parabolic behavior, and increases from HfC2 to reach the maximum on
IrC2. This behavior is similar to that present in the corresponding 5d transition metals. However, not all
compounds, the bulk modulus increases relative to that of the pure elements, as in the case of nitrides.
The values of the bulk modulus increase in: HfC2 (from 109 GPa to 176 Gpa), TaC2 (from 200 GPa to
238 GPa), PtC2 (from 230 to 268), and AuC2 (from 173 to 217). On the other hand, the other two stable
33801-5
L. Mex, A. Aguayo, G. Murrieta
compounds in the pyrite phase show a decrease in the bulk modulus compared to that of pure transition
metals: WC2 from 323 GPa to 266 GPa, and IrC2 from 355 GPa to 279 GPa. This same trend is followed by
the Young’s modulus with an increasing value in the dicarbides with Hf, Ta, Pt and Au.
The generally accepted rule is to associate hard compounds with high bulk and shear modulus values
[21]. However, bulk modulus is not the only mechanical quantity that determines the utility of a mate-
rial for hard coatings. Another important material property to be also considered is toughness, which
is influenced by the degree of plastic deformation (ductility) of the material under mechanical loading.
To analyze the ductility of the compounds, we use the Pettifor’s criterion [22, 23], which states that, for
metallic non-directional bonding compounds, the Cauchy pressure (C12 −C44) value is typically positive.
This region corresponds to a ductile behavior of a material. The other criterion we used was the Pugh’s
modulus ratio G/B, If G/B > 0.57[24], and the materials behave in a brittle manner. In figure 4, we show
the Pugh and Pettifor criteria to map the ductility and brittle behavior for pyrite phase of period VI tran-
sition metal carbides. According to their formulation, the only compound which is not within the ductile
region of the map is AuC2.
Figure 4. Map of brittleness and ductility trends of HfC2, TaC2 , WC2 , IrC2, PtC2, and AuC2. In the figure,
only the transition metal simbol is shown.
4. Conclusion
In summary, we have studied metal carbides TMC2 in the fluorite and pyrite structures using first-
principles calculations. With all the calculations, we conclude that all metal carbides studied with flu-
orite structure are mechanically unstable. On the other hand, the pyrite phase for ReC2 and OsC2 does
not meet the stability criteria as well. The bulk and Young’s moduli value increases to the value of the
pure elements in HfC2, TaC2, PtC2, and Au2, but decreases in WC2 and IrC2. According to the criterion of
brittleness (ductility), all compounds except AuC2 exhibit a ductility behavior.
Acknowledgements
The authors would like to thank Carlos Brito for helpful comments. This work was supported by
Facultad deMatámaticas-UADY under Grant no. FMAT–2012–0007 and CONACyT under Grant no. 025794.
33801-6
Elastic properties of 5d transition-metal carbides: An ab initio study
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Еластичнi властивостi карбiдiв 5d перехiдних металiв:
ab initio дослiдження
Л. Мекс1, А. Агуайо2, Г. Муррiета2
1 Iнженерний факультет, Автономний унiверситет Юкатану, Юкатан, Мексика
2 Факультет математики, Автономний унiверситет Юкатану, Юкатан, Мексика
Проведено систематичнi дослiдження менханiчної стiйкостi карбiдiв перехiдних металiв п’ятої групи
TMC2 (TM = Hf, Ta, W, Re, Os, Ir, Pt, i Au) у пiритовiй i флюоритовiй фазах шляхом обчислення їх ела-
стичних сталих в рамках теорiї функцiоналу густини. Встановлено, що всi карбiди металiв, за винятком
ReC2 i OsC2, є стiйкими у пiритовiй фазi. З iншого боку, всi метали карбiдiв, що вивчалися, виявилися
нестiйкими у флюоритовiй фазi.
Ключовi слова: першопринципнi обчислення, еластичнi стали, твердий матерiал, перехiднi метали
33801-7
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http://dx.doi.org/10.1080/14786440808520496
Introduction
Methods
Results and discussion
Conclusion
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