Materials Sciences and Applicatio ns, 2011, 2, 1383-1391
doi:10.4236/msa.2011.210187 Published Online October 2011 (
Copyright © 2011 SciRes. MSA
Structure and Bonding of Nanolayered Ternary
Abdelkader Yakoubi1*, Hanane Mebtouche1, Mohamed Ameri2, Bachir Bouhafs1
1Modeling and Simulation in Materials Science Laboratory, Physics Department, University of Sidi Bel-Abbes, Sidi Bel-Abbes, Al-
geria; 2Département de Physique, Faculté des Sciences, Université Djillali Liabès, Sidi-Bel-Abbès, Algeria.
Email: *
Received April 2nd, 2011; revised April 25th, 2011; accepted May 25th, 2011.
We have studied the electronic structure and chemical bonding mechanism of nanolayered M2SbP with M = Ti, Zr and
Hf using the full-relativistic of an all-electron full potential linearized augmented-plane-wave (FP-LAPW) method based
on the density functional theory, within the local density approximation scheme for the exchange-correlation potential.
Furthermore, we have to calculate the energy of formation for prove the existence of these compounds experimentally.
Geometrical optimizations of the unit cell are in good agreement with the available theoretical and experimental data.
The bulk modulus of M2SbP conserved as Ti is replaced with Zr, and increases by 8.7% as Ti is replaced with Hf, which
can be understood on the basis of the increased number of valence electrons filling the p-d hybridized bonding states.
The bonding is of covalent-ionic nature with the presence of metallic character. Analyzing the bonding in the binary
MP, it can be concluded that this character is essentially conserved in M2SbP ternaries.
Keywords: Ceramics, Ab-Initio Calculations, Electronic Structure, Crystal Structure, Equations-of-State
1. Introduction
Mn+1AXn (MAX) (n = 1 to 3) phases are a series of ce-
ramics but with a combination of ductility, conductivity
and machinability comparable to metals. M is an early
transition metal, A is an A-group element (mostly IIIA
and IVA) and X is either C or N. These phases have
hexagonal layered structures and belong to the space
group P63/mmc, where in Mn+1Xn layers interleaved with
pure layers of the A-group elements. The structures of
the vast majority of these compounds were determined
by Nowotny [1] and co-workers in the sixties.
The MAX phases are typical representatives of nano-
laminated phases, and according to M. W. Barsoum et al.
[2,3], these phase characterized by high elastic moduli,
are machinable [4], exhibit good damage tolerance [5],
excellent thermal shock resistance [6], and good corro-
sion resistance [7], and they are good thermal and elec-
trical conductors [4]. This unique combination of proper-
ties serves as motivation for fundamental as well as ap-
plied research. MAX phases are used as formers for
healthcare products, hot pressing tools, and resistance
heating elements [8,9], and are suitable for many tech-
nological applications.
Extensive experimental and theoretical studies were
reported on MAX phases because of their special me-
chanical, thermal and electrical properties. A. D. Bor-
tolozo et al. [10] investigated Ti2InC by x-ray diffraction,
magnetic and resistivity measurements. This work sus-
tains the idea of the existence of a new class of super-
conducting materials that crystallizes in the Cr2AlC pro-
T. Scabarozi et al. [11] report on correlations between
thermal expansion, elastic modulii, thermal transport,
specific heat, and electrical transport measurements of
materials within the MAX-phase family. Elastic modulus
measurements are made using an ultrasonic time of flight
technique. Thermal expansion measurements are made
using high-temperature x-ray diffractions.
Also, the electronic properties of these materials have
been studied both theoretically and experimentally [12].
For MAX-phases in thin film form, the processing and
physical properties have been recently reviewed.[13]
Because the MAX-phases crystallize in a hexagonal struc-
ture the anisotropy of its conductivity is of great interest
but it has been difficult to experimentally resolve this
issue [14,15].
This design criteria is not only limited to the carbides
Structure and Bonding of Nanolayered Ternary Phosphides
and nitrides ternary, but it can also be found in ternary
phosphides crystallize in the same structure.
M2SbP (M = Ti, Zr, Hf) compound belong to the
MAX phases [2], the Ti2SbP, Zr2SbP, Hf2SbP compound
have been less investigated compared to the other mem-
bers of this family. Few experimental and theoretical
works have been done to investigate their properties.
Experimentally, H. Boller [15], synthesized this com-
pound by power sintering at 800˚C. Theoretically, D.
Music and co-workers [16] investigated the chemical
bonding and elastic properties of this compound and the
correlation between the electronic structure and elastic
properties of nanolaminates ternary phosphides [17] us-
ing ab initio calculation.
In our paper, we present theoretical results of the bulk
modulus and chemical bonding of M2SbP and see the
effect of a different transition metals element of IVB
group in the electronic properties. The aim of this work is
to evaluate the chemical bonding and the bulk modulus
of our compounds, as the number of valence electrons in
third and fourth periods are increased to gain better insight
into this technologically interesting class of materials.
2. Methodology and Computational Details
In this present work, the calculations of both crystal op-
timization and electronic structures were performed in
the WIEN2K package [18]. The Kohn-Sham equations
were solved by using the highly accurate all-electron
full-potential linearized augmented plane-wave plus
local orbitals (FP-L/APW + lo) method [19,20] in the
framework of density functional theory (DFT) with the
exchange-correlation functional treated in the local den-
sity approximations (LDA) using the scheme of Perdew-
Wang [21]. This method makes no shape approximation
to the potential or the electron density. Within the FP-
LAPW method, the unit cell is divided into nonoverlap-
ping muffin-tin (MT) spheres and an interstitial region.
Inside the muffin-tin sphere of radius RMT, the wave
functions are expanded using radial functions (solution
to the radial Schrödinger equation) times spherical har-
monics up to max, and the expansion of the potential
inside the muffin-tin spheres is carried out up to max.
The parameter max (MT
R is the mallets
muffin-tin spherical radius present in the system and
max is the truncation of the modulus of the recipro-
cal-lattice vector) is used to determine the number of
plane waves needed for the expansion of the wave func-
tion in the interstitial region, while the parameter Gmax is
used to truncate the plane-wave expansion of the poten-
tial and density in the interstitial region. Here, the MT
radii were set to 2.22, 2.4, and 2.7 a.u. for the Ti, Zr, and
Hf atoms, respectively and 2.5, 1.9 a.u. for the Sb and P
atoms respectively. Moreover, we let ,
RK 9.0 min
MT max
R*K 9.0
, max
, and max . The separate en-
ergy of 6.0 Ry was used between valence and core
states. Thus, the Ti: [Ar]3d24s2, Zr: [Kr]4d25s2, and Hf:
[Xe]4f145d26s2, were treated as valence states, while Sb:
[Kr]4d105s25p3 and P: [Ne]3s23p3 were acted as semicore
states with other electrons as core states. The energy
level of unoccupied states was calculated until 7.5Ry
(Emax = 7.5 Ryd). Integrations in the rst Brillouin zone
(FBZ) have been performed using the Monkhorst-Pack
method [22] with 111 special k points [23]. Self-consis-
tency calculation of electronic structures is achieved
when the total-energy variation from iteration to iteration
converged to a 1 mRyd accuracy or better.
In the WIEN2K code, core states were treated at the
fully relativistic approximation with including spin-orbit
interaction [18] as the spin-orbit coupling (SOC) term in
the Hamiltonian for the valence states, HSOC, was treated
as a perturbation in the second variationnel method,
whereas the Dirac equation was solved for the core states.
We compute lattice constants, bulk moduli and the opti-
mized free internal parameters by fitting the total energy
vs. volume curves to the equation of states [24].
The partial density of states and charge-density distri-
butions are obtained using the relaxed structures at the
equilibrium volumes. The input lattice parameters of the
M2SbP studied are taken from Reference [17] and those
of MP in the Hexagonal structure as well as TiSb alloy,
which are used for the comparison purposes, can be
found in References [34,35].
3. Results and Discussion
3.1. Equilibrium, Formation of Energy and
Cohesive Properties
M2SbP (M = Ti, Zr, and Hf) compounds crystallize in the
Cr2AlC crystal structure, with space group P63/mmc
(#194). Its unit cell contains two formula units, and the
atoms occupy the Wyckoff positions 2(a) [(0, 0, 0), (0, 0,
1/2)] for P, 2(d) [(1/3, 2/3, 3/4), (2/3, 1/3, 1/4)] for Sb,
and 4(f) [(1/3, 2/3, z), (2/3, 1/3, z + 1/2), (2/3, 1/3, –z), (1/3,
2/3, –z + 1/2)] for M, where z is the internal free coordi-
nate. The structure is thus defined by two lattice parame-
ters, a and c, and the internal structural parameter, z.
The phosphor (P) atoms fill the octahedral locations
between the M layers [16,17]. The repeating structure of
the planes in the unit cell can be further dened by the
APCMBSbCMAPBMCSbBM close-pack stacking along the
Z-axis [25-27]. The letters A, B, and C stand for the three
distinct positions for atoms to occupy, in the close-
packed <0001> plane [29]. The subscripts denote the
type of atom that sits at the site, e.g. BM means the metal
atom M sits at a B site. The Z-coordinate increases as we
go from left to right.
Copyright © 2011 SciRes. MSA
Structure and Bonding of Nanolayered Ternary Phosphides
Copyright © 2011 SciRes. MSA
parameters and the bulks moduli of M2SbP (they are
treated the potential of exchange-correlation by the GGA)
We have first chosen lattice parameters and internal
parameters to start calculations. We then vary these pa-
rameters until reaching the minimum of energy. On the
scale of meV, the energy bands near the energy gap de-
pend critically on the structural parameters that are not
determined by symmetry. We have therefore performed
detailed structural optimizations of the unit cell geome-
tries as a function of the external stress by minimizing
the total energy. Our results of the calculated lattice con-
stants a and c, bulk modulus and its pressure derivative,
and the optimized free internal parameters are reported in
Table 1. Good agreement between our calculations and
experimental data of H. Boller [15] and theoretical data
of D. Music et al. [16,17] is obtained. Our calculations
are slightly underestimated compared to the experimental
values of as commonly observed in LDA calculations.
The deviation from the experimental values of c and a for
Ti2SbP was 1.03% and 1.65%, respectively. For Zr2SbP,
the change from experimental values was 0.76% and
0.79%, while for Hf2SbP it was 0.61% and 0.79%, re-
spectively [15]. Although the bulk moduli have not been
experimentally measured, but a good agreement is ob-
served with the other first principle data for the lattice
To study stabilities of relative phase for these pho-
sphides hexagonal, we have calculated the energy of
formation (Eform) per atom by using the following equa-
MSbP totalsolid solid solid
E 4E2E2E
E= 8
 (1)
where M designates a metal with Ti, Zr and Hf crystal-
lizing in the hexagonal structure (space group P63/mmc,
prototype Mg) [29-31], Sb is a trigonal structure (space
group R3m, prototype αAs) [32], and P crystallize in
the triclinic structure (space group P1) [33]. There are
two atoms in the unit cell for every metal, the indication
of the stability of these compounds with the following
values according to these calculations of the energy of
formation, for Ti2SbP, Zr2SbP and Hf2SbP are 0.67
eV/atom, 0.79 eV/atom and 0.74 eV/atom respectively,
these are listed in Table 2.
The cohesive energy is a measure of the strength of the
forces that bind atoms together in the solid state and is
Table 1. The equilibrium lattice parameters (a, c and c/a), internal parameter (z), bulk modulus (B) and its pressure deriva-
tive (B’) for M2SbP with M = Ti, Zr, and Hf.
Compounds a (Å) c (Å) c/a z B (GPa) B’
Ti2SbP Present work 3.58 12.42 3.47 0.099 134.6 4.11
Exp [15] 3.64 12.55 3.45 0.100 - -
Ref [16] 3.65 12.63 3.46 0.100 115 4.25
Zr2SbP Present work 3.79 13.14 3.47 0.103 134.5 4.24
Exp [15] 3.82 13.24 3.47 0.103 - -
Ref [16] 3.84 13.44 3.50 0.103 116 4.30
Hf2SbP Present work 3.76 12.96 3.45 0.102 144.4 4.13
Exp [15] 3.78 13.04 3.45 0.102 - -
Ref [16] 3.81 13.21 3.47 0.102 125 4.19
Table 2. The calculated values of the energy of formation (Eform), bulk moduli (B), cohesive energies (Ecoh) and the valence
electron concentration (val-el) for M2SbP.
Ti2SbP Zr2SbP Hf2SbP
Eequilibre (eV) 463670.52774 762143.63421 2012635.855
Eform (eV/atom) 0.69 0.79 0.74
Ecoh (eV/atom) 12.75 13.25 15.44
B (GPa) 134.6 134.5 144.4
Valence electron 88 88 144
Structure and Bonding of Nanolayered Ternary Phosphides
descriptive in studying the phase stability. The cohesive
energy coh of M2SbP is dened as the total energy
of the constituent atoms minus the total energy of the
compound where total refers to the total energy of
M2SbP in the equilibrium conguration and atom ,
, and are the isolated atomic energies of the
cohatom atom atomtotal
pure constituents.
The isolated atomic energies are calculated using the
same WIEN2K code by using the augmented plane-wave
basis. The computed cohesive energies yield shown in
(a) Ti2SbP (b) Zr
(c) Hf2SbP
Figure 1. Full-relativistic LDA + SOC band structur e of (a) Ti2SbP, (b) Zr2SbP and (c) Hf2SbP.
Copyright © 2011 SciRes. MSA
Structure and Bonding of Nanolayered Ternary Phosphides1387
Table 2. 12.75 eV/atom for Ti2SbP, 13.25 eV/atom for
Zr2SbP, and 15.44 eV/atom for Hf2SbP. It shows that the
cohesive energies of the ternary rst-column transition-
metal phosphides increase monotonously, as in the se-
quence Ti2SbP < Zr2SbP < Hf2SbP.
The bulk modulus represents the resistance to volume
change and is related to the overall atomic binding prop-
erties in a material. Therefore, we confirmed in the Table
2, it is expected to change in the same trend of the cohe-
sive energy, according to the transition-metal ion (the M
element) and the valence electrons population of the
3.2. Band Structures and Density of States
We calculated the electronic band structures using the
full relativistic effects (including spin-orbit interaction)
for Ti2SbP, Zr2SbP, and Hf2SbP are given in Figures
1(a), (b) and (c) respectively, at their equilibrium lattice
constants at different high-symmetry points in the Bril-
louin zone.
The band structures also show strong anisotropic fea-
tures with smaller energy dispersion along the c-axis.
The valence and conduction bands overlap considerably
and there is no band gap at the Fermi level. This nding
confirms the metallicity of these materials. The electrical
conductivity is anisotropic for this material, i.e. the elec-
trical conductivity along the c-axis is much lower than
that in the basal planes. The effect of the spin-orbit cou-
pling (SOC) is clear in the Hf2SbP compound because
the hafnium is a strongly correlated element.
The calculated total density of states (TDOS) using the
LDA + SOC for M2SbP, where M = Ti, Zr, and Hf are
presented in Figure 2. It is apparent that these phases are
alike, signifying similarity in chemical bonding. As
shown in the figure, the Fermi level of M2SbP is located
at a pseudogap between the bonding and nonbonding
states. A similar pseudogap is also found in many other
MAX phases [34,35]. Generally, a pseudogap correlates
to the structure stability [36]. So it is believed that the
presence of pseudogap in M2SbP contributed to its phase
stability. At the Fermi level Efermi, the DOS for Ti2SbP,
Zr2SbP, and Hf2SbP, were 2.12, 1.71 and 1.59 states per
unit cell per eV, respectively. Thus there is a small de-
creasing trend in the DOS at Efermi with increasing atomic
numbers of the transition metal (M).
To further elucidate the nature of chemical bonding in
these compounds, we study the partial density of states
(PDOS) of Ti2SbP, a representative, given in Figure 3(a)
Phosphorus does not contribute to the DOS at the Fermi
level and therefore is not involved in the conduction
properties. Ti d electrons are mainly contributing to the
DOS at the Fermi level, and should be involved in the
conduction properties. Sb electrons do not contribute
significantly at the Fermi level. These results are consis-
tent with previous reports on MAX phases [37]. It is
evident that a covalent interaction occurs between the
constituting elements due to the fact that states are de-
generate with respect to both angular momentum and
lattice site. P 3p and Ti 3d as well as Sb 4p and Ti 3d
states are hybridized. Also, due to the difference in elec-
tronegativity between the comprising elements, some
ionic character can be expected. The bonding character
may be described as a mixture of covalent-ionic and, due
to the d resonance in the vicinity of the Fermi level, me-
tallic. The pseudogap, common to all M2SbP phases
studied (see Figure 2), is likely to split the bonding and
antibonding orbitals. This behavior is consistent with
previous reports on MAX phases [37-39].
Furthermore, it has been found for Nb3SiC2 [40] that
the balanced crystal orbital overlap population analysis
[41] is consistent with the PDOS analysis suggesting
splitting in PDOS. However, there is also a difference
with respect to the role of the A element. In the MAX
phases [37,38] the A-M hybridization is weaker than the
Sb-M hybridization in the M2SbP phases. At the same
time, the chemical bonding between M and X elements
Figure 2. Full-relativistic total density of states (TDOS) for
M2SbP with (M = Ti, Zr, Hf).
Copyright © 2011 SciRes. MSA
Structure and Bonding of Nanolayered Ternary Phosphides
(a) (b)
Figure 3. Full-relativistic partial density of states (PDOS) for Ti2SbP, Zr2SbP and Hf2SbP.
Copyright © 2011 SciRes. MSA
Structure and Bonding of Nanolayered Ternary Phosphides1389
in the MAX phases and between M and P in the here
studied phosphides is rather similar (see Figures 3(a),
(b), (c)).
3.3. Charge Densities
The high modulus of Ti2SbP can be explained based on
bonding characters. To visualize the nature of the bond
character and to explain the charge transfer and the
bonding properties of the Ti2SbP, Zr2SbP and Hf2SbP
compounds, we have investigated the eect of the M
states on the total charge densities (see Figure 4). It is
known that MAX phases are usually stacks of ‘hard’
M-X bond and “soft” M-A bond along c direction. For
example, the Ti-C bond strength is much stronger than
Ti-Al bond in Ti2AlC [38]. However, for Ti2SbP (shown
in Figure 4(a)), in addition to the Ti-P hybridization
(covalent bonding), there is also a strong interaction be-
tween Ti and Sb atoms (metallic bonding). This means
that the “soft” M-A bond is strengthened in Ti2SbP,
which is an extraordinary example in MAX phases stud-
ied so far. We believe that this strengthening effect con-
tributed considerably to the increase in bulk modulus of
Ti2SbP. While the more electropositive nature of Sb
conrms the ionic bonding between Ti and Sb. Therefore,
the chemical bonding in Ti2SbP is metallic-covalentionic
in nature.
Figure 4. Electron density distribution of (a) Ti2SbP, (b)
Zr2SbP, (c) Hf2SbP, in the (1120) plane. The electron den-
sity distribution increases from 0.05 (white) to 15.0 (black)
electron/Å3 for Ti2SbP, Zr2SbP, and Hf2SbP.
Figure 5 shows the charge-density contours in the
plane for Ti2SbP Figure 5(a), as well as in the (1120)
(1120) plane for TiP Figure 5(b), where the latter phase
crystallizes in hexagonal structure with space group
P63/mmc [42].
Analyzing the M-P bonding in TiP, it can be con-
cluded that the bonding is characterized by covalent and
ionic contributions and that this character is essentially
conserved in the M2SbP ternaries. While the charge den-
sity distribution of TiP is similar to the one of ZrP and
HfP, the charge distribution of the related ternaries shows
extensive differences. The coupling between the MP
layer and the Sb layer is weaker for Ti2SbP as compared
to Zr2SbP and Hf2SbP. In fact, the bonding between Ti
and Sb in Ti2SbP is similar to the bonding in TiSb (space
group Cmcm) [43], as can be seen in Figure 5(c). Hence,
the structure electronic data presented in Figure 2 can be
understood based on the charge density data discussed
here. These ndings provide a pathway for tailoring the
electronic properties of M2SbP by tuning the valence
electron population. As the valence electron population
of the transition metal M is increased, more charge is
placed in the M-P bonds, which is due to an increase in
the P p-M d hybridization as can be seen in Figure 2,
where the partial density of states data are presented for
Ti2SbP (Figure 3(a)), Zr2SbP (Figure 3(b)), and Hf2SbP
(Figure 3(c)).
Figure 5. Contour plot of the total valence charge densities
for (a) Ti2SbP and (b) TiP in the (112 0 ) pla ne, and (c) TiSb
in the (100) plane.
4. Conclusions
In summary we have studied the electronic structure and
chemical bonding of M2SbP compounds with (M = Ti, Zr
and Hf), by means of ab initio calculations using the full
potential linearized augmented-plane-wave (FP-LAPW)
method. The bulk modulus conserved as Ti is replaced
with Zr, and increases by 8.7% as Ti is replaced with Hf.
Copyright © 2011 SciRes. MSA
Structure and Bonding of Nanolayered Ternary Phosphides
This can be understood since the substitution is associ-
ated with an increased valence electron concentration,
resulting in band filling and an extensive increase in co-
hesion. To study stabilities of relative phase for these
phosphides hexagonal, we have calculated the energy of
formation (Eform) per atom. The energy of formation in-
dicates that these materials do form.
This work is important for basic understanding of
structure and bonding of these phases and may encourage
future experimental research.
5. Acknowledgements
One of us (B. B.) acknowledges the Abdus-Salam Inter-
national Center for Theoretical Physics (Trieste-Italy).
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