First principles study on structural, electronic, elastic and thermal properties of equiatomic m ti (m = fe, co, ni)

Chemistry and Materials Research www.iiste.org
ISSN 2224- 3224 (Print) ISSN 2225- 0956 (Online)
Vol.3 No.8, 2013
22
First Principles Study on Structural, Electronic, Elastic and
Thermal Properties of Equiatomic MTi (M = Fe, Co, Ni)
Nikita Acharya1*
, Bushra Fatima1
, Sunil Singh Chouhan2
and Sankar P. Sanyal1
1. Department of Physics, Barkatullah University, Bhopal, 462026, India
2. Department of Physics, Govt. M.L.B. Girls P.G. College, Bhopal, 462002, India
* E-mail of the corresponding author: acharyaniks30@gmail.com
Abstract
We have investigated the structural, electronic, elastic and thermal properties of MTi (M = Fe, Co and Ni) using
ab-initio full potential linearized augmented plane wave (FP-LAPW) method within generalized gradient
approximation (GGA) and local spin density approximation (LSDA). We have calculated the ground state and
electronic properties such as lattice constant (a0), bulk modulus (B), pressure derivative of bulk modulus (B') and
density of states at Fermi level N(EF) which are in good agreement with experimental and available other
theoretical results. The elastic constants (C11, C12 and C44) and mechanical properties such as Poisson’s ratio (σ ),
Young’s modulus (E), shear modulus (GH), anisotropic factor (A) are also calculated which are agree well with the
experimental and other theoretical results. Ductility for these compounds have been analyzed by Pugh’s rule (B/GH
ratio) and Cauchy pressure (C12 - C44). Our calculated results reveals that NiTi is most ductile amongst the MTi (M
= Fe, Co and Ni) compounds.
Keywords: Ab-initio, electronic properties, elastic properties, thermal properties.
1. Introduction
Intermetallics are short and summarizing designation for the intermetallic phases and compounds which result
from the combination of various metals and form numerous and manifold class of materials. Much of change in
character of intermetallic compounds is due to difference in the chemical bonding that binds the atoms of phase
together. Intermetallic compounds have emerged as materials with vast potential for application in a wide range
of technologically important areas [1]. The enormous potential of intermetallics especially aluminides stems
from their many attractive properties, such as high oxidation, corrosion resistance and relatively low densities,
combined with their ability to retain strength and stiffness at elevated temperatures [2, 3]. Their physical,
electrical, magnetic and mechanical properties are often superior to those of ordinary metals, but their enormous
potential to improve engineering performance remains largely unused because they are brittle and fracture easily
at room temperature. Due to long range ordering and specific properties, the intermetallic alloys are assumed to
fill an existing gap between structural ceramics and classical metallic alloys [4].
The intermetallic compounds of titanium FeTi, CoTi and NiTi have many interesting properties such as high
hardness, melting temperature, shape memory effect, hydrogen capacity per unit weight etc. [5]. These
intermetallic compounds of VIII- group element have unique functional and structural properties also. CsCl-type
(B2-phase) equiatomic alloys generally have low ductility at low temperatures in spite of relatively simple crystal
structure. In particular NiAl and FeAl have high phase stability and exhibit poor ductility at room temperature
[6]. NiTi is well known as shape memory alloy and super elastic material, while FeTi, CoTi are possible to be
used as hydrogen storage materials. CoTi is also known to exhibit positive temperature dependence of yield
strength [6]. In the last few years, the scientific investigations for NiTi have been made extensively from the
aspects of theory and experiments, including the structural, thermodynamic and martensitic (B19')
transformation path etc. [7]. Recently, spectroscopic ellipsometry study of FeTi, CoTi and NiTi alloys have
revealed strong optical transitions and has explained the similarities and differences among optical conductivity
spectra and measured results are much closer to those obtained by full potential linearized augmented plane
wave (FP-LAPW) method [8]. Other studies were devoted to the role of structural disorder and magnetic
properties in CoTi alloys and films [9]. Despite the technological importance of shape memory material and
recent advances the fundamental mechanisms, that governs their unique behavior are not fully known, while the
martensitic transformation governing the thermo-mechanical response of this material (NiTi) at the macroscale is
well known characterized [10]. Kibey et al. [11] have presented energy landscape for martensitic phase
transformation in shape memory NiTi using first principles calculation. Cheng et al. [12] have been carried out
to study the elastic properties and electronic structure of NiTi, CoTi and FeTi using first principles calculations.
Naish et al. [13] have theoretically analyzed possible martensitic phases in the NiTi. Kellou et al. [14] have
reported the electronic properties, bulk surfaces and interfaces of FeTi, CoTi and NiTi alloys using FP-LAPW
method. Zhao et al. [7] have investigated the structural and elastic properties, phase diagram of NiTi alloy from
first principles calculations. The magnetic and electronic properties of CoTi alloys using LMTO method has
been studied by Napierala et al. [15]. Eibler et al. [16] have reported the electronic, structure, chemical bonding

Vol.3 No.8, 2013
23
and spectral properties of FeTi, CoTi and NiTi using self-consistent APW method. Sheng et al. [17] investigated
first principles calculation of intermetallic compounds in FeTiCoNiVCrMnCuAl systems high entropy alloy
using CASTEP code.
In the present work, we have performed a first principles spin polarized calculation of FeTi, CoTi and NiTi,
which crystallize in CsCl-type structure, using density functional theory (DFT) within the both generalized
gradient approximation (GGA) and local spin density approximation (LSDA). We have also calculated the
ductility of MTi (M = Fe, Co, Ni) by B/GH ratio and Cauchy’s pressure and found that all the compounds are
ductile in nature. To the best of our knowledge the thermal properties are reported by us for the first time. We
have also done a comparative study of their structural, electronic, elastic and mechanical properties.
2. Methodology
The first principles calculation of the real material based on the density functional theory is one of the most
powerful tool to understand the electronic structure of these materials. It can give us the information about spin
distribution in magnetic materials which are not measured from experiment. The total energy, ground state
properties and electronic band structures have been computed in spin polarized calculation within GGA and
LSDA approximations using full potential linearized augmented plane wave (FP-LAPW) method as
implemented in the WIEN2k code [18]. Generalized gradient approximation (GGA) has been used for the
exchange and correlation effects [19]. The energy eigen value convergence has been achieved by expanding the
basis function up to RMT*Kmax= 7, where RMT is the smallest atomic sphere radius in the unit cell and Kmax gives
the magnitude of the largest k vector in the plane wave expansion. The valence wave functions inside the spheres
are expanded up to lmax = 10 while the charge density is Fourier expanded up to Gmax = 12. The self consistent
calculations are considered to converge when the total energy of the system is stable within 10-4
Ry. Energy to
separate core and valence state is -6.0 Ry. Integrations in the Brillouin zone were performed using k-points
generated with 10×10×10 mesh grids for all structures.
The elastic moduli require knowledge of the derivative of the energy as a function of the lattice strain. It is well
known that a cubic system has only three independent elastic constants namely C11, C12 and C44. Hence, a set of
three equations is needed to determine all the constants. The first equation involves calculation of bulk modulus
(B), which is related to the elastic constants as:
)2(
3
1
1211 CCB += (1)
The second step involves volume-conservative tetragonal strain given by the following tensor:
( )














−1
1
100
00
00
2
δ+
δ
δ
(2)
where δ = (1+e)-1/3
-1 with e as strain tensor. Application of this strain has an effect on the total energy from its
unstrained value as follows:
)()(3)0()( 32
01211 δδδ OVCCEE ++−+= (3)
where V0 is the volume of the unit cell.
Finally, for the last type of deformation, we use in the volume-conserving rhombohedra strain tensor given by:










111
111
111
3
δ (4)
which transforms the total energy to
)()42(
6
1
)0()( 32
0441211 δδδ OVCCCEE +++++= (5)
The thermal loss mechanisms (temperature dependence) of a material is most suitably described in terms of the
Debye temperature (θD) which is a fundamental parameter closely related to many physical properties such as
elastic constants, specific heat and melting temperature etc. One of the standard methods is to calculate the
Debye temperature from elastic constants data, since θD may be estimated from the average sound velocity vm by
the following equation [20, 21]:

Vol.3 No.8, 2013
24
m
aB
D v
V
n
k
h
3/1
4
3






=
π
θ (6)
where h is Plank’s constant, kB is Boltzmann’s constant, Va is the atomic volume, n is the number of atoms per
formula unit and vm is average sound velocity. The average sound velocity is approximately calculated from [20,
22].
3/1
33
12
3
1
−














+=
lt
m
vv
v (7)
where vt and vl are the transverse and longitudinal sound velocities respectively obtained by using the elastic
constants as follows:
ρ




−++
=
)2(
5
2
11124411 CCCC
vl
(8)
ρ




−+−
=
)2(
5
1
11124444 CCCC
vt
(9)
where C11, C12 and C44 are second order elastic constants and ρ is mass density per unit volume.
3. Result And Discussion
3.1 Structural Properties
The spin polarized calculations are carried out to obtain the total energy of (FeTi, CoTi and NiTi) intermetallic
compounds using FP-LAPW method within GGA approximation. The variation of total energy as a function of
volume has been plotted in Fig. 1 (a – c).
Figure. 1 Equation of states for FeTi (a), CoTi (b) and NiTi (c).
Table 1: Calculated ground state properties for MTi (M = Fe, Co and Ni).
Solids Approximation a0 (Å ) B (GPa) B'
N(EF)
FeTi Pre. GGA 2.96 182.38 5.44 0.16
LSDA 2.90 235.10 4.34 0.19
Expt. 2.97
Theo. 2.88a
192a
0.19b
188c
CoTi Pre. GGA 2.99 173.56 3.96 1.51
LSDA 2.91 180.89 4.41 1.30
Expt. 2.99a
152a
1.66b
Theo. 2.98a
204a
NiTi Pre. GGA 3.01 160.03 4.15 1.57
LSDA 2.94 194.47 4.77 1.51
Expt. 3.01a
142a
1.73b
Theo. 2.99a
191a
a
:Ref[12] b
:Ref[5] c
:Ref[17]
In order to calculate the ground state properties of these intermetallics, the total energies are calculated in B2
phase for different volumes. The calculated total energies are fitted to Birch Murnaghan’s equation of state [23]

Vol.3 No.8, 2013
25
to determine the ground state properties such as lattice constant (a0), bulk modulus (B) and its pressure derivative
(B') at minimum equilibrium volume V0.
(10)
Pressure is obtained by taking volume derivative of the total energy
(11)
The ground state properties are presented in Table 1, and compared with other experimental and theoretical
results. Our calculated bulk modulus (B) slightly differs with the other results reported in the literature which
may be due to the different exchange correlation schemes. To the best of our knowledge the experimental bulk
modulus of FeTi is not reported yet. The order of energetic stability of FeTi, CoTi and NiTi from high to low is :
FeTi > CoTi > NiTi.
3.2 Electronic properties
The calculated band structure of FeTi, CoTi and NiTi intermetallic compounds using FP-LAPW method within
GGA are presented in Fig. 2 ( a, b and c) where Fermi level (EF) is considered at origin. To further understand
the nature of electronic bands structure, we have also calculate the total and partial density of states for these
compounds at ambient pressure and presented in Fig. 3 (a - f). We found similar band profile in both GGA and
LSDA for MTi (M = Fe, Co, Ni) compounds. We therefore present here the band structures in only GGA. In
case of FeTi, the lowest lying bands around -7.4 eV at Г-point are mainly due to ‘s’ states of Fe. The bands
above this and below the Fermi level (EF) around -1.8 eV and -2.2 eV are due to hybridized ‘d’ states of Fe and
Ti. It can be noted from Fig. 2 (a) and 3 (a) that here two bands are crossing the Fermi level at X point which are
mainly due to ‘d’ states of Fe and Ti. It is seen from the Fig. 2 (a) that FeTi is metallic in nature due to strong
hybridization ‘d’ states of Fe and Ti. It is also seen from Fig. 3 (a) the number of DOS at Fermi level N (EF) is
0.16 States/eV (Table 1). In Fig .2 (b) for CoTi the lowest lying band around -7.8 eV at Г- point are mainly due
to Co ‘s’ like states and the bands above this and below the Fermi level around -2.2 eV and -2.8 eV are due to ‘d’
states of Fe and Ti. The bands above the Fermi level are due to Ti ‘d’ states. From Fig. 3 (c ,d) the hybridization
between Co ‘d’ and Ti ‘d’ states from lowest band upto above the Fermi level can also be seen. The finite DOS
at Fermi level N (EF) are found to be 1.51 States/eV for GGA respectively (Table 1).
Similarly, in case of NiTi the lowest lying bands around -7.8 eV at Г -point are mainly due to Ni ‘s’ states and
The cluster of bands just below the Fermi level (EF) around -2.1 eV and -2.7 eV are due to mainly ‘d’ states of
Fe and Ti. It is also seen from Fig. 3(e, f) that there is strong hybridization between Ni-d and Ti-d states. The
number of density of states at Fermi level N (EF) is found to be 1.57 States/eV (Table. 1). From the analysis of
band structure of these compounds it is observed that all the three compounds have similar band structure.
Except FeTi, the band structure of CoTi and NiTi are slightly shifted just below the Fermi level. The presence of
bands at the Fermi level confirms that all the compounds are metallic in nature.
1'
1
)1'(
)/(
'
)(
'
0
0
−
−





+
−
+=
B
BV
B
VV
B
BV
EVE
B








−







= 1
'
)(
'
0
B
V
V
B
B
VP

Vol.3 No.8, 2013
26
Figure. 2 Band structures for FeTi (a), CoTi (b) and NiTi (c)

Vol.3 No.8, 2013
27
Figure. 3 Total and Partial DOS for FeTi (a, b), CoTi (c, d) and NiTi (e, f)
3.3 Elastic Properties
The elastic constants are important parameters that describe the response to an applied macroscopic stress and
especially important as they are related to various solid state phenomena, such as bonding characteristic between
adjacent atomic planes, anisotropic factor of binding and structural stability. We have calculated elastic constants
of FeTi, CoTi and NiTi in B2 phase at ambient pressure by using the method developed by Charpin and

Vol.3 No.8, 2013
28
integrated it in the WIEN2k package [18]. The calculated values of elastic constants are given in Table. 2 along
with the available theoretical and experimental results. It can be noted that our calculated elastic constants satisfy
the stability criterion: C11 > C12, C12 > 0, C44 > 0, C11 + 2C12 > 0, C11 > B > C12 which clearly indicate the stability
of these compounds in B2 phase. Elastic constants play an important role for the determination of the mechanical
properties as discussed in the next section.
Table 2. Calculated elastic constant and Young’s modulus E, shear modulus GH, anisotropic factor A, Poisson’s
ratio σ, B/GH ratio and Cauchy’s pressure C12–C44 for MTi (M = Fe, Co and Ni)
Solids C11
(GPa)
C12
(GPa)
C44
(GPa)
E
(GPa)
GH
(GPa)
A σ C12–C44 B/GH
FeTi(GGA) 372.95 87.10 68.63 237.36 92.49 0.48 0.28 18.47 1.97
LSDA 445.28 130.28 109.80 322.76 126.95 0.69 0.27 20.20 1.85
Theo. 304a
139a
138a
283.58 13.08a
1.64a
0.25 -2 1.69
385c
89.92c
81.13c
262.3c
103.4c
0.54 0.26 8.79 1.82c
CoTi(GGA) 286.51 113.79 74.66 205.74 79.13 0.86 0.29 39.13 2.16
LSDA 210.75 165.97 80.97 133.69 48.55 3.61 0.37 85 3.72
Expt. 203a
129a
68a
143.24 53.26a
1.83a
0.34 61 2.88
Theo. 261a
176a
99a
189.72 70.51a
2.32a
0.34 77 2.89
NiTi(GGA) 195.93 157.59 62.90 109.09 39.14 3.28 0.39 94.68 4.35
LSDA 218.38 184.32 54.92 97.51 34.41 3.22 0.41 129.40 5.68
Exp. 162a
132a
36a
71.74 25.33a
2.40a
0.41 96 5.60
Theo. 218a
178a
71a
119.68 42.87a
3.55a
0.39 107 4.46
a
:Ref[12] c
:Ref[17] Some parameters are derived from experimental and theoretical values of elastic constants.
3.4 Mechanical Properties
Elastic properties can be used to determine mechanical properties such as Young’s modulus (E), shear modulus
(GH), Poisson’s ratio (σ) and anisotropic ratio (A) for useful applications. We have calculated these properties of
FeTi, CoTi and NiTi and presented them in Table 2. Using mechanical properties of these compounds based on
their elastic properties, we have analyzed their ductility using the (B/GH) ratio.
The shear modulus (GH) describes the material’s response to shearing strain using the Voigt-Reuss-Hill (VRH)
method [24-26]. The Hill shear modulus (GH) is given as:
2
RV
H
GG
G
+
= (12)
Where
5
3 441211 CCC
GV
+−
= is the Voigt shear modulus (13)
and
)(34
)(5
121144
121144
CCC
CCC
GR
−+
−
= is the Reuss shear modulus (14)
The Young’s modulus (E) is important for technological and engineering application. Young’s modulus is
defined as the ratio of stress to strain, and is used to provide a measure of the stiffness of the solid, i.e., the larger
value of E, the stiffer is the material. It is reported in the literature that stiffer solids have covalent bonds [27].
Young’s Modulus (E) is given by-
H
H
GB
BG
E
+
=
3
9
(15)
It can be seen from the Table 2 that the highest value of E occurs for FeTi implying stiffer material in nature as
compared to CoTi and NiTi compounds. Another important parameter is the elastic anisotropic factor (A), which
gives a measure of the anisotropy of the elastic wave velocity in a crystal and it is given as:
1211
442
CC
C
A
−
= (16)
which is unity for an isotropic material, anisotropy factor A is listed in Table 2.
The Poisson’s ratio (σ) is given by eq (17) as
)3(2
23
H
H
GB
GB
+
−
=σ (17)
Using the mechanical and physical properties of these compounds based on their elastic properties, we have
analyzed their ductility using the (B/GH) ratio. The shear modulus (GH) [24-26] represents the resistance to
plastic deformation while the bulk modulus (B) represents the resistance to fracture. As suggested by Pugh [28],
if B/GH < 1.75; a material behaves in a brittle manner. Ganeshan et al. [29] have established a correlation

Vol.3 No.8, 2013
29
between the bonding and brittleness/ductility properties. The bond character of cubic compounds is explained
with respect to their Cauchy pressure (C12 - C44). Compound having more positive Cauchy’s pressure tends to
form bonds which are primarily metallic in nature, where as the compounds having more negative Cauchy’s
pressure from bonds which are more angular or covalent in character [30]. Thus the ductile nature of all MTi (M
= Fe, Co and Ni ) can be correlated to their positive cauchy’s pressure and thereby metallic character in their
bonds.
3.5 Thermal properties
With the help of calculated the Young’s modulus (E), Bulk modulus (B0) and the shear modulus (GH), we have
obtained the Debye temperature (θD) by using the average sound velocity vm. At low temperature the vibrational
excitations arise solely from acoustic vibrations. Hence, at low temperature the Debye temperature calculated
from elastic constants. We have calculated the average sound velocities (vm) and Debye temperatures (θD) as well
as the densities for B2 phase by using the calculated elastic constants which are given in Table 3. In the absence
of any measured data in the literature, they could not be compared. Hence, our results can be considered as a
prediction for these properties of intermetallic compounds and it will testify future experimental work.
Table 3. Calculated longitudinal ν l, transverse νt , average elastic wave velocities νm, Debye Temperature θD for
MTi (M = Fe, Co and Ni)
Solids ρ*103
(kg/m3
) ν l (ms-1
) νt (ms-1
) νm (ms-1
) θD (K)
FeTi (GGA) 5.064 7867 4406 4900 352.20
LSDA 5.413 8670 4880 5425 398.66
Other Theo. 5.539 7916 4583 5083 376.43
Other Theo. 5.091 7989 4551 5054 366.63
CoTi (GGA) 5.091 7377 3947 4405 314.08
LSDA 5.530 6825 3225 3626 265.78
Expt. 5.098 6684 3302 3704 264.21
Other Theo. 5.149 7711 3851 4317 308.99
NiTi (GGA) 4.955 6826 3026 3413 241.35
LSDA 5.345 6821 2727 3087 223.86
Expt. 4.985 5988 2352 2664 188.78
Other Theo. 5.350 6954 3075 3468 251.57
These values are derived from experimental and theoretical values of elastic constants.
4. Conclusion
First principles calculation have been performed on MTi (M = Fe, Co and Ni) intermetallic compounds by FP-
LAPW method using DFT with both GGA and LSDA. Our results on the structural, electronic, elastic,
mechanical and thermal properties are in good agreement with other experimental and available other theoretical
results. The calculated elastic constants show that FeTi, CoTi and NiTi are elastically stable in B2 phase. Using
these elastic constants the shear modulus (GH) , poisson’s ratio (σ), young’s modulus (E) and anisotropy factor
(A) are also reported. The electronic band structures show the metallic character for all compounds. In the
present study we found B/GH ratio > 1.75 and C12-C44 > 0 which implies that all these compounds are ductile in
nature and NiTi have an excellent ductility amongst all the compounds. We have also analyzed that the more
delocalized bands are present in NiTi at Fermi level (EF) as compared to FeTi and CoTi. The thermal properties
are also reported for these compounds.
Acknowledgement
The authors are thankful to University Grants Commission (UGC), New Delhi for the financial support. SPS is
thankful to CSIR and UGC (SAP), New Delhi for financial assistances.
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