Ab initio calculations of the thermodynamicsand phase diagram of ScSb
Min Teng • Hongzhi Fu • Jianwu Feng •
WenFang Liu • Gao Tao
Received: 9 January 2012 / Accepted: 22 May 2012 / Published online: 1 June 2012
� Springer Science+Business Media, LLC 2012
Abstract The hydrostatic pressure-induced transition
phase of ScSb from the sixfold-coordinated NaCl-type (B1)
to the eightfold-coordinated CsCl-type (B2) is investigated
within the Perdew–Burke–Ernzerhof (PBE) form of gen-
eralized gradient approximation. It is found that the tran-
sition pressure from the B1 to B2 phase is at 42.8 GPa
which agrees with experiment. Through the quasi-har-
monic Debye model, the dependences of the relative vol-
ume V on the pressure P, the thermal expansion, the
Gruneisen parameter ratio, (c - c0)/c0, the Debye tem-
perature H, and heat capacity CV on the pressure P and
temperature T are estimated. The quasi-harmonic Debye
model for the first time is used to predict phase diagram of
B1 ? B2.
Introduction
As one of the group-III–V transition metals, the scandium
compounds have recently attracted particular interest
because of various anomalous physical properties, in
respect of structural, magnetic, and phonon properties [1].
By use of synchrotron radiation, Hayashi et al. [2] have
studied powder X-ray diffraction of ScSb and YSb with the
NaCl-type structure up to 45 GPa at room temperature and
found the B1–B2 transition at high pressures. ScSb and
YSb compounds crystallize in B1 structure at ambient
conditions. Under pressure, ScSb have been found to begin
a first-order phase transition from sixfold-coordinated
NaCl-type (B1) to the eightfold-coordinated CsCl-type
(B2) at around 28 GPa. The low- and high-pressure
structures coexist between 28 and 43 GPa. For YSb, the
low- and high-pressure structures coexist between 26
and 36 GPa [2]. Previous theoretical study using a
Trouiller–Martins non-local norm-conserving pseudopo-
tentials predicted the transition phase pressures to be 16
and 22.5 GPa for ScSb and YSb, respectively [3].
Recently, Chen et al. [1], theoretically studied the high
pressure-induced phase transition of YSb and ScSb com-
pounds using DFT–GGA method. It was found that the
phase transition from the B1 to B2 structures began to
occur at around 29 and 40 GPa for YSb and ScSb com-
pounds, respectively. Bouhemadou and Khenata [4] and
Maachou et al. [5] investigated the structural, elastic, and
high pressure properties in ScSb, using the full-potential
augmented plane wave plus local orbital’s method
(FP-APW ? lo), and they show the phase transitions of
ScSb from B1 to B2 at 39.78 GPa. In view to the inter-
action potential with long-range Coulomb, van der Waals
interaction and the short-range repulsive interaction up to
second-neighbor ions within the Hafemeister and Flygare
M. Teng
Department of Information Technology, Henan Judicial Police
Vocational College, Zhengzhou 450011, People’s Republic
of China
H. Fu (&) � J. Feng
College of Physics and Electronic Information, Luoyang Normal
College, Luoyang 471022, People’s Republic of China
e-mail: [email protected]
H. Fu
National Laboratory of Superhard Materials, Jilin University,
Changchun 130012, People’s Republic of China
W. Liu
College of Chemistry and Chemical Engineering, Luoyang
Normal College, Luoyang 471022, People’s Republic of China
G. Tao
Institute of Atomic and Molecular Physics, Sichuan University,
Chengdu 610065, People’s Republic of China
123
J Mater Sci (2012) 47:6673–6678
DOI 10.1007/s10853-012-6605-x
approach, Varshney et al. [6] analyze the structural as well
as elastic properties in yttrium and scandium antimonides.
They concluded that the observed structural transformation
shall be responsible for the transfer of electrons from Sb s
and p like states to the Y(Sc)-d like states continuously
under pressure.
Due to the absence of theoretical calculations and
experimental measurements on the thermodynamic prop-
erties of ScSb and in order to study its equation of states
(EOSs), we have performed first-principles study of its
structural and thermodynamic properties under high pres-
sure. Our main objectives in this paper are three. First, we
want to characterize the B1 and B2 structures using ultra-
soft pseudopotential introduced by Vanderbilt [7] and the
generalized gradient approximation (GGA) [8]. Second, we
will determine the thermodynamic aspects of both struc-
tures through the quasi-harmonic Debye model [9], with
particular attention to the calculations of the quasi-har-
monic Debye model, the volume V, the thermal expansion,
the Gruneisen parameter ratio, (c - c0)/c0, the Debye
temperature H, and heat capacity CV on the pressure P and
temperature T. The last objective of this work is to char-
acterize the phase diagram of B1 ? B2. As known, the
electronic properties and structural phase transformation of
insulators and semiconductors are interesting classical
problems in the solid-state physics. High-pressure research
on structural phase transformations and behavior of mate-
rials under compression, have become quite interesting in
the recent few years as it provides insight into the nature of
the solid-state theories, and determine the values of fun-
damental parameters.
Computational details
The thermal properties and elastic properties calculations
are performed using the pseudopotential plane-wave
method within the framework of the density functional
theory and implemented through the Cambridge Serial
Total Energy Package (CASTEP) Program [10, 11]. This
technique has become widely recognized as the method of
choice for computational solid structural properties inves-
tigations [12]. The thermodynamic properties for ScSb are
calculated by the quasi-harmonic Debye model [9]. The
exchange correlation energy is described in the GGA using
the Perdew–Burke–Ernzerhof (PBE) functional [8]. The Sc
(3d14s2) and Sb (4d105s2p3) states are treated as valence
electrons. Interactions of electrons with ion cores are pre-
sented by the norm-conserving pseudopotential for all
atoms. In all the high precision calculations, the cutoff
energy of the plane-wave basis set is 320 eV for ScSb. The
special points sampling integration over the Brillouin zone
are carried out using the Monkhorst–Pack method with a
8 9 8 9 8 special k-point mesh. The kinetic energy cutoff
and mesh of k points are optimized by performing self-
consistent calculations. The self-consistent is considered to
be converged when the total energy is 10-6 eV/atom.
These parameters are sufficient in leading to well-con-
verged total energy and elastic stiffness coefficients
calculations.
To investigate the thermodynamic properties of ScSb,
we apply the quasi-harmonic Debye model, in which the
nonequilibrium Gibbs function G*(V;P,T) can be written in
the form of [9, 13–16]
G�ðV; P; TÞ ¼ EðVÞ þ PV þ AVibðhðVÞ; TÞ; ð1Þ
where E(V) is the total energy per unit cell, PV corresponds
to the constant hydrostatic pressure condition, and
AVib(H(V);T) is the vibration term, which can be written as
AVibðhðVÞ; TÞ ¼ nKT9h8Tþ 3 lnð1� e�h=TÞ � D
hT
� �� �;
ð2Þ
where n is the number of atoms in the molecule, and the
Debye integral D(H/T) is defined as [9]
DðyÞ ¼ 3
y3
Zy
0
x3
ex � 1dx: ð3Þ
The heat capacity CV and the thermal expansion (a) are
expressed as
CV ¼ 3nK 4Dðh=TÞ � 3h=T
eh=T � 1
� �; ð4Þ
a ¼ cCV
BTV; ð5Þ
where c is the Gruneisen parameter defined as
c ¼ � dlnhðVÞdlnV
: ð6Þ
The Debye temperature, H in Eq. (6), is related to an
average sound velocity, since the vibrations of the solid are
considered as elastic waves in Debye’s theory. For ScSb
crystal, the Debye temperature can be estimated from the
average sound velocity vm, using the following equation
[17]
h ¼ h
kB
3nNAq4pM
� �1=3
vm; ð7Þ
where h is Planck’s constant, kB is Boltzmann’s constant,
NA is Avogadro’s number, M is the molecule mass, q is
the density, and the average sound velocity vm is
approximately given by [18]
6674 J Mater Sci (2012) 47:6673–6678
123
vm ¼1
3
2
v3s
þ 1
v3p
!" #�13
; ð8Þ
where vp and vs are the longitudinal and transverse elastic
wave velocities, respectively, which can be obtained from
Navier’s equation [17]
vp ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiBS þ
4
3G
� �=q
s; vs ¼
ffiffiffiffiffiffiffiffiffiG=q
p; ð9Þ
where G is the shear modulus and BS is the adiabatic bulk
modulus.
The adiabatic bulk modulus BS is approximated by the
static compressibility [8]
BS � BðVÞ ¼ Vd2EðVÞ
dV2
� �: ð10Þ
Results and discussion
The structural properties are the first step to understand the
material properties with a microscopic point of view. The
calculated equilibrium lattice constants a, zero-pressure
bulk modulus B0 and its pressure derivation B00 from the
Birch–Murnaghan EOS [19] are listed in Table 1, which is
good agreement with other theoretical works [1, 3–6] and
the revealed experimental data [2].
As known, there are two basic methods to obtain the zero-
temperature transition pressure for alloys. The first one is the
slope of the common tangent of the B1 and B2 E–V curves.
The second one is Gibbs free energy, G = H - TS
= E ? PV - TS, i.e., at the phase transition pressure Pt,
Gibbs free energies of B1 and B2 phases attaining the same.
The Gibbs free energy DG (= GB1 - GB2) as a function of
pressure P is illustrated in Fig. 1. The pressure causing
DG approaching to zero is the phase transition pressure (Pt).
The calculated transition pressure (B1 to B2) is
Pt = 42.8 GPa, greater than those predicted by Refs. [1, 3–
6] and close to the experiment data 43 GPa [2]. The calcu-
lated equilibrium lattice constants a, zero-pressure bulk
modulus B0 and its pressure derivation B00 are also consistent
with other theoretical and experimental results available
(Table 1).
The equations of state (EOS) of ScSb in both B1 and B2
phases are obtained using the quasi-harmonic Debye model
[9]. We illustrate the normalized primitive cell volume
V dependences on pressure P in Fig. 2, together with the
experiment data given by Hayashi et al. [2]. Obviously, in
B1 phase, below 20 GPa, our results almost are identical
with Hayashi et al. [2]. Only when pressure goes higher,
the curve of V–P becomes slightly smaller than that of
Hayashi et al. (Fig. 2). We attribute this to applying the
different pseudopotentials. In the investigated pressure the
curve of V–P shows steeper, indicating that ScSb is com-
pressed much more easily at higher pressure. At transition
pressure Pt, our calculation shows that there is volume
change between B1 and B2, which (B1 to B2) decreases
2.79 A3 close to the 2.42 A3 proposed by Maachou et al.
[5], and the structural change from NaCl-type to the CsCl-
type structure occurs with the volume collapse of about
5.7 % in keeping with 5.2 % given by Varshney et al. [6].
The negligible changes of Sc–Sc and Sc–Sb bond distances
with the applied pressures are observed in Fig. 3, together
with the experiment data given by Hayashi et al. [2]. This
result is important since to a first approximation the
vibration frequencies are dependent on bond distances. As
expected experimentally and theoretically, the Sc–Sc and
Sc–Sb bond lengths decrease with pressure. It is shown that
our results are coordinate with the experiment data in the
applied pressures (Fig. 3). We can also note that the Sc–Sc
Table 1 The lattice constants a (A), bulk modulus B0 (GPa) and its pressure derivation B00 (GPa), elastic constants Cij (GPa) of the B1 and B2
structures of ScSb at P = 0 and T = 0, together with the transition pressures Pt (GPa)
This work Other theoretical calculations Experiments
B1 structure
a 5.883 5.887 [5], 5.87 [1], 5.797 [3], 5.793[4]LDA, 5.93 [4]GGA, 5.93 [4]GGA 5.851 [2]
B0 67 70.90 [5], 66.84 [1], 71.3 [3], 78.4 [4]LDA, 64.58 [4]GGA, 58 [6] 58 ± 3 [2]
B00 4.09 3.47 [5], 4.14 [1], 3.65 [3], 3.88 [4]LDA, 3.86 [4]GGA 9.5 ± 0.8 [2]
C11 155GGA, 170LDA 186 [4]LDA, 155 [4]GGA
C12 23.18GGA, 24.2LDA 21 [4]LDA, 20 [4]GGA
C44 23.30GGA, 26.8LDA 26 [4]LDA, 22 [4]GGA, 18 [6]
Pt 42.8 39.78 [5], 40 [1], 16 [3], 35 [4], 31 [6] 43 [2]
B2 structure
a 3.617 3.622 [5], 3.61 [1], 3.61 [3], 3.55 [4]LDA, 3.639 [4]GGA 3.66 [2]
B0 70 69.78 [5], 68.14 [1], 68.14 [3], 83.35 [4]LDA, 67.39 [4]GGA
B00 3.98 3.77 [5], 4.03 [1], 4.03 [3], 3.46 [4]LDA, 3.74 [4]GGA
J Mater Sci (2012) 47:6673–6678 6675
123
bonds shorten slightly faster than the Sc–Sb bonds (Fig. 3),
which shows that there exist a high antiphase boundary
(APB) energy and a fairly large elastic shear anisotropy.
The high APB energy is attributed to the directional
bonding between Sc and Sb electrons.
The thermal expansion parameter a is thought to be
described the alteration in a frequency of the crystal lattice
vibration based on the lattice’s increase or decrease in
volume as a result of temperature changes. It is directly
related to the EOS. We have determined the pressure
dependence of a for both B1 and B2 phase shown in Fig. 4.
It can be seen that the thermal expansion parameter, a,
decreases nonlinearly with the pressure for both B1 and B2
phase. However, the thermal expansion parameter in higher
temperature decreases rapidly with pressure than that in
lower temperature. At Pt, the thermal expansion parame-
ters jump up by 46.1 and 57.7 % at 300 and 2000 k,
respectively.
The variations of the heat capacity CV with pressure
P for the B1 and B2 phase of ScSb are shown in Fig. 5.
They are normalized by (CV - CV0)/CV0, where CV and
CV0 are heat capacity at any pressure P and zero pressure.
The heat capacities decrease almost linearly with the
applied pressures, as shows the fact that the vibration fre-
quency of the particles in ScSb changes with pressure and
temperature. However, at Pt, the higher temperature, the
smaller decrease of CV can be seen.
Anharmonic properties of solids are customarily
described in terms of the Gruneisen parameter c, which can
also be calculated from knowledge of the phonon fre-
quencies as a function of the crystal volume V. We
have determined the pressure dependence of Gruneisen
Fig. 1 The Gibbs free energy DG (= GB1 - GB2) as a function of
pressure P for ScSb
Fig. 2 The volume–pressure diagram of the B1 and B2 structures of
ScSb at various pressures
Fig. 3 The changes of Sc–Sc and Sc–Sb bond distances with the
applied pressures
Fig. 4 The thermal expansion versus temperature and pressure for
ScSb
6676 J Mater Sci (2012) 47:6673–6678
123
parameter c for both B1 and B2, which are shown in Fig. 6.
It is shown that below the phase transition pressure Pt, the
Gruneisen parameter ratios (in B1 phase), (c - c0)/c0,
where c and c0 are Gruneisen parameter at any pressure
P and zero pressure, decrease (nonlinearly) with the pres-
sure, and the higher temperature, the faster decrease of cappears. However, above the phase transition pressure Pt,
the Gruneisen parameter ratios (in B2 phase at 300 and
1800 K), (c - c0)/c0, decrease (nearly linearly) almost in
the same slope with the pressure. At Pt, the ratios of
(c - c0)/c0, jump up by 0.74 % and jump down by
15.53 % at the temperature 300 and 1800 K, respectively.
The pressure dependence of the calculated Debye tem-
perature hD is shown in Fig. 7. For both B1 and B2, hD
increases (nonlinearly in B1 and almost linearly in B2)
with the pressure. At zero pressure, hD is found to be
279.8 K which is about 28.5 % greater than the obtained
for constituent atom Sb (200 K). When the pressure is near
Pt the Debye temperature is higher in B2 phase than that in
B1 phase.
To predict the phase diagram at high temperatures and
pressures, we employ quasi-harmonic Debye model [8],
which has been successfully applied to investigate the
thermodynamic properties of PtN2 [14], TiAl [15], NiAl
[16], MgTe [20], ZrC [21], and so on. The pressure
dependence of the Gibbs free energy difference between
B1 and B2 structures (DG = GB1 - GB2) of ScSb at dif-
ferent temperatures is displayed in Fig. 8. At 0 K (not
shown in Fig. 8), the calculated phase transition pressure
(B1 to B2) is 42.8 GPa, close to the experimental result
Fig. 5 The variations of heat capacity with pressure P. They are
normalized by (CV - CV0)/CV0, where CV and CV0 are heat capacity
at any pressure P and zero pressure at the temperatures of 300 and
1800 K, respectively
Fig. 6 The variations of the Gruneisen parameter ratios with pressure
P. They are normalized by (c - c0)/c0, where c and c0 are Gruneisen
parameter at any pressure P and zero pressure at the temperatures of
300 and 1800 K, respectively
Fig. 7 Variation of Debye temperature (hD) with pressure
Fig. 8 Calculated Gibbs energy difference between B1 and B2 phase
of ScSb (DG = GB1 - GB2) as a function of pressure at different
temperatures
J Mater Sci (2012) 47:6673–6678 6677
123
43 GPa [2].The B1 crystal phase is thermodynamically and
mechanically stable at zero pressure. Conversely, the B2
phase is unstable. As pressure increases, beyond the phase
transition pressure (Pt), the B2 system becomes mechani-
cally and thermodynamically stable while the B1 phase
remains unstable up to the greatest pressure studied. The
increase of temperature results in an increase of DG. At
400 K, DG is already negative at low pressures and, thus,
the B1 structure appears in the calculated phase diagram.
At the same time, DG increases under compression and
eventually becomes positive indicating a stabilization of
the B2 structure, and at higher temperatures the transition
pressure of the B1 ? B2 decreases even further.
Conclusions
We have investigated the hydrostatic pressure-induced
transition phases of ScSb from the B1 to B2 phase by the
ab initio plane-wave pseudopotential density functional
theory method using package CASTEP. It is found that the
transitional phase occurs at 42.8 GPa from the B1 to B2
phase according to the usual condition of equal enthalpy.
Through the quasi-harmonic Debye model, we have suc-
cessfully obtained the dependences of the volume V, the
thermal expansion parameter a, the Gruneisen parameter
ratios, (c - c0)/c0, the Debye temperature H, and heat
capacity CV on the pressure P and temperature T. To pre-
dict phase transition B1 to B2 at high temperatures, we use
the quasi-harmonic Debye model and obtain the phase
diagram of ScSb. More experiments are necessary to fur-
ther make clear the structural sequence of ScSb.
Acknowledgements This project was supported by the National
Natural Science Foundation of China under Grant No. 40804034, and
by the Natural Science Foundation of the Education Department of
Henan province of China under Grant No. 2009B590001 and by
Henan Science and Technology Agency of China under Grant No.
092102210314 and by Open Project of State Key Laboratory of
Superhard Materials (Jilin University) No. 201107.
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